aa r X i v : . [ m a t h . L O ] M a y STOCHASTICITY AS DENSITY
JUSTIN MILLER
Abstract.
Intrinsic density was introduced by Astor to study asymptoticcomputability. Intrinsically small sets, those of intrinsic density zero, serve asthe basis for generalizing classical asymptotic computability to its permutation-invariant form. The existence of intrinsic density, however, is a measure of lackof information more generally. It is already known that having intrinsic densityone half is equivalent to injection and permutation stochasticity, for example,and that sufficiently random sets have intrinsic density one half. We shall studystochasticity in a more general form through the lens of intrinsic density. Weshall first build tools to create examples of sets with defined intrinsic densityto prove that every real in the unit interval can be achieved as the intrinsicdensity of a set of natural numbers. Once we have developed these tools, weshall use them to study the classical notions of Church stochasticity and MWCstochasticity. We shall generalize these to a notion of density and prove thatevery real in the unit interval can also be achieved as the “MWC-density,” andthus “Church-density,” of a set.
Keywords: intrinsic density, stochasticity, randomness, computability, martin-gale 1.
Introduction
Intrinsic density originated in an attempt to resolve some objections about clas-sical asymptotic computability, which uses sets of density zero as error sets incomputations. We briefly recall the notion of (asymptotic) density in the naturalnumbers:
Definition 1.1.
Let A ⊆ ω . • The density of A at n is ρ n ( A ) = | A ↾ n | n , where A ↾ n = A ∩{ , , , . . . , n − } . • The upper density of A is ρ ( A ) = lim sup n →∞ ρ n ( A ) . • The lower density of A is ρ ( A ) = lim inf n →∞ ρ n ( A ) . • If ρ ( A ) = ρ ( A ) = α , we call α the density of A and denote it by ρ ( A ) . Remark.
We shall follow the convention, unless otherwise stated, that capitalEnglish letters represent sets of natural numbers and the lowercase variant, indexedby a subscript of natural numbers, represents the elements of the set. As an exam-ple, if E is the set of even numbers, then e n = 2 n . If F is the set of factorials, then f n = ( n + 1)!. ( f n is not n ! because 0! = 1! = 1.) Recall that the principal functionfor a set A , p A , is defined via p A ( n ) = a n .Using this representation, it is not hard to see the following characterization ofupper and lower density: The author would like to thank his advisor, Dr. Peter Cholak, for the advice, discussion, andsupport that made this project possible.Partially supported by NSF-DMS-1854136.
Lemma 1.2.
Let A ⊆ ω be { a < a < a < . . . } . Then • ρ ( A ) = lim sup n →∞ n +1 a n +1 • ρ ( A ) = lim inf n →∞ na n Proof.
Note that if A ↾ n + 1 has a 0 in the final bit, then ρ n ( A ) = | A ↾ n | n > | A ↾ n | n + 1 = ρ n +1 ( A )Therefore, to compute the upper density it suffices to check only those numbers n for which A ↾ n has a 1 as its last bit. Those numbers are exactly a n + 1 by thedefinition of a n , and | A ↾ a n + 1 | = n + 1. Therefore { n +1 a n +1 } n ∈ ω is a subsequence of { ρ n ( A ) } n ∈ ω which dominates the original sequence, so ρ ( A ) = lim sup n →∞ ρ n ( A ) =lim sup n →∞ n +1 a n +1 .Similarly, to compute the lower density it suffices to check only the numbers n such that the final digit of A ↾ n is a 0, but the final digit of A ↾ n + 1 is a 1. (Thatis, if there is a consecutive block of zeroes in the characteristic function of A , weonly need to check the density at the end of the block when computing lower den-sity, as each intermediate point of the zero block has a higher density than the end.)These numbers are exactly a n by definition, and | A ↾ a n | = n . Therefore { na n } n ∈ ω is a subsequence of { ρ n ( A ) } n ∈ ω which is dominated by the original sequence, so ρ ( A ) = lim inf n →∞ ρ n ( A ) = lim inf n →∞ na n . (cid:3) One potential objection with using density 0 sets as error sets is that there aremany computable sets of density zero, so one is able to make any information thatone desires be “almost” computable by hiding it within a small computable set. Tocombat this, Astor [1] introduced intrinsic density, which requires that sets havethe same asymptotic density under any computable permutation:
Definition 1.3. • The absolute upper density of A is P ( A ) = sup { ρ ( π ( A )) : π a computable permutation }• The absolute lower density of A is P ( A ) = inf { ρ ( π ( A )) : π a computable permutation }• If P ( A ) = P ( A ) = α , we call α the intrinsic density of A and denote it by P ( A ) . Interestingly, this turns out to be a robust measure of lack of information. If aset X has intrinsic density, then we cannot computably shrink or enlarge parts ofit with a permutation to change the density. If we knew where elements of X couldbe found, then we could build a permutation that sent them to a set of density 1or 0. This intuition has a formal counterpart: Astor [2] proved that any set whichhas intrinsic density must be of high or DNC degree, i.e. must be sufficiently non-computable. Sets of intrinsic density 0, also known as intrinsically small sets, wereexplored by Astor in [1] and [2] and the author in [5]. Exploring intrinsic smallnessinherently explores sets of intrinsic density 1 because intrinsic density is preservedby complementation. Interestingly, intrinsic densities between 0 and 1, especially , are linked to stochasticity and randomness.Stochasticity and randomness are well-established and related notions which also TOCHASTICITY AS DENSITY 3 measure lack of information, and turn out to have strong ties to intrinsic density.Stochasticity represents the idea that we cannot select bits from an infinite sequenceof 0’s and 1’s in such a way that the ratio of 1’s to the number of bits is not , i.e.we cannot make a selection procedure that is able to find more 1’s or 0’s. Definition 1.4.
Let C be a class of pairs of (partial) functions s, i : 2 ω → ω . For ( s, i ) ∈ C and A ∈ ω , if s ( A ) and i ( A ) are both defined and s ( A ) is infinite then welet S s,i ( A ) represent { n : p s ( A ) ( n ) ∈ i ( A ) } . Then X is C -stochastic, or stochasticfor C , if, for every ( s, i ) ∈ C , either S s,i ( A ) is undefined, s ( A ) is finite, or we have ρ ( S s,i ( A )) = 12 We call such a C a class of stochasticity. Here s is a function which determines which bits to select, and i is a functionwhich interprets A . We think of a set A as a sequence of infinitely many 0-1 valuedcoin flips. The selection function s tells us which coins to select and which onesto ignore, and the interpretation i function tells us how or in what order to countthe selected coins. Our goal is to find 1-valued coins as much as possible. Oftenwe are particularly interested in stochasticity with respect to certain narrow classeswhere the selection functions are continuous and the interpretations are simplyreorderings. Remark.
It is important to note that this general definition of stochasticity is notfound in the literature. However, we may recover all known important classes ofstochasticity as special cases of it.
A monotone selection function is a function f : 2 <ω → { , } . That is, f looksat a finite binary string and decides if it wants to select (i.e. return 1) the followingbit or not based on the previous bits. Given a selection function f , it induces amap f : 2 ω → ω that is defined via f ( A ) = { n : f ( A ↾ n ) = 1 } for all A . (We shallabuse notation and allow f to represent both a monotone selection function andthe induced map on Cantor space.) If C is the class of ( f, id ) where f is a partialcomputable monotone selection function and id is the identity, then C -stochasticityis found in the literature and is called von Mises-Wald-Church stochasticity, orMWC stochasticity. If we restrict this C to only the total f , then the correspondingnotion is called Church stochasticity. In both cases, we may use the results of thefirst n bits to computably determine whether or not we want to select the n + 1-stbit, but all of the bits we select must be counted in order.Using our coin analogy, for MWC stochasticity and Church stochasticity, all ofthe coins have been covered by cups. We must choose whether or not we think thefirst coin is 1-valued before looking under any cups. Then we look under the firstcup and see if we are right or wrong, and we use this information moving forward.Having revealed the first n coins, we must choose whether or not to select the n +1stcoin (i.e. determine if we think it is 1-valued) prior to revealing it.Another historically important notion of stochasticity is KL-stochasticity, also knownas Kolmogorov-Loveland stochasticity. This notion is similar to MWC-stochasticity,however we are allowed to select bits out of order rather than being forced to choosewhether or not to select the n -th bit only after seeing the first n bits. Formally, JUSTIN MILLER a finite assignment is an element σ of ( ω × { , } ) <ω satisfying that each n ∈ ω appears in the first element of at most one pair of σ , that is a finite sequence ofpairs ( a, b ) for distinct natural numbers a and not necessarily distinct b ∈ { , } .A finite assignment encodes finitely many bits of a set A much like a finite binarystring, however finite assignments are not required to define initial segments of A .We define FA to be the set of all finite assignments. A scan rule is a partial com-putable function s : F A → ω which satisfies that for all σ ∈ F A , neither ( s ( σ ) , s ( σ ) ,
1) is in σ . A scan rule determines how we select the next bit based onthe information of finitely many bits, possibly out of order: having seen the a -thbit is a b for each ( a, b ) ∈ σ , s ( σ ) is the index of the next bit we wish to check.Given a set A and a scan rule s , we recursively define s A ( n ) to represent the resultof the first n bits of A selected by s : s A (0) = ∅ and s A ( n + 1) = s A ( n ) ⌢ ( s ( s A ( n )) , A ( s ( s A ( n ))))We consider pairs ( s, c ) of a scan rule s and a choice function c : F A → { , } . Thescan rule determines how to look at the bits, and the choice function c determineswhich bits to select, i.e. if c ( σ ) = 1 then we would like to count the s ( σ )-th bit. Re-visiting the coin analogy, the coins are once again hidden under cups. s determinesthe order in which we look under the cups, and c determines whether we want toselect the next coin based on those already revealed.A pair ( s, c ) induces a pair of maps s KL , c KL : 2 ω → ω via c KL ( A ) = { n : c ( s A ( n )) = 1 } and s KL ( A ) = { n : A ( s A ( n )) = 1 } That is, c KL ( A ) represents the set of bits of A selected by c and s KL ( A ) repre-sents those bits rearranged into the order in which they were selected. Then if C is the class for stochasticity containing ( c KL , s KL ) for all scan rules s and choicefunctions c , C -stochasticity is Kolmogorov-Loveland stochasticity, also known asKL-stochasticity.One particularly important weakening of KL-stochasticity for our study of intrinsicdensity is injection stochasticity. Injection stochasticity is induced by all total com-putable injective functions f via the pairs (ˆ1 , f − ). (In other words, the selectionfunction which counts every coin and the interpretation functions which order thecoins according to f .) That is, a set A is injection stochastic if ρ ( f − ( A )) = for all total computable injective f . Permutation stochasticity, as expected, is thesubclass where f is required to be a permutation. Using this definition, Astor firstobserved the following: Lemma 1.5 (Astor [1] Lemma 4.2) . A set A is injection stochastic if and only ifit is permutation stochastic.Proof. If A is injection stochastic, it is trivially permutation stochastic.Suppose that A is permutation stochastic. Then ρ ( π ( A )) = for every com-putable permutation π . Let f be a total computable injective function and let F = { n ! : n ∈ ω } . Define π f via π f ( n ) = f ( n ) if n F and f ( n ) is not in π f ([0 , n )), and the least element of the complement of π f ([0 , n )) otherwise. As π f TOCHASTICITY AS DENSITY 5 is a computable permutation, so is π − f and thus ρ ( π − f ( A )) = .Now notice that π − f ( A ) ↾ n differs from f − ( A ) ↾ n by at most 2 | F ↾ n | , asthere can only be disagreement on F and f − ( π f ( F )). In fact, there are two typesof disagreement. In the first, we specifically mapped π f ( n !) to something otherthan f ( n !), which can only happen within F . In the second, k is not a factorialbut f ( k ) ∈ π f ([0 , k )) because of some n ! < k . Thus the set of disagreements hasdensity zero because F does, so ρ ( f − ( A )) = ρ ( π − f ( A )) = 12 (cid:3) Remark.
In his version of the proof, Astor used the squares instead of the factori-als. In general for all of our arguments which use a computable density set, anyset with such properties will suffice. We shall endeavor to use the factorials in allsuch proofs for clarity. It is immediate from the definition that permutation stochasticity is exactlyintrinsic density . Therefore, this lemma shows that injection stochasticity alsocorresponds to intrinsic density . Unlike stochasticity, intrinsic density is definedfor any real α in the unit interval as opposed to . We therefore posit the followinggeneralization of Definition 1.4: Definition 1.6.
Let C be a class of stochasticity. Then X has C -density α for α ∈ [0 , if, for every ( s, i ) ∈ C , either S s,i ( A ) is undefined, s ( A ) is finite, or wehave ρ ( S S,i ( A )) = α We shall address the question of which intrinsic (injection-) densities are achiev-able in Section 3, and we shall apply the tools from that section to MWC- andChurch-densities in Section 5. One important trait of Church-density is that if A has Church-density α , then ρ ( A ) = α because the selection function ˆ1 which selectsevery bit is a total computable monotone selection function and S ˆ1 ,id ( A ) = A . (Itfollows immediately from the fact that MWC-density is defined for a larger classfor stochasticity that the same is true of MWC-density.)Randomness is well-studied and more well-known than stochasticity, so we shallonly provide a cursory overview. (For a more in-depth review of randomness aswell as stochasticity, see Downey-Hirschfeldt [3].) While there are many notions ofrandomness, we shall only need 1-Randomness, also known as Martin-L¨of Random-ness, for our purposes. There are many equivalent ways of defining randomness,and we shall recall two. Definition 1.7.
A martingale is a function m : 2 <ω → R ≥ such that m ( σ ) = 12 m ( σ
0) + 12 m ( σ for all σ . A supermartingale is a function s : 2 <ω → R ≥ with s ( σ ) ≥ s ( σ
0) + 12 s ( σ JUSTIN MILLER for all σ . A (super)martingale m succeeds on a set X if lim sup n →∞ m ( X ↾ n ) = ∞ . X is 1-Random if no computably enumerable supermartingale succeeds on it. Martingales capture the unpredictability of random sets: we could not win arbi-trarily large amounts of money betting on the bits of X in any c.e. or computableway. An alternative yet equivalent formulation of randomness is the measure-theoretic approach, which is based upon the intuition that if a set is random thenit should avoid all small sets which can be described with computable approxima-tions. Definition 1.8.
A Martin-L¨of (ML) test is a sequence {U i } i ∈ ω of uniformly Σ classes with µ ( U i ) ≤ − i for all i . (Here µ is the usual Lebesgue measure on Cantorspace.) A set X passes {U i } i ∈ ω if X T i ∈ ω U i . X is 1-Random if it passes everyMartin-L¨of test. While historically the study of algorithmic randomness began with respect tothe Lebesgue or “fair coin” measure, more recent work has focused on studyingrandomness with respect to other measures. It is not difficult to see how Definition1.8 generalizes to an arbitrary computable measure. This was first studied byReimann and Slaman in [7].
Definition 1.9.
Let ν be a computable measure on Cantor space. A ν -Martin-L¨oftest is a sequence {U i } i ∈ ω of uniformly Σ classes with ν ( U i ) ≤ − i for all i . Aset X passes {U i } i ∈ ω if X T i ∈ ω U i . X is 1-Random with respect to ν if it passesevery ν -Martin-L¨of test. Given the equivalence of Definition 1.7 and Definition 1.8, one might hope tofind a generalization for Definition 1.7 that is equivalent to Definition 1.9. It isnot necessarily immediate how one might do this for an arbitrary measure, butthe definition of a martingale from probability theory informs us for the followingspecial class of measures.
Definition 1.10.
Let < p < be a rational number. Given a finite binary string σ , [ σ ] ⊆ ω represents the basic open set of extensions of σ . µ p is the measure onCantor space such that for any σ ∈ <ω , µ p ([ σ ]) = p |{ n< | σ | : σ ( n )=1 }| (1 − p ) |{ n< | σ | : σ ( n )=0 }| Note that µ is the usual Lebesgue measure. More generally, µ p is the p -biased“coin flip” measure, or the measure induced by the Bernoulli probability with pa-rameter p . A martingale in the probability theoretic set requires that within asequence of random variables, the expected value of the next variable given knowninformation must equal the expected value of the of the current variable. We trans-late this into our setting in the following definition. Definition 1.11.
Let < p < be rational. A p -martingale is a function m :2 <ω → R ≥ such that m ( σ ) = (1 − p ) m ( σ
0) + pm ( σ for all σ . A p -supermartingale is a function s : 2 <ω → R ≥ with s ( σ ) ≥ (1 − p ) s ( σ
0) + ps ( σ for all σ . A p -(super)martingale m succeeds on a set X if lim sup n →∞ m ( X ↾ n ) = ∞ . A set X is p -1-Random if no computably enumerable p -supermartingalesucceeds on it. TOCHASTICITY AS DENSITY 7
A slight modification of the standard proof that randomness for (super)martingalesis the same as randomness for Martin-L¨of tests (as found in Downey-Hirschfeldt[3] Section 6.3.1, referencing work of Ville [10] and Schnorr [8]) shows that 1-Randomness with respect to µ p is equivalent to p -1-Randomness. Theorem 1.12 (Essentially Ville [10]) . Let < p < be rational. Given σ ∈ <ω ,we abuse notation and write µ p ( σ ) to denote µ p ([ σ ]) . Let m be a p -(super)martingale. • If σ ∈ <ω and S is a prefix-free set of extensions of σ , then Σ τ ∈ S µ p ( τ ) m ( τ ) ≤ µ p ( σ ) m ( σ ) • Let R n = { X : ∃ k m ( X ↾ k ) ≥ n } . Then µ p ( R n ) ≤ m ( ∅ ) n .Proof. • Note that it suffices to only consider finite sets S , as if S is infiniteand Σ τ ∈ S µ p ( τ ) m ( τ ) > µ p ( σ ) m ( σ ), there is some finite subset of S alsoexhibiting this property.We argue by induction on | S | . For | S | = 1, let τ (cid:23) σ , i.e. τ = σγ for some γ ∈ <ω . Note by induction and the definition of a p -(super)martingalethat µ p ( γ ) m ( τ ) ≤ m ( σ ). Therefore µ p ( τ ) m ( τ ) = µ p ( σ ) µ p ( γ ) m ( τ ) ≤ µ p ( σ ) m ( σ )Now suppose | S | = k + 1 and the induction hypothesis holds for all i ≤ k .Let γ (cid:23) σ be maximal such that τ (cid:23) γ for all τ ∈ S . Then let S ⊆ S be theset of all τ ∈ S with τ (cid:23) γ S = S \ S . (Note that for all τ ∈ S , τ (cid:23) γ γ is maximal such that all τ ∈ S are extensions of γ , both | S | ≤ k and | S | ≤ k . Therefore, the induction hypothesis impliesthat Σ τ ∈ S µ p ( τ ) m ( τ ) ≤ µ p ( γ m ( γ τ ∈ S µ p ( τ ) m ( τ ) ≤ µ p ( γ m ( γ τ ∈ S µ p ( τ ) m ( τ ) = Σ τ ∈ S µ p ( τ ) m ( τ ) + Σ τ ∈ S µ p ( τ ) m ( τ ) ≤ µ p ( γ m ( γ
0) + µ p ( γ m ( γ µ p that µ p ( γ
0) = (1 − p ) µ p ( γ ) and µ p ( γ
1) = pµ p ( γ ), so by the properties of a p -(super)martingale we haveΣ τ ∈ S µ p ( τ ) m ( τ ) ≤ µ p ( γ )((1 − p ) m ( γ
0) + pm ( γ ≤ µ p ( γ ) m ( γ )The base case proved that µ p ( γ ) m ( γ ) ≤ µ p ( σ ) m ( σ ), so this concludes theinduction. • Let S be a prefix-free set which induces the Σ -class R n with all τ ∈ S satisfying m ( τ ) ≥ n . By definition, µ p ( R n ) = Σ τ ∈ S µ p ( τ )As each τ ∈ S satisfies m ( τ ) ≥ n ,Σ τ ∈ S µ p ( τ ) ≤ Σ τ ∈ S m ( τ ) n µ p ( τ ) JUSTIN MILLER
Finally, we may apply the first part with σ = ∅ to obtainΣ τ ∈ S m ( τ ) n µ p ( τ ) ≤ µ p ( ∅ ) m ( ∅ ) n = m ( ∅ ) n (cid:3) Theorem 1.13 (Essentially Schnorr [8]) . X is p -1-Random if and only if it is1-Random with respect to µ p .Proof. Let m be a c.e. p -(super)martingale. Without loss of generality, assume m ( ∅ ) = 1. Let U n = { X : ∃ k m ( X ↾ k ) ≥ n } . This is a µ p -Martin-L¨of test byTheorem 1.12, and it is immediate that X ∈ T n ∈ ω U n if and only if m succeeds on X .Let { U n } n ∈ ω be a µ p -Martin-L¨of test with { S n } n ∈ ω the uniform sequence of c.e.prefix-free finite binary strings which induces { U n } n ∈ ω . We shall define c.e. p -martingales m n via the following procedure: If we see σ enter S n at some stage,then add 1 to m ( τ ) for all τ (cid:23) σ . For γ ≺ σ , add µ p ( σ ) µ p ( γ ) to m n ( γ ). Then it is im-mediate from this definition that m e : 2 <ω → R ≥ is a c.e. function. Furthermore,note that it is a p -martingale: let σ ∈ <ω . As S n is prefix-free, if σ (cid:23) τ ∈ S n , then pm n ( σ
1) + (1 − p ) m n ( σ
0) = p + (1 − p ) = 1 = m n ( σ )by construction. Otherwise, m n ( σ ) = Σ τ ∈ S n ,τ ≻ σ µ p ( τ ) µ p ( σ ) = 1 µ p ( σ ) Σ τ ∈ S n ,τ ≻ σ µ p ( τ )by definition. Note that for i = 0 , m n ( σi ) = Σ τ ∈ S n ,τ (cid:23) σi µ p ( τ ) µ p ( σi )as if σi ∈ S n then m n ( σi ) = 1 = µ p ( σi ) µ p ( σi ) . Therefore pm n ( σ
1) + (1 − p ) m n ( σ
0) = p ( Σ τ ∈ S n ,τ (cid:23) σ µ p ( τ ) µ p ( σ
1) ) + (1 − p )( Σ τ ∈ S n ,τ (cid:23) σ µ p ( τ ) µ p ( σ
0) ) = pµ p ( σ
1) ( Σ τ ∈ S n ,τ (cid:23) σ µ p ( τ )) + 1 − pµ p ( σ
0) ( Σ τ ∈ S n ,τ (cid:23) σ µ p ( τ )) =1 µ p ( σ ) ( Σ τ ∈ S n ,τ (cid:23) σ µ p ( τ )) + 1 µ p ( σ ) ( Σ τ ∈ S n ,τ (cid:23) σ µ p ( τ )) =1 µ p ( σ ) ( Σ τ ∈ S n ,τ (cid:23) σ µ p ( τ ) + Σ τ ∈ S n ,τ (cid:23) σ µ p ( τ )) = 1 µ p ( σ ) Σ τ ∈ S n ,τ ≻ σ µ p ( τ ) = m n ( σ )Thus m n is a p -martingale, and { m n } n ∈ ω is a uniformly c.e. collection of p -martingales. Furthermore, m n ( ∅ ) = Σ τ ∈ S n µ p ( τ ) ≤ − n , so m = Σ n ∈ ω m n is a c.e. p -martingale by a slight modification of Proposition 6.3.2 of Downey-Hirschfeldt[3]. Finally, it follows that m succeeds on X if and only if X ∈ T n ∈ ω U n . (cid:3) Astor [1] proved that 1-Random sets have density by referring to Propositions3.2.13 and 3.2.16 of Nies [6], which state that 1-Randoms must have density andthat they are closed under permutations. We shall provide a direct proof of thestronger fact that p -1-Randoms have intrinsic density p in Section 3. He merelystated that Schnorr randoms (a weaker notion of randomness) also have intrinsic TOCHASTICITY AS DENSITY 9 density without proof. However, the above facts for 1-Randoms are also true ofSchnorr randoms as seen in Propositions 3.5.12 and 3.5.21 of Nies [6], so Schnorrrandoms also have intrinsic density .All sets known to have defined intrinsic density previously had intrinsic density0, 2 − n for n ∈ ω , or 1. We shall show that we can in fact achieve any real in theunit interval using intrinsic density. To do so, we shall build some tools to developnew sets with defined intrinsic density from old ones. (Our proof that every real isrealized can be modified to work with Astor’s results, however his results only workwith random objects: We build tools that work off of intrinsic density alone.) Wewould like to take a set A of intrinsic density α and a set B of intrinsic density β and somehow code B and A in such a way that we are left with a set which has newintrinsic density obtained as some function of α and β . However, we cannot hopeto code things in a nice computable way that allows us to recover the original sets,as intrinsic density was defined with the intention of blocking computable codingin the setting of asymptotic computability.Our main technique will involve proving that two sets A and B cannot have differ-ent intrinsic densities by creating a computable permutation which sends A to B modulo a set of density zero. The following lemma shows that if we can do this,then the density of the image of A is the same as the density of B , and thereforethat they cannot have different intrinsic densities. Lemma 1.14. If ρ ( H ) = 0 , then ρ ( X \ H ) = ρ ( X ∪ H ) = ρ ( X ) and ρ ( X \ H ) = ρ ( X ∪ H ) = ρ ( X ) .Proof. Notice that ρ n ( X ) = ρ n ( X \ H ) + ρ n ( X ∩ H )By definition. Therefore ρ ( X ) = lim sup n →∞ ρ n ( X ) = lim sup n →∞ ρ n ( X \ H ) + ρ n ( X ∩ H )By subadditivity of the limit superior, ρ ( X ) ≤ lim sup n →∞ ρ n ( X \ H ) + lim sup n →∞ ρ n ( X ∩ H )As ρ ( H ) = 0 and X ∩ H ⊆ H , ρ ( X ) ≤ lim sup n →∞ ρ n ( X \ H ) = ρ ( X \ H )However, ρ ( X \ H ) ≤ ρ ( X ) because X \ H ⊆ X , so ρ ( X ) = ρ ( X \ H ) as desired.The argument for the union and the argument for lower density are functionallyidentical. (For the union we use X ∪ H , X , and H \ X in place of X , X \ H , and X ∩ H respectively.) (cid:3) We begin by illustrating why the classical operations for combining two sets failto yield new intrinsic densities in Section 2. Section 3 will introduce new toolsfor obtaining sets and use them to show that every real in the unit interval canbe obtained as an intrinsic density. We shall close some natural gaps that arisein Sections 2 and 3 in Section 4, then we will conclude in Section 5 by reviewingstochasticity via our results of Section 3 and show that the entire unit interval isachieved via MWC-density and Church-density. The Failure of Classical Coding
The Join.
As mentioned previously, we would like to find some operationthat takes sets A and B of intrinsic density α and β respectively and outputs anew set with intrinsic density which is given as a function of α and β . The mostcommon operation for combining two sets in computability theory is the join. It iseasy to show that if A has density α and B has density β , then A ⊕ B has density α + β . However, this is not so simple in the case of intrinsic density. Lemma 2.1. If P ( A ) = P ( B ) , then A ⊕ B does not have intrinsic density.Proof. We shall proceed by showing that there is a computable permutation whichsends A ⊕ B to A modulo a set of density 0, and similarly for B . Then the upper(and lower) density of A ⊕ B under these permutations will match that of A and B respectively. Therefore if these densities are different, the density of A ⊕ B is notinvariant under computable permutation.Let F = { n ! : n ∈ ω } and G = F . For any fixed computable permutation π ,there is another computable permutation ˆ π defined via enumerating the odds ontothe factorials in order and enumerating the evens onto the nonfactorials accordingto the ordering induced by π . That is, ˆ π (2 n + 1) = f n and ˆ π (2 n ) = g π ( n ) .Then as F has density 0, Lemma 1.14 shows ρ (ˆ π ( A ⊕ B )) = ρ (ˆ π ( A ⊕ B ) \ F )As the image of the odds under ˆ π is a subset of F ,ˆ π ( A ⊕ B ) \ F = ˆ π ( A ⊕ ∅ )and ρ (ˆ π ( A ⊕ B )) = ρ (ˆ π ( A ⊕ ∅ ))Notice that ˆ π ( A ⊕ ∅ ) is just π ( A ) with each element n increased by | F ↾ n | . Thus ρ n ( π ( A )) ≥ ρ n (ˆ π ( A ⊕ ∅ )) ≥ | π ( A ) ↾ n | − | F ↾ n | n As F is the factorials, the final expression tends to ρ n ( π ( A )) in the limit, so we seethat ρ (ˆ π ( A ⊕ ∅ )) = ρ ( π ( A ))and ρ (ˆ π ( A ⊕ B )) = ρ (ˆ π ( A ⊕ ∅ )) = ρ ( π ( A )) ρ (ˆ π ( A ⊕ B )) = ρ ( π ( A )) by a nearly identical argument.In particular, P ( A ⊕ B ) ≥ P ( A ) and P ( A ⊕ B ) ≤ P ( A ) because we are tak-ing the limit superior and inferior over all computable permutations, of whichˆ π is but one. (Basically, ˆ π sends A ⊕ B to π ( A ) modulo a set of density zero,so the intrinsic upper (lower) density of A ⊕ B cannot be smaller (larger) thanthe intrinsic upper (lower) density of A .) Reversing the use of the evens and theodds in the definition of ˆ π , we get that the same is true for B in place of A , so P ( A ⊕ B ) ≤ min ( P ( A ) , P ( B )) and P ( A ⊕ B ) ≥ max ( P ( A ) , P ( B )). Therefore if P ( A ) = P ( B ), P ( A ⊕ B ) = P ( A ⊕ B ). (cid:3) TOCHASTICITY AS DENSITY 11
The Cartesian Product.
Another classical candidate would be the Carte-sian product A × B . However, this is even less reliable than the join. Whether ornot A × B even has asymptotic density related to the density of A and the densityof B can depend on the selected pairing function. For example, if h , i : ω → ω is apairing function, consider the function f : ω → ω defined via f ( i, n ) = h i − , n i !for i > f (0 , n ) = s n where S is the set of nonfactorials. Then f has all of the properties we desire in apairing function, i.e. it is a computable bijection with computable inverse between ω and ω . Using f as a pairing function, A × B (as a set of codes for pairs h a, b i , a ∈ A and b ∈ B ) would have density equal to that of B if 0 ∈ A and density 0otherwise. Removing or adding a single element from A never changes the density,let alone the intrinsic density, but we could toggle the upper density of A × B be-tween 0 and ρ ( B ) by toggling whether or not 0 is in A .Even if we fix a pairing function h , i which does respect the density of A and B , the above f shows that this will not extend to intrinsic density: As f and h , i are both computable and have computable inverse, there is a permutation π suchthat π ( h n, m i ) = f ( n, m ). Then π ( A × B ) will be as in the previous paragraph, so A × B cannot have intrinsic density determined by the intrinsic densities of A and B .These methods seem like they should generalize to any attempt at “nicely” coding A and B into computable sets in such a way that we can easily recover them. Thisintuition will be formalized in Theorem 3.3.3. Achieving Different Intrinsic Densities
The methods of Section 2 illustrate why coding methods that enumerate a setonto a computable one are insufficient. As long as we computably know where oneof our sets A is being coded, there is a permutation which can make the resultingset look like A modulo a set of density 0, so the best case scenario is that theresulting set can have the same intrinsic density as the original sets. We there-fore must use a coding method which is not inherently computable to achieve ourgoals of changing the density. For example, Astor [1] proved that if A has intrinsicdensity α and B is 1-Random relative to A , then A ∩ B has intrinsic density α .The reliance on randomness here is not ideal: intrinsic density is itself not a goodnotion of randomness as there are sets with defined intrinsic density which can becomputed by arbitrarily small subsets: Let A be 1-Random and let X = A and X n +1 = X n ⊕ X n . By Lemma 3.9 and Theorem 4.1 below, X k will be a set ofintrinsic density , but { n : 2 k n ∈ X k } = A , so there is a subset with density k +1 which computes all of X k .It would be preferable if we could create tools that work based solely on condi-tions on intrinsic density as opposed to conditions of randomness. We can in factdo this. The following coding methods are natural and computable in A and B ,but do not allow us to recover A or B easily, and so do not fall prey to the methodsof the previous section. We shall show in this section that these tools generalize the previous known results and do not require appeals to randomness. We shallthen apply them to achieve every real in the unit interval. (This part will use ran-domness, but only as an easy way to obtain sets with the right properties: the coretheorems work regardless of how random a set is.) Definition 3.1.
Let A and B be sets of natural numbers. • B ⊲ A , or B into A , is { a b < a b < a b < . . . } That is,
B ⊲ A is the subset of A obtained by taking the “ B -th elements of A .” • B ⊳ A , or B within A , is { n : a n ∈ B } That is,
B ⊳ A is the set X such that X ⊲ A = A ∩ B . With
A ⊲ B , we are simply thinking of A as a copy of ω as a well-order and B ⊲ A is the subset corresponding to B under the order preserving isomorphismbetween A and ω . The intuition for why this might work for our purposes is that ifa computable permutation on ω could change the size of a copy of B living inside A , then it must have been able to change the size of B or A to begin with. We shallsee below that this intuition is correct and B ⊲ A will work elegantly with intrinsicdensity, multiplying the intrinsic densities of A and B as long as some conditionsare met.We first make a few elementary observations: • For all A , A = A ⊲ ω = ω ⊲ A = A ⊳ ω . • For all A and B and any i , a i is either in B or B . Therefore i is either in B ⊳ A or B ⊳ A respectively, so (
B ⊳ A ) ⊔ ( B ⊳ A ) = ω . • If A is intrinsically small, then so is X ⊲ A for any X , as intrinsic smallnessis closed under subsets. The same is not true for X ⊳ A , as in general it isnot necessarily a subset of A or X . • If B ∩ C = ∅ , then ( B ⊲A ) ∩ ( C ⊲A ) = ∅ . Furthermore, A = ( X ⊲A ) ⊔ ( X ⊲A ). • Following the notation introduced for stochasticity in Definition 1.4, S s,i ( A ) = { n : p s ( A ) ( n ) ∈ i ( A ) } = i ( A ) ⊳ s ( A ) • ⊲ is associative, i.e. B ⊲ ( A ⊲ C ) = (
B ⊲ A ) ⊲ C : By definition, ( A ⊲ C ) = { c a < c a < c a < . . . } and thus B ⊲ ( A ⊲ C ) = { c a b < c a b < c a b < . . . } Similarly, (
B ⊲ A ) = { a b < a b < a b < . . . } , and therefore by definition( B ⊲ A ) ⊲ C = { c a b < c a b < c a b < . . . }• ⊳ is not associative: Consider the set of evens E , the set of odds O , and theset N of evens which are not multiples of 4. Then( O ⊳ N ) ⊳ E = ∅ ⊳ N = ∅ However,
O ⊳ ( N ⊳ E ) =
O ⊳ O = ω TOCHASTICITY AS DENSITY 13 • ⊲ and ⊳ do not associate with each other in general: B ⊲ ( A ⊳ ( B ⊲ A )) =
B ⊲ ω = B but ( B ⊲ A ) ⊳ ( B ⊲ A ) = ω Similarly,
B ⊳ ( A ⊲ B ) = ω , but ( B ⊳ A ) ⊲ B is a subset of B .The following theorem is quite intuitive and allows us to use a single set withdefined intrinsic density to find new ones, however these new sets will have thesame intrinsic density as the original set. We shall first prove a technical lemma toaid in our proofs. Lemma 3.2.
Let f , f , . . . , f k be a finite collection of injective computable func-tions and let C be a computable set. Then there is a computable set H ⊆ C suchthat ρ ( f i ( H )) = 0 for all i .Proof. Let h = c . Then given h n , define h n +1 to be the least element c of C with f i ( c ) ≥ h n ! for all i . Then set H = { h < h < h < . . . } . Then ρ ( f i ( H )) = 0 forall i because | f i ( H ) ↾ n | ≤ |{ n ! : n ∈ ω } ↾ n | . (cid:3) Theorem 3.3.
Let C be computable and P ( A ) = α . Then P ( A ⊳ C ) = α .Proof. Under the map which takes c n to n , A ∩ C is mapped to A ⊳ C . Howeverunless C is ω , this is not a permutation. Using Lemma 3.2, we are able to massagethis map into a permutation which takes c n to n modulo a set of density 0. Thenunder this permutation, A ∩ C (and A ) goes to A ⊳ C modulo a set of density 0.Therefore if
A⊳C did not have intrinsic density α , A could not either by Lemma 1.14.Formally, assume P ( A ⊳ C ) = α . Suppose π is a computable permutation with ρ ( π ( A ⊳ C )) > α . Let f : C → ω be defined via f ( c n ) = n . Then f ( A ∩ C ) = A ⊳ C : A ∩ C A ⊳ C π ( A ⊳ C ) f π By Lemma 3.2, there is H ⊆ C computable with ρ ( π ( f ( H ))) = 0. Define π f : ω → ω via π f ( n ) = f ( n ) for n ∈ C \ H , and for n ∈ C ⊔ H define π f ( n ) to bethe least element of f ( H ) not equal to π f ( j ) for some j < n . As f agrees with π f on C \ H , π f (( A ∩ C ) \ H ) = f ( A ∩ C ) \ f ( H ) = ( A ⊳ C ) \ f ( H )Therefore by applying π , π ( π f (( A ∩ C ) \ H )) = π (( A ⊳ C ) \ f ( H )) = π ( A ⊳ C ) \ π ( f ( H ))Using the above equality, ρ ( π ( π f (( A ∩ C ) \ H ))) = ρ ( π ( A ⊳ C ) \ π ( f ( H )))As ρ ( π ( f ( H ))) = 0, we can apply Lemma 1.14 and see ρ ( π ( A ⊳ C ) \ π ( f ( H ))) = ρ ( π ( A ⊳ C ))As ( A ∩ C ) \ H ⊆ A , ρ ( π ( π f ( A ))) ≥ ρ ( π ( π f (( A ∩ C ) \ H ))) = ρ ( π ( A ⊳ C ))However, we assumed that ρ ( π ( A ⊳ C )) > α , so ρ ( π ( π f ( A ))) > α . As π ◦ π f is acomputable permutation, this implies P ( A ) = α . This proves that if π is a computable permutation with ρ ( π ( A ⊳ C )) > α , then P ( A ) = α . If there is no such permutation, there must be a computable permu-tation π with ρ ( π ( A ⊳ C )) < α because we assumed that P ( A ⊳ C ) = α . Thenbecause ( π ( A ⊳ C )) ⊔ ( π ( A ⊳ C )) = π (( A ⊳ C ) ⊔ ( A ⊳ C )) = π ( ω ) = ω we have ρ n ( π ( A ⊳ C )) = 1 − ρ n ( π ( A ⊳ C )) for all n . Therefore by the subtractionproperties of the limit superior, ρ ( π ( A ⊳ C )) ≥ − ρ ( π ( A ⊳ C ))As we assumed ρ ( π ( A ⊳ C )) < α ,1 − ρ ( π ( A ⊳ C )) > − α Thus ρ ( π ( A⊳C )) > − α . We now apply the previous case to get that P ( A ) = 1 − α ,which automatically implies P ( A ) = α . (cid:3) We obtain an alternate proof of Lemma 2.1 as a corollary of this result.
Corollary 3.4. (Lemma 2.1) If P ( A ) = P ( B ) , then A ⊕ B does not have intrinsicdensity.Proof. Suppose A ⊕ B has intrinsic density γ . Let E be the set of even numbersand O the set of odd numbers. By Theorem 3.3, P (( A ⊕ B ) ⊳ E ) = P (( A ⊕ B ) ⊳ O ) = γ However ( A ⊕ B ) ⊳ E = A and ( A ⊕ B ) ⊳ O = B , so P ( A ) = P ( B ) = γ . (cid:3) In addition to giving a much simpler proof of Lemma 2.1, this result is confirmingwhat we might suspect given the results of Section 2: we cannot achieve sets of newintrinsic density by enumerating sets of intrinsic density along computable sets, asthe resulting set must have the same intrinsic density if it has intrinsic density atall. Therefore we need to turn our attention to coding within noncomputable sets.We now make an observation about the asymptotic density of
B ⊲ A , which will becritical for investigating its intrinsic density.
Lemma 3.5. • ρ ( B ⊲ A ) ≤ ρ ( B ) ρ ( A ) . • ρ ( B ⊲ A ) ≥ ρ ( B ) ρ ( A ) .Proof. By Lemma 1.2, ρ ( B ⊲ A ) = lim sup n →∞ n + 1 a b n + 1 = lim sup n →∞ n + 1 a b n + 1 · n →∞ n + 1 a b n + 1 · b n + 1 b n + 1By the submultiplicativity of the limit superior, ρ ( B ⊲ A ) ≤ (lim sup n →∞ b n + 1 a b n + 1 )(lim sup n →∞ n + 1 b n + 1 ) = (lim sup n →∞ b n + 1 a b n + 1 ) ρ ( B )Now { b n +1 a bn +1 } n ∈ ω is a subsequence of { n +1 a n +1 } n ∈ ω , solim sup n →∞ b n + 1 a b n + 1 ≤ lim sup n →∞ n + 1 a n + 1 = ρ ( A ) TOCHASTICITY AS DENSITY 15
Therefore ρ ( B ⊲ A ) ≤ ρ ( B ) ρ ( A ) as desired.The case for the limit inferior is nearly identical, reversing ≤ to ≥ and using super-multiplicativity along with the corresponding identity from Lemma 1.2. (cid:3) Corollary 3.6. If ρ ( A ) = α and ρ ( B ) = β , then ρ ( B ⊲ A ) = αβ . Therefore, if
B ⊲ A has intrinsic density, its intrinsic density must be the productof the densities of A and B . Our next goal is to prove that B ⊲ A does indeedhave defined intrinsic density with sufficient assumptions on A and B . Recall thata set X has Y -intrinsic density, or intrinsic density relative to Y , if its density isinvariant under all Y -computable permutations as opposed to just the computableones. We use P Y ( X ) to denote the Y -intrinsic density of X if it exists. Theorem 3.7. If P ( A ) = α and P A ( B ) = β , then P ( B ⊲ A ) = αβ .Proof. The proof is very similar to the proof of Theorem 3.3, however we shallpresent it fully here without referring to techniques from that proof, as it is quitetechnical. Here the idea is that for any fixed computable permutation π , there isan A -computable permutation which sends B to π ( B ⊲ A ) ⊳ π ( A ) modulo a set ofdensity 0. Therefore if π witnesses that B ⊲ A does not have intrinsic density αβ ,i.e. π ( B ⊲ A ) does not have density αβ , and A has intrinsic density α , Lemma 3.5will show that π ( B ⊲ A ) ⊳ π ( A ) does not have density β , and thus B does not have A -intrinsic density β .Formally, assume P ( A ) = α . Assume that P ( B ⊲ A ) = αβ . We shall show that P A ( B ) = β . First suppose that there is some computable permutation π such that ρ ( π ( B ⊲ A )) > αβ . We shall let π ( A ) = { p < p < p < . . . } . Let f : A → ω be de-fined via f ( a n ) = n and g : π ( A ) → ω via g ( p n ) = n , i.e. f maps A to its indices and g maps π ( A ) to its indices. Then f ( B ⊲ A ) = B and g ( π ( B ⊲ A )) = π ( B ⊲ A ) ⊳ π ( A ): B ⊲ A π ( B ⊲ A ) B π ( B ⊲ A ) ⊳ π ( A ) πf g Note by Lemma 3.5 that ρ ( π ( B ⊲ A ) ⊳ π ( A )) > β : From the definition,( π ( B ⊲ A ) ⊳ π ( A )) ⊲ π ( A )) = π ( B ⊲ A )and ρ ( B ⊲ A ) > αβ by assumption. ρ ( π ( A )) = α because P ( A ) = α , so ρ ( π ( B ⊲A ) ⊳ π ( A )) ≤ β would contradict Lemma 3.5.From this point forward we shall let X = π ( B ⊲ A ) ⊳ π ( A )for the sake of readability.By Lemma 3.2 relativized to A and applied to g ◦ π , there is an A -computableset H ⊆ A such that: ρ ( g ( π ( H ))) = 0We shall now define permutations which preserve the properties of f and g out-side of H . Define π f : ω → ω via π f ( k ) = f ( k ) for k ∈ A \ H , and for k ∈ A ⊔ H , let π f ( k ) be the least element of f ( H ) not equal to π f ( m ) for some m < k . Define π g : ω → ω similarly using π ( A ), π ( H ), and g ( π ( H )) in place of A , H , and f ( H )respectively. Then π f and π g are A -computable because H , f , and g are, and it isa permutation because f and g are bijections (from A and π ( A ) to ω respectively)which have been modified to be total without violating injectivity or surjectivity.Now we shall compute π g ( π ( π − f ( B \ f ( H )))). As f ( B ⊲ A ) = B and f agreeswith π f on H , π − f ( B \ f ( H )) = ( B ⊲ A ) \ H Furthermore π (( B ⊲ A ) \ H ) = π ( B ⊲ A ) \ π ( H )As g ( π ( B ⊲ A )) = X and π g agrees with g on π ( H ), π g ( π ( B ⊲ A ) \ π ( H )) = g ( π ( B ⊲ A )) \ g ( π ( H )) = X \ g ( π ( H ))Thus π g ( π ( π − f ( B \ f ( H )))) = X \ g ( π ( H )). As ρ ( g ( π ( H )) = 0, Lemma 1.14 shows ρ ( X \ g ( π ( H ))) = ρ ( X )By the definition of X , ρ ( X ) = ρ ( π ( B ⊲ A ) ⊳ π ( A ))which is greater than β by the above. As B \ f ( H ) ⊆ B , π g ( π ( π − f ( B \ f ( H )))) ⊆ π g ( π ( π − f ( B )))and thus ρ ( π g ( π ( π − f ( B )))) ≥ ρ ( π g ( π ( π − f ( B \ f ( H )))))Therefore ρ ( π g ( π ( π − f ( B )))) ≥ ρ ( π ( B ⊲ A ) ⊳ π ( A )) > β As π g ◦ π ◦ π − f is an A -computable permutation, P A ( B ) = β .Therefore we have proved that if there is some computable permutation π suchthat ρ ( π ( B ⊲ A )) > αβ , then P A ( B ) = β . If there is no such permutation, thenthere must be a computable permutation π such that ρ ( π ( B ⊲ A )) < αβ because weassumed P ( B ⊲ A ) = αβ . As A = ( B ⊲ A ) ⊔ ( B ⊲ A ), π ( A ) = π ( B ⊲ A ) ⊔ π ( B ⊲ A ).Therefore ρ ( π ( B ⊲ A )) = ρ ( π ( A ) \ π ( B ⊲ A ))The fact that ρ n ( π ( A )) = ρ n ( π ( B ⊲ A )) + ρ n ( π ( A ) \ π ( B ⊲ A )) combined with theproperties of the limit superior with regards to subtraction implies ρ ( π ( A ) \ π ( B ⊲ A )) ≥ ρ ( π ( A )) − ρ ( π ( B ⊲ A ))We know that ρ ( π ( A )) = α because P ( A ) = α . As we assumed that ρ ( π ( B ⊲ A )) <αβ , ρ ( π ( α )) − ρ ( π ( B ⊲ A )) > α − αβ = α (1 − β )Bringing this together, ρ ( π ( B ⊲ A )) > α (1 − β )Thus we can apply the first case of the proof to show that P A ( B ) = 1 − β , whichautomatically implies P A ( B ) = β , so we are done. (cid:3) Notice that the previous two theorems prove a more general form of Astor’sresult:
TOCHASTICITY AS DENSITY 17
Corollary 3.8. If P ( A ) = α and P A ( B ) = β , then P ( A ∩ B ) = αβ . Remark.
Astor [1] proved this for the special case when A has intrinsic density α and B is 1-Random relative to A , which by Lemma 3.9 below implies B has A -intrinsic density .Proof. By definition, A ∩ B = ( B ⊳ A ) ⊲ A As P A ( B ) = β , Theorem 3.3 relativized to A shows that P A ( B ⊳ A ) = β . Thereforewe can apply Theorem 3.7 to A and B ⊳ A to get that P (( B ⊳ A ) ⊲ A ) = P ( A ∩ B ) = αβ (cid:3) With these tools in hand, we may now look towards showing everything in theunit interval is realized as the intrinsic density of a set of natural numbers. To dothis, we would like to have a countable collection of sets which all have intrinsicdensity relative to each other so that we may apply Theorem 3.7. Sufficientlyrandom sets turn out to be a simple way to get such a collection. As mentionedabove, the following lemma is a stronger form of the previously known result that1-Randoms have intrinsic density . The proof outline follows that of Astor [1]for the special case, and the techniques are similar to those found in Nies [6] andDowney-Hirschfeldt [3]. Lemma 3.9.
Let < p < be rational. If X is p - -Random, then X has intrinsicdensity p .Proof. We shall first show that p -random sets must have density p . This is naturalwhen one considers the martingale approach to randomness: If we expect the ratioof ones to be larger than p , then we shall bet more of our capital on ones. If wedo so carefully, then our betting strategy will succeed on sets with sufficiently largeupper density.Formally, we define a family of martingales such that at least one will succeedon any set with upper density greater than p . Let 0 < α < − p be rational andconsider the martingale M α : 2 <ω → Q defined via: • M α ( ∅ ) = 1 • M α ( σ
0) = (1 − α − p ) M α ( σ ) • M α ( σ
1) = (1 + αp ) M α ( σ )It is immediate that M α is a computable p -martingale from definition. If n σ = |{ k < | σ | : σ ( k ) = 1 }| , M α ( σ ) = (1 + αp ) n σ (1 − α − p ) | σ |− n σ Let p < ǫ ≤
1. If ρ | σ | ( σ ) ≥ ǫ , then n σ ≥ ǫ | σ | and M α ( σ ) ≥ (1 + αp ) ǫ | σ | (1 − α − p ) (1 − ǫ ) | σ | = ((1 + αp ) ǫ (1 − α − p ) − ǫ ) | σ | Notice that for a fixed ǫ , α can be chosen such that (1 + αp ) ǫ (1 − α − p ) − ǫ >
1: As α < − p , 1 − α − p >
0, so we can take the logarithm. (1 + αp ) ǫ (1 − α − p ) − ǫ > and only if ǫ log(1 + αp ) + (1 − ǫ ) log(1 − α − p ) > − α − p ) > ǫ (log(1 − α − p ) − log(1 + αp ))As 1 − α − p < αp > − α − p ) − log(1 + αp ) < − α − p )log(1 − α − p ) − log(1 + αp ) < ǫ By L’Hˆopital’s Rule, the limit of the left hand side as α approaches 0 is p . As ǫ > p , there is α close enough to 0 such that this is true, and thus such that(1 + αp ) ǫ (1 − α − p ) − ǫ > α , M α succeeds on any set X whose upper density is greater than ǫ , as this implies that there are infinitely many n such that M α ( X ↾ n ) ≥ ((1 + α ) ǫ (1 − α ) − ǫ ) n . Therefore, for any X with ρ ( X ) > p , there is an ǫ > p with ρ ( X ) ≥ ǫ . The corresponding M α thus succeeds on X . Additionally, for any set X with lower density less than p , the same analysis can be applied to the complement.By switching the roles of (1 + αp ) and (1 − α − p ) in the construction of M α , we obtaina computable p -martingale which succeeds on X . Therefore any p -1-Random (infact any p -computably random) set must have density p .Now we shall show that p -1-Random sets are also closed under permutation, com-pleting the proof. Here the classical notion of martingales does not work as well, aspermutations do not select bits monotonically in general like martingales do. How-ever, it is not difficult to see that permutations preserve µ p , so we shall prove thisresult using the measure notion of randomness, which is enough due to Theorem1.13.Given σ ∈ <ω , consider [ σ ] = { X ∈ ω : σ (cid:22) X } . For π a computable per-mutation, let [ π ( σ )] = { X ∈ ω : X ( π ( n )) = σ ( n ) for all n < | σ |} Notice that [ π ( σ )] is open. Furthermore, let k = max n< | σ | { π ( n ) } . Then P σ = { τ ∈ k +1 : τ ( π ( n )) = σ ( n ) for all n < | σ |} is a prefix-free set which defines [ π ( σ )]. Then for all σ it follows from the definitionof [ π ( σ )] that µ ([ π ( σ )]) = Σ τ ∈ P σ µ p ( τ ) = µ p ( σ ) Σ γ ∈ k +1 −| σ | µ p ( γ ) = µ ([ σ ])If {U i } i ∈ ω is a µ p -Martin-L¨of test, then let V i be defined via V i = [ σ ∈ U i [ π ( σ )] TOCHASTICITY AS DENSITY 19
By the above, µ p ( V i ) = µ p ( U i ), so {V i } i ∈ ω is also a µ p -Martin-L¨of test because π is computable. X passes {U i } i ∈ ω if and only if π ( X ) passes {V i } i ∈ ω by definition.Therefore if Y is not p -1-Random, then π − ( Y ) is not p -1-Random either. Thusthe p -1-Randoms are closed under computable permutation as desired. (cid:3) Lemma 3.10.
There is a countable, disjoint sequence of sets { A i } i ∈ ω such that P ( A i ) = i +1 . Furthermore, lim n →∞ P ( F i>n A i ) = 0 .Proof. Recall that given a set X , X [ i ] denotes the i -th column of X , i.e. { n : h i, n i ∈ X } . Let X ⊆ ω be 1-Random. Then for all i , X [ i ] is 1-Random relativeto L j = i X [ j ] . (Essentially Van Lambalgen [9], Downey-Hirschfeldt [3] Corollary6.9.6) Note that the proof of Lemma 3.9 relativizes to the fact that Z -1-Randomshave Z -intrinsic density easily. In particular, taking a single 1-Random automat-ically gives us infinitely many mutually 1-Random sets. Using these together withTheorem 3.7, we can construct the desired sequence, where the mutual randomnessensures us that the conditions of the theorem are met.Let B = ω . Given B n , let A n = X [ n ] ⊲ B n and B n +1 = X [ n ] ⊲ B n Note that for all i , B i +1 ⊆ B i and A i ∩ B i +1 = ∅ , as B i +1 = X [ i ] ⊲ B i and A i = X [ i ] ⊲ B i . Then for i < j , A i ∩ A j = ∅ because A j ⊆ B j ⊆ B i +1 . Thus { A i } i ∈ ω is disjoint. We now verify that P ( A i ) = i +1 and P ( B i ) = i by induction. P ( B ) = P ( ω ) = 1, and B is computable. Suppose that B i is L jn A i ) = 0 must be true for any such collectionof sets, as lim n →∞ P ( F i ≤ n A i ) = 1. (cid:3) Jockusch and Schupp [4] proved that asymptotic density enjoys a restricted formof countable additivity: if there is a countable sequence { S i } i ∈ ω of disjoint sets suchthat ρ ( S i ) exists for all i and lim n →∞ ρ ( G i>n S i ) = 0then ρ ( G i ∈ ω S i ) = ∞ Σ i =0 ρ ( S i ) The intrinsic density analog of this results follows immediately from the fact thatpermutations preserve disjoint unions. That is, if there is a countable sequence { S i } i ∈ ω of disjoint sets such that P ( S i ) exists for all i andlim n →∞ P ( G i>n S i ) = 0then P ( G i ∈ ω S i ) = ∞ Σ i =0 P ( S i )This together with the previous lemma allows us to construct a set with intrinsicdensity r for any r ∈ [0 , Corollary 3.11.
Every real in [0 , is realized as the intrinsic density of some setof natural numbers.Proof. Let r ∈ [0 , B r ⊆ ω be the set whose characteristic function isidentified with the binary expansion that gives r , i.e. the set of all n such that the n -th bit in the binary expansion for r is a one. Now let { A i } i ∈ ω be as in Lemma3.10. Let X r = F n ∈ B r A n . Note thatlim n →∞ P ( G i ∈ B r ,i>n A i ) = 0Because F i ∈ B r ,i>n A i ⊆ F i>n A i and lim n →∞ P ( F i>n A i ) = 0. By the fact thatcountable unions sum intrinsic densities and the definition of X r , P ( X r ) = Σ n ∈ B r P ( A n ) = Σ n ∈ B r n +1 By the definition of the binary expansion, P ( X r ) = Σ n ∈ B r n +1 = r (cid:3) Notice that Corollary 3.11 can be relativized to a fixed oracle X . Lemma 3.9and Van Lambalgen’s theorem both relativize, so Lemma 3.10 does as well. Thissuffices to relativize Corollary 3.11: in general, B r cannot be taken to have anyspecial relationship with X , as it is unique for a fixed r . However, whether ornot B r is computable from the oracle has no basis on X r because each A n has X -intrinsic density.4. Filling the Gaps for Intrinsic Density
Our work in the previous two sections left some gaps, which we address here.We showed in Section 2 that if the join of two sets has intrinsic density, then eachset has the same intrinsic density. We now prove the converse using the definitionsfrom Section 3.
Theorem 4.1.
Suppose P ( A ) = P ( B ) = α . Then P ( A ⊕ B ) = α . TOCHASTICITY AS DENSITY 21
Proof.
We shall use a technical lemma to complete the proof. Let E represent theeven numbers, and let O represent the odd numbers. Lemma 4.1.1 will prove that,for any computable permutation π , ρ ( π ( A ⊕ B ) ⊳ π ( E )) = ρ ( π ( A ⊕ B ) ⊳ π ( O )) = α To show this we shall give a computable permutation which sends A to π ( A ⊕ B ) ⊳ π ( E ) modulo a set of density zero. We will do this by first showing thereis an obvious computable injective function which takes A to π ( A ⊕ B ) ⊳ π ( E ),then use the techniques from Section 3 to massage it into a suitable permutation.We can use the same method to send B to π ( A ⊕ B ) ⊳π ( O ) modulo a set of density 0.From there, we will use Lemma 4.1.1 to show that ρ ( π ( A ⊕ B )) = α , proving thetheorem. Note that we cannot use Theorem 3.3 to obtain the above facts becausewe are trying to prove that A ⊕ B has intrinsic density α . Lemma 4.1.1.
Let π be a computable permutation and let A and B be as in thestatement of Theorem 4.1. Then ρ ( π ( A ⊕ B ) ⊳ π ( E )) = ρ ( π ( A ⊕ B ) ⊳ π ( O )) = α Proof.
Let h : π ( E ) → ω send the n -th element of π ( E ) to n (i.e. the inverse of theprincipal function), and let d : ω → E be defined via d ( n ) = 2 n . Then notice that d ( A ) = A ⊕ ∅ . Furthermore, observe that for any X ⊆ π ( E ), h ( X ) = X ⊳ π ( E ) bythe definition of h and the within operation. Therefore h ( π ( d ( A ))) = h ( π ( A ⊕ ∅ )) = π ( A ⊕ ∅ ) ⊳ π ( E )As π ( A ⊕ B ) ∩ π ( E ) ⊆ π ( A ⊕ ∅ ), π ( A ⊕ ∅ ) ⊳ π ( E ) = π ( A ⊕ B ) ⊳ π ( E )Thus h ( π ( d ( A ))) = π ( A ⊕ B ) ⊳ π ( E ). We shall now use the techniques of Section 3to change h and d into permutations which preserve the relevant densities.By Lemma 3.2, there is a computable set H ⊆ π ( E ) with ρ ( h ( H )) = 0. Nowdefine the computable permutation π h via π h ( n ) = h ( n ) for n ∈ π ( E ) \ H , andhave π h enumerate π ( O ) ⊔ H onto h ( H ) in order. Similarly, define the computablepermutation π d via π d ( n ) = d ( n ) for n ∈ ω \ d − ( π − ( H )), and have π d enumerate d − ( π − ( H )) onto O ⊔ π − ( H ).As π d agrees with d on d − ( π − ( H )), we now see that π d ( A \ π − d ( π − ( H ))) = ( A ⊕ ∅ ) \ π − ( H )Furthermore, applying π shows that π ( π d ( A \ π − d ( π − ( H )))) = π (( A ⊕ ∅ ) \ π − ( H )) = π ( A ⊕ ∅ ) \ H As π h agrees with h on π ( E ) \ H and h ( π ( A ⊕ ∅ )) = π ( A ⊕ B ) ⊳ π ( E ), we have π h ( π ( A ⊕ ∅ ) \ H ) = ( π ( A ⊕ B ) ⊳ π ( E )) \ h ( H )Therefore ( π ( A ⊕ B ) ⊳ π ( E )) \ h ( H ) ⊆ π h ( π ( π d ( A ))) and π h ( π ( π d ( A ))) ⊆ ( π ( A ⊕ B ) ⊳ π ( E )) ∪ h ( H ).By choice of H , ρ ( h ( H )) = 0, so Lemma 1.14 shows that ρ ( π h ( π ( π d ( A )))) = ρ (( π ( A ⊕ B ) ⊳ π ( E )) \ h ( H )) = ρ ( π ( A ⊕ B ) ⊳ π ( E )) and ρ ( π h ( π ( π d ( A )))) = ρ (( π ( A ⊕ B ) ⊳ π ( E )) \ h ( H )) = ρ ( π ( A ⊕ B ) ⊳ π ( E ))Therefore, as P ( A ) = α and π h ◦ π ◦ π d is a computable permutation, ρ ( π ( A ⊕ B ) ⊳ π ( E )) = α A nearly identical argument with O in place of E and B in place of A shows similarlythat ρ ( π ( A ⊕ B ) ⊳ π ( O )) = α (cid:3) We shall now show that this implies that ρ ( π ( A ⊕ B )) = α . Consider ρ n ( π ( A ⊕ B )). By definition, ρ n ( π ( A ⊕ B )) = | π ( A ⊕ B ) ↾ n | n As ω = π ( E ) ⊔ π ( O ), | π ( A ⊕ B ) ↾ n | n = | π ( A ⊕ B ) ∩ π ( E ) ↾ n | + | π ( A ⊕ B ) ∩ π ( O ) ↾ n | n The latter expression can be rewritten as | π ( E ) ↾ n || π ( E ) ↾ n | · | π ( A ⊕ B ) ∩ π ( E ) ↾ n | n + | π ( O ) ↾ n || π ( O ) ↾ n | · | π ( A ⊕ B ) ∩ π ( O ) ↾ n | n Let m be the largest number such that the m -th element of π ( E ) is less than n ,and let k be the analogous number for π ( O ). Now notice that | π ( A ⊕ B ) ∩ π ( E ) ↾ n || π ( E ) ↾ n | = ρ m ( π ( A ⊕ B ) ⊳ π ( E ))and | π ( A ⊕ B ) ∩ π ( O ) ↾ n || π ( O ) ↾ n | = ρ k ( π ( A ⊕ B ) ⊳ π ( O ))by the definition of the within operation. Therefore, we can rewrite ρ n ( π ( A ⊕ B ))as ρ m ( π ( A ⊕ B ) ⊳ π ( E )) · ρ n ( π ( E )) + ρ k ( π ( A ⊕ B ) ⊳ π ( O )) · ρ n ( π ( O ))Using the fact that ρ n ( π ( E )) + ρ n ( π ( O )) = 1, ρ n ( π ( A ⊕ B )) = ρ m ( π ( A ⊕ B ) ⊳π ( E )) · ρ n ( π ( E ))+ ρ k ( π ( A ⊕ B ) ⊳π ( O )) · (1 − ρ n ( π ( E )))Rearranging, this is equal to ρ k ( π ( A ⊕ B ) ⊳ π ( O )) + ρ n ( π ( E )) · ( ρ m ( π ( A ⊕ B ) ⊳ π ( E )) − ρ k ( π ( A ⊕ B ) ⊳ π ( O )))Taking the limit as n goes to infinity, m and k both go to infinity. Thus ρ m ( π ( A ⊕ B ) ⊳ π ( E )) − ρ k ( π ( A ⊕ B ) ⊳ π ( O ))goes to 0 by Lemma 4.1.1. As ρ n ( π ( E )) is bounded between 0 and 1 by definition,the second term vanishes. Thereforelim n →∞ ρ n ( π ( A ⊕ B )) = lim n →∞ ρ k ( π ( A ⊕ B ) ⊳ π ( O )) = ρ ( π ( A ⊕ B ) ⊳ π ( O )) = α as desired. (cid:3) Lemma 2.1 and Theorem 4.1 can easily be generalized.
TOCHASTICITY AS DENSITY 23
Definition 4.2.
Let H be a computable, infinite, co-infinite set. Then the H -joinof A and B , denoted by A ⊕ H B , is ( A ⊲ H ) ⊔ ( B ⊲ H )Notice that A ⊕ B = A ⊕ E B . Furthermore, there is a computable permutation π that sends E to H and O to H in order. Therefore π ( A ⊕ B ) = A ⊕ H B , so thegeneralizations of Lemma 2.1 and Theorem 4.1 follow without needing to reworkthe proofs.Recall that Theorem 3.7 says if P ( A ) = α and P A ( B ) = β , then P ( B ⊲ A ) = αβ .Whether or not either of these conditions can be weakened or dropped is a naturalquestion. It is immediate that we cannot drop the requirement that A has intrinsicdensity: P A ( ω ) = 1 for any A , so ω always satisfies the requirements on B , but ω ⊲ A = A , so A must have intrinsic density. Similarly, B ⊲ ω = B for any B , so B must have intrinsic density. Therefore the only possible weakening of Theorem 3.7would be to require P ( B ) = β as opposed to P A ( B ) = β . However, this fails in astrong way. Lemma 4.3.
Let P ( A ) = . Then P ( A ⊕ A ) = but A ⊲ ( A ⊕ A ) does not haveintrinsic density.Proof. Note that A ⊕ A has intrinsic density by Theorem 4.1 as P ( A ) = implies P ( A ) = .Let E represent the set of even numbers. Notice that A ⊕ A contains exactlyone of 2 k or 2 k + 1 for all k ∈ ω . Therefore the n -th element of A ⊕ A is 2 n if n ∈ A and 2 n + 1 if n A . Thus E ⊳ ( A ⊕ A ) = A by definition. By the properties of the within operation, A ⊲ ( A ⊕ A ) = ( E ⊳ ( A ⊕ A )) ⊲ ( A ⊕ A ) = E ∩ ( A ⊕ A ) = A ⊕ ∅ By Lemma 2.1, however, A ⊕ ∅ does not have intrinsic density. (cid:3) Note that we cannot generalize this result to A ⊕ H A in general, specifically itis not always true that H ⊳ ( A ⊕ H A ) = A : consider H the set of naturals con-gruent to 2 modulo 3, and let A be a set containing 0 but not containing 1. Then0 H ⊳ ( A ⊕ H A ) because p A ⊕ H A (0) = 1 and 1 H . Thus H ⊳ ( A ⊕ H A ) = A aswitnessed by 0.Recall that Theorem 3.3 says that if P ( A ) = α and C is computable, then A ⊳ C also has intrinsic density α . It is natural to wonder if this is symmetric, does C ⊳ A have intrinsic density? The proof of Lemma 4.3 shows that it is possible for
C ⊳ A to have intrinsic density. However, this is not true in general, as
C ⊳ ω = C . It isnot obvious what can be said, if anything, about when C ⊳ A has intrinsic density.Future work exploring this may reveal something interesting about the structureof sets with intrinsic density: let P ( A ) > C be coinfinite, computable with P ( C ⊳ A ) >
0. Such sets would witness the failure of the weak version of Theorem3.7, as (
C ⊳ A ) ⊲ A = C ∩ A and no subset of a computable set can have intrinsicdensity greater than zero. Applications to Classical stochasticity
We shall now apply the tools of Section 3 to determine which MWC and Church-densities can be achieved. It turns out that the into and within operations behavesimilarly for MWC and Church densities as they do for intrinsic density. Through-out this section it is important to remember that C -density requires us to measurethe density of S s,i ( A ) for all ( s, i ) ∈ C which, as per the remark following Definition3.1, satisfies S s,i ( A ) = { n : p s ( A ) ( n ) ∈ i ( A ) } = i ( A ) ⊳ s ( A )where s is the selection function that determines which bits we are interested inand i is the interpretation function which determines how to read those bits. ForMWC-density, the selection function is induced by a monotone selection function f (a computable function from finite binary strings to { , } ) and the interpretationfunction is the identity, so we are required to measure the density of A ⊳ f ( A ) forall monotone selection functions f to compute MWC-density and Church-density.Throughout this section, we shall focus on MWC-density, however, all of our re-sults will go through for Church-density as well: we will often be given a monotoneselection function and need to modify it to suit our needs. Our modification willnever make a total monotone selection function not total, so the result will hold inthe Church-density case as well.It is not immediate that the same basic properties of intrinsic density apply ingeneral, however we can recover some facts. Lemma 5.1.
Let A have MWC-density α . Then A has MWC-density − α .Proof. Let f be a computable monotone selection function. Define f : 2 <ω → { , } via f ( σ ) = f (1 − σ ), where 1 − σ = τ ∈ | σ | with τ ( n ) = 1 − σ ( n ) for all n < | σ | .Then f ( A ↾ n ) = f ( A ↾ n ). As A has MWC-density α and f is a computablemonotone selection function, either f ( A ) is finite or ρ ( A ⊳ f ( A )) = α . If the former,then f ( A ) is also finite. If the latter, then A ⊳ f ( A ) = A ⊳ f ( A ) = A ⊳ f ( A )Therefore ρ ( A ⊳ f ( A )) = ρ ( A ⊳ f ( A )) = 1 − ρ ( A ⊳ f ( A )) = 1 − α as desired. (cid:3) Lemma 5.2.
Suppose C is computable and A has MWC-density α . Then A ⊳ C has MWC-density α .Proof. Let f be a computable monotone selection function. Define ˆ C : 2 <ω → <ω via ˆ C ( σ ) = τ with τ ∈ max ( n : c n < | σ | )+1 and τ ( i ) = σ ( c i ) for all i < | τ | . Notice thatˆ C ( X ↾ c n ) = ( X ⊳ C ) ↾ n by definition.Now define f C : 2 <ω → { , } via f C ( σ ) = 1 if and only if | σ | = c i for some i and f ( ˆ C ( σ )) = 1. As C is computable, ˆ C is computable and thus f C is a computablemonotone selection function. We now show that A ⊳ f C ( A ) = ( A ⊳ C ) ⊳ f ( A ⊳ C ).We shall show that (
A ⊳ C ) ⊳ f ( A ⊳ C ) ⊆ A ⊳ f C ( A ) with a sequence of if andonly ifs, therefore proving the reverse as well. n is in ( A ⊳ C ) ⊳ f ( A ⊳ C ) if and only
TOCHASTICITY AS DENSITY 25 if the n -th element of f ( A ⊳ C ) is in
A ⊳ C , i.e. the n -th k with f (( A ⊳ C ) ↾ k ) = 1is in A ⊳ C . This occurs if and only if c k ∈ A . Now note that f C ( A ) is the set ofall c i such that f ( ˆ C ( A ↾ c i )) = f (( A ⊳ C ) ↾ i ) = 1, so k is as above if and only if c k ∈ f C ( A ) and c k ∈ A . Note that c k must be the n -th element of f C ( A ) because k was the n -th number with f (( A ⊳ C ) ↾ k ) = 1, so n ∈ A ⊳ f C ( A ).As A has MWC-density α , ρ (( A ⊳ C ) ⊳ f ( A ⊳ C )) = ρ ( A ⊳ f C ( A )) = α As f was arbitrary, A ⊳ C also has MWC-density α . (cid:3) To prove the analog of Theorem 3.7 for MWC-density, we require more relativiza-tion. We shall see that this is a theme with MWC-density compared to intrinsicdensity. Unlike intrinsic density, where the selection and interpretation functionsact independently of the input set, MWC-density can change the selected bits basedon finitely much of the input set. This means that if B is related to A in somepredictable fashion, then a monotone selection rule may be able to use informationfrom B to predict bits of A . Assuming the sets have MWC-density relative to eachother will avoid this issue as using B as an oracle will allow us to simulate an in-put set involving B , and vice versa for A . We shall see some consequences of thisdistinction after Theorem 5.4. To prove this theorem, however, we shall requirethe following technical observation. The proof is merely obtained by unravelingdefinitions, but we provide it for clarity as the definitions can be cumbersome. Lemma 5.3.
Let A , B and C be sets. Then ( A ⊳ C ) ⊳ ( B ⊳ C ) =
A ⊳ ( B ∩ C ) Proof.
By definition,
A ⊳ ( B ∩ C ) = { n : p B ∩ C ( n ) ∈ A } That is, it is the set of all n such that the n -th element of B ∩ C is in A .Similarly, by definition( A ⊳ C ) ⊳ ( B ⊳ C ) = { n : p B⊳C ( n ) ∈ A ⊳ C } That is, it is the set of all n such that the n -th element of B ⊳ C is in
A ⊳ C .However, if k ∈ A ⊳ C for some k , this by definition means c k ∈ A . Therefore if n ∈ ( A ⊳ C ) ⊳ ( B ⊳ C ), this translates to c p B⊳C ( n ) ∈ A . As p B⊳C ( n ) is the n -thelement of B ⊳ C , c p B⊳C ( n ) is the n -th element of C which is in B . Another way tophrase this is that c p B⊳C ( n ) is the n -th element of B ∩ C . This confirms that thesets are identical. (cid:3) It is worth noting a corollary of this lemma which we will not need yet seemsdifficult to obtain via intuition: As intersection is symmetric,
A ⊳ ( B ∩ C ) = A ⊳ ( C ∩ B )Therefore applying Lemma 5.3 once on each side tells us that( A ⊳ C ) ⊳ ( B ⊳ C ) = (
A ⊳ B ) ⊳ ( C ⊳ B )Now we are ready to prove the analog of Theorem 3.7.
Theorem 5.4.
Suppose that A has MWC-density α relative to B and B has MWC-density β relative to A . Then B ⊲ A has MWC-density αβ .Proof. The proof is similar to the proof of Theorem 3.7, however there is an extraconsideration for MWC-density because the selected bits can depend on the input.In Theorem 3.7, π ( B ⊲ A ) is a subset of π ( A ), so we send B to π ( B ⊲ A ) ⊳ π ( A )(modulo a set of density zero) and apply Lemma 3.5. However, we don’t know ingeneral if A ⊳ f ( A ) contains ( B ⊲ A ) ⊳ f ( B ⊲ A ) because f ( B ⊲ A ) need not be a subsetof f ( A ), so we first construct a B -computable monotone selection function f B suchthat f B ( A ) = f ( B ⊲ A ) and therefore
A ⊳ f B ( A ) is a superset of ( B ⊲ A ) ⊳ f ( B ⊲ A ).Then because A has MWC-density α relative to B , A ⊳ f B ( A ) will have density α . From there we shall borrow the proof idea of Theorem 3.7, namely we shallconstruct an A -computable monotone selection function f A such that B ⊳ f A ( B ) = (( B ⊲ A ) ⊳ f ( B ⊲ A )) ⊳ ( A ⊳ f B ( A ))Again, as B has MWC-density β with respect to A , B ⊳ f A ( B ) will have density β .We may then apply Lemma 3.5 to show that ( B ⊲ A ) ⊳ f ( B ⊲ A ) has density αβ asdesired.Formally, let f be a computable monotone selection function. If f ( B ⊲ A ) is finite orundefined, we are done. If not, define f B : 2 <ω → { , } via f B ( σ ) = f ( B⊲σ ), where
B ⊲ σ ∈ | σ | is defined as one might expect: B ⊲ σ ( n ) = 1 if and only if σ ( n ) = 1 and n is the b i ’th m such that σ ( m ) = 1 for some i ∈ ω . As ( X ⊲ Y ) ↾ n = X ⊲ ( Y ↾ n ),it is immediate that f B ( A ) = { n : f B ( A ↾ n ) = 1 } = { n : f ( B ⊲ ( A ↾ n )) = 1 } = { n : f (( B ⊲ A ) ↾ n ) = 1 } = f ( B ⊲ A )Therefore, as
B ⊲ A ⊆ A ,( B ⊲ A ) ⊳ f ( B ⊲ A ) = (
B ⊲ A ) ⊳ f B ( A ) ⊆ A ⊳ f B ( A )Let X = (( B ⊲ A ) ⊳ f ( B ⊲ A )) ⊳ ( A ⊳ f ( B ⊲ A ))We shall construct an A -computable monotone selection function f A such that B ⊳ f A ( B ) = X via Lemma 5.3.Let f A : 2 <ω → { , } be defined via f A ( σ ) = f ( σ ⊲ A ), where σ ⊲ A = τ ∈ a | σ | isdefined via τ ( n ) = 1 if and only if n = a m and σ ( m ) = 1 for some m < | σ | . Wenow claim that B ⊳ f A ( B ) = ( B ⊲ A ) ⊳ ( A ∩ f ( B ⊲ A )).If n ∈ ( B ⊳ A ) ⊳ ( A ∩ f ( B ⊲ A )), then the n -th element of A ∩ f ( B ⊲ A ) is in
B ⊲ A by the definition of the within operation. This implies it is of the form a m for m ∈ B , where m is the n -th number k such that a k ∈ A ∩ f ( B ⊲ A ). As a m isin f ( B ⊲ A ), by the definition of f A this implies that m is the n -th number with f (( B ⊲ A ) ↾ a m ) = f (( B ↾ m ) ⊲ A ) = f A ( B ↾ m ) = 1Thus m is the n -th element of f A ( B ), and it lies in B , so m = p f A ( B ) ( n ) ∈ B .Therefore n ∈ B ⊳ f A ( B ). As n was arbitrary,( B ⊳ A ) ⊳ ( A ∩ f ( B ⊲ A )) ⊆ B ⊳ f A ( B ) TOCHASTICITY AS DENSITY 27
This argument reverses, so B = ( B ⊲ A ) ⊳ ( A ∩ f ( B ⊲ A )).Therefore, X = (( B ⊲ A ) ⊳ f ( B ⊲ A )) ⊳ ( A ⊳ f ( B ⊲ A )) = (
B ⊲ A ) ⊳ ( A ∩ f ( B ⊲ A )) =
B ⊳ f A ( B )The first equality is by definition, the second is by Lemma 5.3, and the final is fromthe previous paragraph. This implies X ⊲ ( A ⊳ f B ( A )) = ( B ⊳ f A ( B )) ⊲ ( A ⊳ f B ( A ))As A has MWC-density α with respect to B and f B ( A ) = f ( B⊲A ), ρ ( A⊳f ( B⊲A )) = α . As B has MWC-density β with respect to A , ρ ( B ⊳ f A ( B )) = β . Therefore byLemma 3.5, ρ (( B ⊳ f A ( B )) ⊲ ( A ⊳ f ( B ⊲ A ))) = ρ ( B ⊳ f A ( B )) ρ ( A ⊳ f ( B ⊲ A )) = αβ Finally, recall from the definition of X that X ⊲ ( A ⊳ f ( B ⊲ A )) = (
B ⊲ A ) ⊳ f ( B ⊲ A ).Therefore ρ (( B ⊲ A ) ⊳ f ( B ⊲ A )) = ρ ( X ⊲ ( A ⊳ f ( B ⊲ A ))) = αβ as desired. (cid:3) Following Theorem 3.7, we were able to obtain as an easy corollary that if A has intrinsic density α and B has intrinsic density β relative to A , then A ∩ B hasintrinsic density αβ . The proof simply observed that B ⊳ A had intrinsic density β relative to A via the relativized form of Theorem 3.3 and then applied Theorem3.7 because ( B ⊳ A ) ⊲ A = A ∩ B .Unfortunately, the same proof will not work in the MWC-density case. Theo-rem 5.4 requires relativization in both directions, and while the relativized formof Theorem 5.2 ensures that B ⊳ A has MWC-density β relative to A , it does notensure that A has MWC-density α relative to B ⊳ A , so we cannot apply Theorem5.4 as we wish. This remains an open question which we shall state completely inQuestion 6.5.Fortunately, we can recover the intersection property for relatively MWC-densesets using an alternate proof.
Lemma 5.5. If A has MWC-density α relative to B and B has MWC-density β relative to A , then A ∩ B has MWC-density αβ .Proof. Let f be a computable monotone selection function. If f ( A ∩ B ) is finite,then we are done. Otherwise, consider ( A ∩ B ) ⊳ f ( A ∩ B ). Define the B -computablemonotone selection function f B : 2 <ω → { , } via f B ( σ ) = 1 if and only if f ( σ ∩ B ) = 1, where σ ∩ B = τ ∈ | σ | is given by τ ( n ) = 1 if and only if σ ( n ) = 1 and B ( n ) = 1. Then clearly f B ( A ) = f ( A ∩ B ), so A ⊳ f B ( A ) = A ⊳ f ( A ∩ B ). As A hasMWC-density α relative to B , ρ ( A ⊳ f B ( A )) = ρ ( A ⊳ f ( A ∩ B )) = α We shall now construct an A -computable monotone selection function f A such that B ⊳ f A ( B ) = ( B ⊳ f ( A ∩ B )) ⊳ ( A ⊳ f ( A ∩ B ))via Lemma 5.3. Define f A : 2 <ω → { , } via f A ( σ ) = 1 if and only if f ( A ∩ σ ) = 1 and | σ | ∈ A ,where A ∩ σ is defined similarly to σ ∩ B in the obvious way. Then it followsimmediately that f A ( B ) = A ∩ f ( A ∩ B ), so B ⊳ f A ( B ) = B ⊳ ( A ∩ f ( A ∩ B )By Lemma 5.3, B ⊳ ( A ∩ f ( A ∩ B ) = ( B ⊳ f ( A ∩ B )) ⊳ ( A ⊳ f ( A ∩ B ))Therefore by the properties of the within operation we have that( B ⊳ f A ( B )) ⊲ ( A ⊳ f B ( A )) = ( B ⊳ f ( A ∩ B )) ⊳ ( A ⊳ f ( A ∩ B )) ⊲ ( A ⊳ f ( A ∩ B )) =( B ⊳ f ( A ∩ B )) ∩ ( A ⊳ f ( A ∩ B )) = ( A ∩ B ) ⊳ f ( A ∩ B )By Lemma 3.5, ρ (( B ⊳ f A ( B )) ⊲ ( A ⊳ f B ( A )) = ρ (( A ∩ B ) ⊳ f ( A ∩ B )) = αβ As f was arbitrary, A ∩ B has MWC-density αβ . (cid:3) Where Lemma 4.1 says that in a specific sense intrinsic density is ignorant ofinternal structure, the opposite is true of MWC-density. In fact, the analog ofLemma 4.1 for MWC-density fails in very strong fashion.
Lemma 5.6.
Suppose that A has MWC-density α for ≤ α < . Then A ⊕ A doesnot have MWC-density.Proof. Let E be the set of even numbers and let O be the set of odd numbers.Define f : 2 <ω → { , } via f ( σ ) = 1 if | σ | ∈ O and σ ( | σ | −
1) = 1 and f ( σ ) = 0otherwise. Then for any A , f ( A ⊕ A ) = A ⊲ O . Therefore( A ⊕ A ) ⊳ f ( A ⊕ A ) = ( A ⊕ A ) ⊳ ( A ⊲ O ) = ω so ρ (( A ⊕ A ) ⊳ f ( A ⊕ A )) = 1However as A has MWC-density α <
1, it has density α and A ⊕ A has density α . Therefore A ⊕ A cannot have MWC-density as its asymptotic density does notmatch the density of ( A ⊕ A ) ⊳ f ( A ⊕ A ). (cid:3) Not only does the join fail to behave for MWC-density, but we shall in fact seethat the union does not behave either. The difficulty lies in the fact that the bitsselected by f on A ⊔ B need not be the union of the bits selected by f on A andthe bits selected by f on B in general. On one hand, it is not difficult to provethat if A has MWC-density α relative to B and B has MWC-density β relative to A with A and B disjoint, then A ⊔ B has MWC-density α + β : given a monotoneselection function f , there is a B -computable monotone selection function f B suchthat f B ( A ) = f ( A ⊔ B ) and similarly there is an A -computable monotone selectionfunction f A such that f A ( B ) = f ( A ⊔ B ). Then( A ⊔ B ) ⊳ f ( A ⊔ B ) = ( A ⊳ f ( A ⊔ B )) ⊔ ( B ⊳ f ( A ⊔ B )) = ( A ⊳ f B ( A )) ⊔ ( B ⊳ f A ( B ))by the properties of the within operation. Therefore ρ (( A ⊔ B ) ⊳ f ( A ⊔ B )) = ρ ( A ⊳ f B ( A )) + ρ ( B ⊳ f A ( B )) = α + β However, Lemma 5.5 ensures that A ∩ B = ∅ implies that one of A or B has MWC-density 0 under these assumptions, so this result cannot be used to obtain newMWC-densities as the disjoint unions of sets with others. TOCHASTICITY AS DENSITY 29
One may think to drop the requirements that A and B have MWC-density rel-ative to one another, therefore disallowing the use of Lemma 5.5 and avoiding thisproblem. However, the union still need not have MWC-density. The followinglemmas will allow us to construct such an example. Lemma 5.7.
There exists an infinite set A ≤ T ∅ ′ such that A has MWC-density(and therefore Church-density) .Proof. The construction is similar in principle to the jump strategy for constructingintrinsically small sets from [5]. However, the details are more technical due to thefact that monotone selection rules can change their behavior on different inputswhereas permutations cannot. We cannot simply choose large enough elements toenter our set, as a given monotone selection rule may refuse to act until it sees anelement enter the set. We utilize the power of the jump to determine if, for a givenmonotone selection function f , it is possible to force a large gap into A ⊳ f ( A ) andensure the density is small. If it is not possible, then we do not allow anything into A ⊳ f ( A ) at all until a large gap appears naturally. If no such gap appears, then A ⊳ f ( A ) will be finite and we succeed.Formally, let f i be an enumeration of the partial computable monotone selectionfunctions. The basic module for ensuring that ρ ( A ⊳ f i ( A )) = 0 for this specific f i is as follows: After seeing the n -th 1 enter A ⊳ f i ( A ) at σ s , we do not allow another1 to enter until we see n f i ( σ ) ↓ uniformly in σ and i .) We will attempt to achievethis by picking some m such that f i ( σ s k ) = 1 for n k ’s less than m and setting σ s +1 = σ s m
10. The jump can determine if such an m exists.Suppose we have defined σ (cid:22) A and there is no m such that f i ( σ m ) = 1. Thenwe cannot force anything into A ⊳ f i ( A ) without adding extra 1’s to A , potentiallyadding some 1’s to f j ( A ) for some j = i . To fix this issue, we say f i is pausedfor σ if there does not exist an m such that f i ( σ m ) = 1. As mentioned above, ∅ ′ can determine if f i is paused for σ . When determining how to extend σ s to σ s +1 = σ s m
10, if f i ( σ s m
1) = 1, then σ s +1 puts a 0 into A ⊳ f i ( A ). If not, thennothing changes. In both cases, no 1’s are added to A ⊳ f i ( A ) by σ s . We continueand ask if f i is paused for σ s +1 . Either we will eventually see enough 0’s enter A ⊳ f i ( A ) after some number of stages and be allowed to add a 1, or this will nothappen and f i ( A ) will be finite. We succeed in both cases.We say f i is almost paused for σ if there is some k such that f i is paused for σ k and σ k does not put enough zeroes into A ⊳ f i ( A ). Here, to say f i is pausedmeans we cannot force another 0 into A ⊳ f i ( A ) by only adding 0’s to A . To say f i is almost paused means we may be able to force some zeroes into A ⊳ f i ( A ), but wecannot force enough zeroes into A ⊳ f i ( A ). (Being almost paused resembles a Σ question, but the bound on the number of zeroes necessary reduces it to a questionthe jump can answer: we can ask if f i is paused for σ s : If so, then it is almostpaused. If not, then extend to σ s k , where k witnesses that f i is not paused for σ s . This adds a zero to A ⊳ f i ( A ). Now ask if f i is paused for σ s k and repeat.Eventually we will either reach a point where it is paused or we will put enoughzeroes into A ⊳ f i ( A ).) Finally, we describe the construction using this module on all i simultaneously:at stage s , we consider only the i ≤ s . Using the jump, determine those i which arealmost paused at stage s and ignore them. For the remaining i , we may choose m large enough such that σ s +1 = σ s m
10 puts enough zeroes into
A ⊳ f i ( A ) to ensure n zeroes are enumerated before the n + 1-st 1, where n is the current number ofones in A ⊳ f i ( A ). As we are ignoring all of the almost paused selection functions,we can always extend to σ s +1 and thus A is infinite. Furthermore, ρ ( A ⊳ f i ( A )) = 0for all i as either f i ( A ) is finite or ρ n ( A ⊳ f i ( A )) ≤ k +1 k + k +1 for increasing k . (If thereare k + 1 ones in | A ⊳ f i ( A ) ↾ n | , then there are at least k zeroes between the finaltwo.) (cid:3) Lemma 5.8. If A has MWC-density and g is an increasing, total, computablefunction, then B = { a n + g ( n ) : n ∈ ω } also has MWC-density .Proof. We argue by contrapositive. Let f be a monotone selection function suchthat ρ ( B ⊳ f ( B )) >
0. We shall construct a monotone selection function ˆ f suchthat ρ ( A ⊳ ˆ f ( A )) ≥ ρ ( B ⊳ f ( B )) > σ ∈ <ω , let σ < σ < · · · < σ k represent all indices on which σ is 1.Define g ( σ ) to be τ ∈ | σ | + g ( k +1) with τ ( i ) = 1 if and only if i = σ j + g ( j ) for some j ≤ k . Finally, define ˆ f : 2 <ω → { , } via ˆ f ( σ ) = 1 if and only if f ( g ( σ )) = 1.Suppose that n ∈ B ⊳ f ( B ). Then p f ( B ) ( n ) = a k + g ( k ) for some k ∈ ω . In particu-lar, f ( B ↾ a k + g ( k )) = 1. Therefore, as g ( A ↾ a k ) = B ↾ a k + g ( k ) by the definitionof g ( σ ), we have ˆ f ( A ↾ a k ) = f ( g ( A ↾ a k )) = f ( B ↾ a k + g ( k )) = 1Finally, notice that the m such that p ˆ f ( A ) ( m ) = a k must be less than or equal to n because each element of ˆ f ( A ) corresponds to an element of f ( B ) but not necessarilyvice versa. It follows that ρ ( A ⊳ ˆ f ( A )) ≥ ρ ( B ⊳ f ( B )), as each element of B ⊳ f ( B )corresponds to an element of A ⊳ ˆ f ( A ) which is no larger. As ρ ( B ⊳ f ( B )) >
0, weare done. (cid:3)
Lemma 5.9.
There exists a set A such that A and A ⊲ A both have MWC-density,but A ⊔ ( A ⊲ A ) does not.Proof. By Lemma 5.7, there is an infinite set X with MWC-density 0. By Lemma5.8, A = { x n + n : n ∈ ω } also has MWC-density 0. Notice that a n +1 − a n = x n +1 + ( n + 1) − x n − n = x n +1 − x n + 2 n + 1 > n + 1It follows that the a n -th element of A is a n + n + 1. (The only way this could notbe the case is if a n +1 ≤ a n + n + 1.) Therefore A ⊲ A = { a n + n + 1 : n ∈ ω } , so ithas MWC-density 0 by Lemma 5.8.Let f : 2 <ω → { , } be defined via f ( σ ) = 1 if and only if m < | σ | is the largestnumber with σ ( m ) = 1, σ has 2 k + 1 1’s, and | σ | = m + k + 1. It is immediate that f is a total monotone selection function, and furthermore f ( A ⊔ ( A ⊲ A )) =
A ⊲ A : bythe above, A ⊔ ( A ⊲ A ) alternates between elements of A and elements of A ⊲ A . The
TOCHASTICITY AS DENSITY 31 elements of A signal where elements of A ⊲ A will sit, allowing f to select exactlythose elements. Therefore( A ⊔ ( A ⊲ A )) ⊳ f ( A ⊔ ( A ⊲ A )) = ( A ⊔ ( A ⊲ A )) ⊳ ( A ⊲ A ) = ω Thus A ⊔ ( A ⊲ A ) does not have MWC-density 0. (cid:3)
This shows that disjoint unions in general need not sum MWC-densities.Another potential solution to the problem of misbehaving unions is to remove the re-quirement that the sets be disjoint: if A has MWC-density α relative to B and B hasMWC-density β relative to A , then must A ∪ B have MWC-density α + β − αβ ? (Theinclusion exclusion principle implies that ρ n ( A ∪ B ) = ρ n ( A ) + ρ n ( B ) − ρ n ( A ∩ B ).Together with Lemma 5.5, this suggests that the MWC-density of A ∪ B must be α + β − αβ if it has MWC-density at all.) It turns out that this is true. Lemma 5.10.
Suppose A has MWC-density α relative to B and B has MWC-density β relative to A . Then A ∪ B has MWC-density α + β − αβ .Proof. Let f be a computable monotone selection function. If f ( A ∪ B ) is finite,we are done. Otherwise, consider ( A ∪ B ) ⊳ f ( A ∪ B ). By definition, ρ (( A ∪ B ) ⊳ f ( A ∪ B )) = lim n →∞ ρ n (( A ∪ B ) ⊳ f ( A ∪ B ))By the inclusion-exclusion principle and the properties of the within operation, ρ n (( A ∪ B ) ⊳f ( A ∪ B )) = ρ n ( A⊳f ( A ∪ B ))+ ρ n ( B ⊳f ( A ∪ B )) − ρ n (( A ∩ B ) ⊳f ( A ∪ B ))Let f A : 2 <ω → { , } be defined via f ( σ ) = 1 if and only if f ( σ ∪ A ) = 1, where σ ∪ A = τ ∈ | σ | with τ ( n ) = 1 if and only if σ ( n ) = 1 or A ( n ) = 1. Let f B bedefined similarly for B in place of A .As A has MWC-density α relative to B and B has MWC-density β relative to A , ρ ( A ⊳ f B ( A )) = ρ ( A ⊳ f ( A ∪ B )) = lim n →∞ ρ n ( A ⊳ f ( A ∪ B )) = α and ρ ( B ⊳ f A ( B )) = ρ ( B ⊳ f ( A ∪ B )) = lim n →∞ ρ n ( B ⊳ f ( A ∪ B )) = β Therefore, what remains is to use an argument similar to that for Lemma 5.5 tohandle the intersection.Define ˆ f A : 2 <ω → { , } via ˆ f A ( σ ) = 1 if and only if f A ( σ ) = 1 and | σ | ∈ A .Then it follows immediately thatˆ f A ( B ) = A ∩ f A ( B ) = A ∩ f ( A ∪ B )so B ⊳ ˆ f A ( B ) = B ⊳ ( A ∩ f ( A ∪ B )By Lemma 5.3, B ⊳ ( A ∩ f ( A ∪ B ) = ( B ⊳ f ( A ∪ B )) ⊳ ( A ⊳ f ( A ∪ B ))Therefore by the same argument as in Lemma 5.5 we have ρ (( B ⊳ f A ( B )) ⊲ ( A ⊳ f B ( A )) = ρ (( A ∩ B ) ⊳ f ( A ∪ B )) = αβ Thus we have lim n →∞ ρ n (( A ∩ B ) ⊳ f ( A ∪ B )) = αβ , and it follows that ρ (( A ∪ B ) ⊳ f ( A ∪ B )) =lim n →∞ ρ n ( A⊳f ( A ∪ B ))+ lim n →∞ ρ n ( B⊳f ( A ∪ B )) − lim n →∞ ρ n (( A ∩ B ) ⊳f ( A ∪ B )) = α + β − αβ as desired. As f was arbitrary, A ∪ B has MWC-density α + β − αβ . (cid:3) Note that if A has MWC-density α relative to B and B ⊳ A (whether this latterrelativization is implied by the other conditions is essentially Question 5.3) and B has MWC-density β relative to A , then Lemma 5.10 can be obtained as an easycorollary of Lemma 5.17: A ∪ B = A ⊔ ( B ∩ A ) = A ⊔ (( B ⊳ A ) ⊲ A )In addition to the general union, we can show that a specific type of disjoint unioncombines MWC-densities in the same way. The format and disjointness of thisspecial form is more useful for our attempts to translate the proof of Corollary 3.11to MWC-density, which we will discuss in Subsection 5.1. Differences in how theunion behaves between the two notions of density will make it difficult to utilizethis proof idea, so we shall show that every real is obtained as the MWC-densityof some set in a different fashion.We instead turn our attention to modifying Lemma 3.9. In the case of intrinsicdensity, this only showed that rational numbers can be achieved as the intrinsicdensity of a set. We shall modify this to work for all reals, which requires firsttweaking our definitions for randomness to accommodate arbitrary reals. Definition 5.11.
Let < r < be an irrational number, and let { p n } n ∈ ω be asequence of nonzero rationals with p n ≤ r for all n and lim n →∞ p n = r . Given afinite binary string σ , [ σ ] ⊆ ω represents the basic open set of extensions of σ . Weuse µ r to represent the measure on Cantor space such that for any σ ∈ <ω , µ r ([ σ ]) = ( Π { n : σ ( n )=1 } p n )( Π { n : σ ( n )=0 } (1 − p n )) We say a set X is µ r -escaping if it passes every { p n } n ∈ ω -relative µ r -Martin-L¨of-test. First notice that this is a generalization of Definition 1.10 in the case for rational p with the sequence with p n = p for all n . Secondly, notice that we are abusingnotation by referring to this measure with only the subscript r , as the measurecan vary wildly using different choices of the sequence { p n } n ∈ ω . We refer to setswhich pass every µ r -Martin-L¨of test as µ r -escaping rather than µ r -Random becausethis dependence upon which sequence is chosen and the noncomputability of thesequence for most r seems to make this a poor notion of algorithmic randomnessin general. It is open whether or not one can make this definition more robust insuch a way that these issues disappear, which is Question 6.1 below. Definition 5.12.
Let < r < be an irrational number, and let { p n } n ∈ ω bea sequence of nonzero rationals with p n ≤ r for all n and lim n →∞ p n = r . An r -martingale is a function m : 2 <ω → R ≥ such that m ( σ ) = (1 − p | σ | ) m ( σ
0) + p | σ | m ( σ TOCHASTICITY AS DENSITY 33 for all σ . An r -supermartingale is a function s : 2 <ω → R ≥ with s ( σ ) ≥ (1 − p | σ | ) s ( σ
0) + p | σ | s ( σ for all σ . An r -(super)martingale m succeeds on a set X if lim sup n →∞ m ( X ↾ n ) = ∞ . A set X is r -escaping if no { p n } n ∈ ω -computably enumerable r -supermartingalesucceeds on it. The same comments for Definition 5.11 apply to Definition 5.12. Furthermore,as the modified versions of Theorem 1.12 and Theorem 1.13 work with the proofsmodified in the obvious way, a set X is µ r -escaping if and only if it is r -escaping. Lemma 5.13.
Suppose that
X r -escaping for some irrational r and the sequence { p n } n ∈ ω . Then X has MWC-density r .Proof. We shall argue by contrapositive. Let f be a monotone selection functionand suppose that ρ ( X ⊳f ( X )) > ǫ > r for some rational ǫ . It is sufficient to considerthis case, as if ρ ( X ⊳ f ( X )) < r then ρ ( X ⊳ f ( X )) > − r , where f is the mono-tone selection function obtained by flipping the bits then applying f . Additionally,without loss of generality we may assume in the partial case that f ( X ↾ n ) ↓ for all n : given f , we know that infinitely often ρ n ( X ⊳ f ( X )) > ǫ . Therefore wheneverwe have some σ witnessing this fact by stage s , we may force f ( τ ) to converge to 0for all τ (cid:22) σ which have not converged by stage s .We shall construct an r -(super)martingale which succeeds on X using f . Let α be as in the proof of Lemma 3.9 for r and ǫ . (Here we are letting r take the placeof p in Lemma 3.9: The proof of the existence of α did not rely on the rationalityof p .) Define m : 2 <ω → { , } as follows: • m ( ∅ ) = 1. • If f ( σ ) = 1, let m ( σ
1) = (1 + αp | σ | ) m ( σ ) and m ( σ
0) = (1 − α − p | σ | ) m ( σ ). • If f ( σ ) = 0, let m ( σ
1) = m ( σ
0) = m ( σ ). • If f ( σ ) ↑ , let m ( σ
1) = m ( σ
0) = 0.Note that m is a { p n } n ∈ ω -c.e. r -(super)martingale. Furthermore, as f ( X ↾ k ) ↓ forall k , m ( X ↾ n ) = 0 for all n . Thus m ( X ↾ n ) = ( Π { k
1, so sup n →∞ m ( X ↾ n ) = ∞ becausethere are infinitely many such s . Thus m succeeds on X and therefore X is not r -escaping. (cid:3) Lemma 5.14.
Suppose X is p - -Random for some rational < p < . Then X has MWC-density p .Proof. The proof for Lemma 5.13 with r = p and p n = p for all n proves thisresult. (cid:3) Note that it is not immediately clear if Lemma 5.13 works for intrinsic density,i.e. it is unknown if all r -escaping sets have intrinsic density r . Clearly r -escapingsets have asymptotic density r as a Corollary of Lemma 5.13, but it is unclear ifcomputable permutations preserve the property of being r -escaping. In Lemma 3.9,to prove permutation invariance we relied upon the fact that µ p for rational p isinvariant under computable permutation, but this need not be true an arbitrary µ r .We just need to demonstrate that r -escaping sets exist and we shall have achievedevery real as the MWC-density of some set. Lemma 5.15.
Let < r < be an irrational number, and let { p n } n ∈ ω be asequence of nonzero rationals with p n ≤ r for all n and lim n →∞ p n = r . Then r -escaping sets exist.Proof. The standard proof for the existence of a universal Martin-L¨of test rela-tivizes to show there is there is a universal µ r -Martin-L¨of test relative to { p n } n ∈ ω .Therefore µ r -escaping sets have µ r -measure 1 and therefore exist. As we observedthat µ r escaping sets are exactly the r -escaping sets, r -escaping sets exist. (cid:3) Corollary 5.16.
Every real in the unit interval is achieved as the MWC-densityof some set.Proof.
By Lemma 5.7 combined with Lemma 5.1, 0 and 1 are realized as the MWC-densities of sets. By Lemma 5.14, all rationals in the unit interval are as well.Lemma 5.13 combined with Lemma 5.15 covers the irrationals. (cid:3)
Translating the Proof of Corollary 3.11.
While Corollary 5.16 showedthat MWC-density achieves every real in the unit interval, it relied upon the ex-istence of seemingly arbitrary and poorly understood sets. Corollary 3.11 had amuch more elegant proof that used the well-understood concept of 1-Randomnessto explicitly construct sets of arbitrary intrinsic density. In this subsection we shalldiscuss the attempts to translate this proof into the MWC-density case and whythey fall short. As mentioned previously, we can prove that a certain special formof disjoint union sums MWC-densities.
Lemma 5.17.
Suppose that A has MWC-density α relative to B and B has MWC-density β relative to A . Then A ⊔ ( B⊲A ) has MWC-density α + β (1 − α ) = α + β − αβ .Proof. Let f be a monotone selection function. We wish to show that ρ (( A ⊔ ( B ⊲ A ) ⊳ f ( A ⊔ ( B ⊲ A ))) = α + β − αβ By the properties of the within operation,( A ⊔ ( B ⊲ A ) ⊳ f ( A ⊔ ( B ⊲ A )) = (
A ⊳ f ( A ⊔ ( B ⊲ A ))) ⊔ (( B ⊲ A ) ⊳ f ( A ⊔ ( B ⊲ A )))
TOCHASTICITY AS DENSITY 35 so ρ (( A ⊔ ( B ⊲A ) ⊳f ( A ⊔ ( B ⊲A ))) = ρ (( A⊳f ( A ⊔ ( B ⊲A ))))+ ρ ((( B ⊲A ) ⊳f ( A ⊔ ( B ⊲A ))))Therefore, we shall first construct a B -computable monotone selection function f B such that f B ( A ) = f ( A ⊔ ( B ⊲ A )). Then
A ⊳ f ( A ⊔ ( B ⊲ A )) =
A ⊳ f B ( A )and therefore because A has MWC-density α with respect to B , we have ρ ( A ⊳ f ( A ⊔ ( B ⊲ A ))) = ρ ( A ⊳ f B ( A )) = α Define f B : 2 <ω → { , } via f B ( σ ) = 1 if and only if f ( σ ⊔ ( B ⊲ σ )) = 1, where σ ⊔ ( B⊲σ ) is defined to be τ ∈ | σ | with τ ( k ) = 1 if and only if σ ( k ) = 1 or k is the b i -th 0 in σ for some i . From this definition, it is immediate that f B ( A ) = f ( A ⊔ ( B⊲A ))as desired.It remains to show that ρ (( B ⊲ A ) ⊳ f ( A ⊔ ( B ⊲ A ))) = β (1 − α ) = β − βα We would like to use Theorem 5.4 here, however we cannot because
B⊲A w not haveMWC-density relative to A . To fix this, we will mimic the proof of Theorem 5.4,that is we shall construct a B -computable monotone selection function g B such that g B ( A ) = f ( A ⊔ ( B ⊲A )). Then
A⊳g B ( A ) will be a superset of ( B ⊲A ) ⊳f ( A ⊔ ( B ⊲A ))with density 1 − α because A has MWC-density α relative to B . Then there is some X such that X ⊲ ( A ⊳ g B ( A )) = ( B ⊲ A ) ⊳ f ( A ⊔ ( B ⊲ A ))Finally, it suffices to construct an A -computable monotone selection function g A such that B ⊳ g A ( B ) = X : X will then have density β due to the fact that B hasMWC-density β relative to A and Lemma 3.5 will ensure that ρ (( B ⊲ A ) ⊳ f ( A ⊔ ( B ⊲ A ))) = ρ ( X ⊲ ( A ⊳ g B ( A ))) = β (1 − α )as desired.Define g B : 2 <ω → { , } via g B ( σ ) = 1 if and only if f B ( σ ) = 1, where σ isdefined to be τ ∈ | σ | with τ ( k ) = 1 if and only if σ ( k ) = 0. Then g B ( A ) = f B ( A ) = f ( A ⊔ ( B ⊲ A ))Let g A : 2 <ω → { , } be defined via g A ( σ ) = f ( A ⊔ ( σ ⊲ A ), where A ⊔ ( σ ⊲ A ) = τ ∈ p A ( | σ | ) is defined via τ ( n ) = 1 if and only if n ∈ A or n = p A ( k ) for some k < | σ | and σ ( k ) = 1. We now claim that B ⊳ g A ( B ) = X .Recall that X is (( B ⊲ A ) ⊳ f ( A ⊔ ( B ⊲ A ))) ⊳ ( A ⊳ g B ( A ))As mentioned above, g B ( A ) = f ( A ⊔ ( B ⊲ A )), so we may apply Lemma 5.3 toobtain X = ( B ⊲ A ) ⊳ ( A ∩ f ( A ⊔ ( B ⊲ A )))Suppose n ∈ X . By the definition of X , p A ∩ f ( A ⊔ ( B⊲A )) ( n ) ∈ B ⊲ A
That is, the n -th element of A ∩ f ( A ⊔ ( B ⊲ A )) is in
B ⊲ A . Therefore, it isof the form p A ( b k ) for some k . Furthermore, p A ( b k ) ∈ f ( A ⊔ ( B ⊲ A )), so bydefinition f (( A ⊔ ( B ⊲ A )) ↾ p A ( b k )) = 1. This then implies, by the definition of g A , that g A ( B ↾ b k ) = 1. Therefore b k ∈ g A ( B ), and p − g A ( B ) ( b k ) ∈ B ⊳ f A ( B ).Finally, note that p − g A ( B ) ( b k ) = n because every element of g A ( B ) is an element of A ∩ f ( A ⊔ ( B ⊲ A )) by definition, and b k corresponds to the n -th such one. Therefore n ∈ B ⊳ g A ( B ). This argument reverses, so B ⊳ g A ( B ) = X . (cid:3) Lemma 5.14 relativizes in straightforward fashion. As a result, the proof ofLemma 3.10 immediately lifts to prove an analog for MWC-density: There is adisjoint sequence of sets { A i } i ∈ ω such that each A i has MWC-density i +1 relativeto the others which can be obtained using the into operation and Van Lambalgen’sTheorem. (Theorem 5.4 requires more relativization than Theorem 3.7, but the factthat Theorem 5.4 itself relativizes ensures that the same proof technique applies.)Unfortunately, the fact that unions do not preserve MWC-density in general meansthat given a real r , we do not know that the infinite union of the A i ’s correspondingto the binary expansion of r will have MWC-Density. In the finite case, however,Lemma 5.17 will ensure the union has the desired MWC-density. Lemma 5.18.
Let X be 1-Random and let { A i } i ∈ ω be constructed from X as inLemma 3.10. If D is a finite set of natural numbers, then F i ∈ D A i has MWC-density Σ i ∈ D i +1 .Proof. Essentially, each F i ∈ D A i is composed of finitely many unions of the formfound in Lemma 5.17 and finitely many applications of the into operation. VanLambalgen’s theorem will ensure we have all of the necessary relativizations neces-sary to use Lemma 5.17 and Theorem 5.4 to reduce the number of unions by one.Combined with induction on the size of the union, this will prove the result.Recall that we defined A = X [0] and A i = X [ i ] ⊲ X [ i − ⊲ . . . ⊲ X [0] for i >
0. Therefore G i ∈ D A i = G i ∈ D X [ i ] ⊲ X [ i − ⊲ . . . ⊲ X [0] (If i = 0 or i = 1 then we take X [ i − ⊲ . . . ⊲ X [0] to mean ω and X [0] respectivelyto ensure that this does indeed match the definition of A i from Lemma 3.10.)We argue by induction on the size of D . If D is a singleton, then its member is ofthe form X [ i ] ⊲X [ i − ⊲. . .⊲X [0] for some i . By Van Lambalgen’s Theorem, each X [ j ] is 1-Random relative to the join of the others, and therefore by Lemma 5.14 eachhas MWC-density relative to the join of the others. Thus X [ i ] ⊲ X [ i − ⊲ . . . ⊲ X [0] has MWC-density i +1 by Theorem 5.4. This concludes the base case.Now suppose it holds that for any 1-Random X and any finite set D of sizeless than or equal to n , F i ∈ D A i has MWC-density Σ i ∈ D i +1 . Now suppose D has size n + 1. First consider the case when 0 ∈ D . Then using the fact that TOCHASTICITY AS DENSITY 37 ( A ⊔ B ) ⊲ C = ( A ⊲ C ) ⊔ ( B ⊲ C ) and the associativity of the into operation, G i ∈ D A i = X [0] ⊔ ( G i ∈ D,i> X [ i ] ⊲ X [ i − ⊲ . . . ⊲ X [0] ) = X [0] ⊔ (( G i ∈ D,i> X [ i ] ⊲ X [ i − ⊲ . . . X [1] ) ⊲ X [0] )Let Y be defined via Y [ i ] = X [ i +1] . Y is 1-Random relative to X [0] by Van Lam-balgen’s Theorem. Thus by the relativized induction hypothesis, G i ∈ D,i> X [ i ] ⊲ X [ i − ⊲ . . . X [1] = G i ∈ D,i> Y [ i − ⊲ Y [ i − ⊲ . . . Y [0] has MWC-density Σ i ∈ D,i> i relative to X [0] . Finally, Lemma 5.17 then impliesthat X [0] ⊔ (( G i ∈ D,i> X [ i ] ⊲ X [ i − ⊲ . . . X [1] ) ⊲ X [0] )has MWC-density12 + ( Σ i ∈ D,i> i )(1 −
12 ) = 12 + ( Σ i ∈ D,i> i +1 ) = Σ i ∈ D i +1 as desired.Now suppose that j > D . Then we have G i ∈ D A i = ( X [ j ] ⊲ X [ j − ⊲ . . . ⊲ X [0] ) ⊔ ( G i ∈ D,i>j X [ i ] ⊲ X [ i − ⊲ . . . ⊲ X [0] ) =( X [ j ] ⊲ ( X [ j − ⊲ . . .⊲ X [0] )) ⊔ ( G i ∈ D,i>j ( X [ i ] ⊲ X [ i − ⊲ . . . ⊲ X [ j ] ) ⊲ ( X [ j − ⊲ . . . ⊲ X [0] )) =( X [ j ] ⊔ ( G i ∈ D,i>j X [ i ] ⊲ X [ i − ⊲ . . . ⊲ X [ j ] )) ⊲ ( X [ j − ⊲ . . . ⊲ X [0] )Let Y be defined via Y [ i ] = Y [ i + j ] and ˆ D = { n − j : n ∈ D } . Then Y is 1-Randomby Van Lambalgen’s Theorem and ˆ D is a set of size n which contains 0. Thereforewe can apply the relativized version of the previous case to see that X [ j ] ⊔ ( G i ∈ D,i>j X [ i ] ⊲ X [ i − ⊲ . . . ⊲ X [ j ] ) = Y [0] ⊔ ( G i ∈ ˆ D,i> Y [ i ] ⊲ Y [ i − ⊲ . . . ⊲ Y [0] )has MWC-density Σ i ∈ ˆ D i +1 = Σ i ∈ D i +1 − j = Σ i ∈ D j i +1 relative to X [ j − ⊲ . . . ⊲ X [0] . As X [ j − ⊲ . . . ⊲ X [0] has MWC-density j relative to X [ j ] ⊔ ( F i ∈ D,i>j X [ i ] ⊲ X [ i − ⊲ . . . ⊲ X [ j ] ) by Van Lambalgen’s Theorem and multipleiterations of the relativized form of Theorem 5.4, it follows that( X [ j ] ⊔ ( G i ∈ D,i>j X [ i ] ⊲ X [ i − ⊲ . . . ⊲ X [ j ] )) ⊲ ( X [ j − ⊲ . . . ⊲ X [0] )has MWC-density ( Σ i ∈ D j i +1 ) 12 j = Σ i ∈ D i +1 as desired. This completes the induction. (cid:3) Unfortunately, it remains open whether or not this can be extended to infiniteunions of this form, which is Question 6.7 below. The difficulty lies once again inthe fact that the input set can change which bits are and are not selected. In theory,given any 0 < r < B r as in Corollary3.11, for any ǫ > N such that F n ∈ B r ,n We set out to study which reals in the unit interval could be achieved as theintrinsic density of some set. As having intrinsic density is a type of stochastic-ity, it was natural to ask the same question for other notions of stochasticity fromthe literature. We introduced the into and within operations as tools of study, andthey turned out to form a calculus of sorts over the subsets of the natural numbersfor multiple notions of density. As a result, we succeeded in showing that the wholeunit interval is realized using intrinsic density in Section 3. We were also able toprove that MWC-density and Church-density realize the entire unit interval usinga less constructive approach in Section 5.We believe there is significant room for future work. We did not investigate full KL-density in this paper, but we believe the methods used to study MWC-density couldbe modified to do so. Little is known about the computability-theoretic strength ofthe various sets discussed throughout this paper save the immediately obtainablefacts such as A ⊲ B ≤ T A ⊕ B . In addition to these broader areas available tostudy, there are some interesting questions about randomness within the lens ofthese results.Not much is known about Definitions 5.11 and 5.12 other than that they seemto represent a non-robust notion of randomness: For any set of natural numbers X and any r , we could choose a sequence { p n } n ∈ ω as in these definitions whichcomputes X . It is immediate then that X is not r -escaping with respect to thissequence, so the notion of r -escaping is far from robust. Question 6.1. Is there an improvement to Definitions 5.11 and 5.12 which capturesthe idea of being r -1-Random for irrational r in the same sense as the definitionfor rational r , i.e. one which does not rely on the chosen sequence or uses a nicecanonical sequence? Stochasticity and randomness are closely related. The following question ofCholak asks if there is a lens through which they are one and the same, whichcould potentially provide a very compelling answer to Question 6.1. Question 6.2 (Cholak) . Is there a natural class for stochaticity C for which thesets with C -density are exactly the 1-Randoms? Is there one for which the setswith C -density p are exactly the p -1-Randoms for any rational < p < ? If so, isthere such a class where every real in the unit interval is the C -density of some set? TOCHASTICITY AS DENSITY 39 It is important that we stipulate natural here: We could do something mean-ingless such as define the selection function s : 2 ω → ω via s ( X ) = ω if X is p -1-Random and s ( X ) = X otherwise. Then C = { ( s, id ) } would technically besuch a class for stochasticity, however it is clearly not useful or interesting in anyway.The remaining questions are of a more technical nature relating to our variousresults.For intrinsic density, we proved that P ( A⊳C ) = α if C is computable and P ( A ) = α .It is known that C ⊳ A does not necessarily have intrinsic density in general as wit-nessed by A = ω , which leads to the following question. Question 6.3. Are there conditions on A such that, for computable C , C ⊳ A hasintrinsic density? For a discussion on the applications of this question, see the end of Section 4.In proving Theorem 5.4, we used more relativization than was necessary in theintrinsic density analog Theorem 3.7. However, it is not known whether this isnecessary or merely useful. Question 6.4. Is the relativization optimal in Theorem 5.4? That is, are there sets A and B such that B has MWC-density β relative to A and A has MWC-density α but B ⊲ A does not have MWC-density αβ ? The same proof that showed the relativization used in Theorem 3.7 is optimalwill not work for Theorem 5.4 because A ⊕ A will not have MWC-density.We could not directly lift the proof that the intersection of two intrinsically densesets multiplied the intrinsic densities of the sets to the case of MWC-density due todifferent relativization requirements between Theorem 3.7 and its analog Theorem5.4. A positive resolution to the following question would allow us to do this. Question 6.5. If A has MWC-density α and C is computable, does A have MWC-density α relative to C ⊳ A ? If so, does this relativize? If this is not true, is itat least the case that whenever A has MWC-density α relative to B and B hasMWC-density relative to A , does A have MWC-density α relative to B ⊳ A ? Our usual techniques do not suffice to answer this question, as they are focused onusing oracles, or relativized information, to answer questions about non-relativizedMWC-density. This question requires us to answer a question about MWC-densityrelative to a specific set using non-relativized information.If this is true and relativizes, or the weaker formulation is true, then whenever A has MWC-density α relative to B and B has MWC-density β relative to A , A ∩ B would have MWC-density αβ as a corollary of Theorem 5.4 and A ∪ B wouldhave MWC-density α + β − αβ as a corollary of Lemma 5.17. (Recall that both ofthese facts are true, but they required separate proofs.)Lemma 5.9 demonstrated that MWC-density is not preserved by unions. Both setsin the given counterexample had MWC-density 0, and relied nontrivially on this fact to encode information about each other. This strategy would not necessarilywork in sets of positive density, which leads to the following question: Question 6.6. Are there disjoint sets A and B with MWC-density α > and β > respectively such that A ⊔ B does not have MWC-density α + β ? In Subsection 5.1 we discussed the difficulty in translating the proof of Corollary3.11 into the intrinsic density case. Question 6.7. Given a sequence { A n } n ∈ ω as constructed in Lemma 3.10, let Is it the case that every set of intrinsic density α has C -density α ?That is, for any set A with intrinsic density α , is it the case that f ( A ) ⊳ range ( f ) has density α for all total computable injective functions f ? Note that this is similar to a question asked by the author in [5]: is it the casethat for every intrinsically small set A and total computable injective function f , f ( A ) is intrinsically small? References [1] Eric P. Astor. Asymptotic density, immunity, and randomness. Computability , 4(2):141-158,2015. ISSN 2211-3568. doi: 10.3233/COM-150040.[2] Eric P. Astor. The computational content of intrinsic density. ArXiv , arXiv:1708.04267, 2017.v2[3] Rodney G. Downey and Denis R. Hirschfeldt. Algorithmic randomness and complexity. Theoryand Applications of Computability , Springer, 2010.[4] Carl G. Jockusch, Jr. and Paul E. Schupp. Generic computability, Turing degrees, and asymp-totic density. Journal of the London Mathematical Society , 85(2):472–490, 2012. ISSN 0024-6107.doi.org/10.1112/jlms/jdr051.[5] Justin Miller. Intrinsic smallness. ArXiv , arXiv:1909.00050, 2019. v1.[6] Andr´e Nies. Computability and randomness. Oxford Logic Guides , Oxford University Press,2009.[7] Jan Reimann and Theodore A. Slaman. Probability measures and effective randomness. ArXiv ,arXiv:0707.1390, 2007. v1 .[8] C.-P. Schnorr. A unified approach to the definition of random sequences. Mathematical SystemsTheory , 5(3):246-258, 1971. ISSN: 0025-5661 DOI: 10.1007/BF01694181[9] M. van Lambalgen. The axiomatization of randomness. The Journal of Symbolic Logic ,55(3):1143-1167, 1990. 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