Combinatorial and number-theoretic properties of generic reals
aa r X i v : . [ m a t h . L O ] F e b COMBINATORIAL AND NUMBER-THEORETIC PROPERTIES OFGENERIC REALS
WILL BRIAN AND MOHAMMAD GOLSHANI
Abstract.
We discuss some properties of Cohen and random reals. We show that theybelong to any definable partition regular family, and hence they satisfy most “largeness”properties studied in Ramsey theory. We determine their position in the Mahler’s clas-sification of the reals and using it, we get some information about Liouville numbers.We also show that they are wild in the sense of o-minimality, i.e., they define the set ofintegers. introduction In this paper we study Cohen and random reals, and ask questions about what sort of realsthey are. Do they satisfy the conclusion of Hindman’s theorem? How badly approximableare they by algebraic numbers? What of the same questions, but with Cohen reals? AreCohen or random reals wild in the sense of o-minimality?Let N denote the natural numbers (without 0). We say that F is a semifilter of infinitesubsets of N if it is closed under taking supersets. Such collections are also sometimescalled Furstenberg families , or simply families . A semifilter F is partition regular if for every A ∈ F , and every partition of A into finite sets A , A , . . . , A n , there is some i ≤ n such that A i ∈ F . Roughly, a partition regular semifilter is a collection of sets that can be considerednot small. Such collections are also sometimes called co-ideals .For example, call r ⊆ N large if Σ n ∈ r n = ∞ . A famous conjecture of Erd¨os–Tur´an saysthat if r is large, then it contains arbitrary long arithmetic progressions. Both the semifilterof large sets and the semifilter of sets containing arbitrarily long arithmetic progressions arepartition regular. (The second assertion is the famous theorem of van der Waerden). Key words and phrases.
Cohen and random reals, partition regular families, Mahler’s classification ofreals, Hausdorff dimension, o-minimality.The second author’s research has been supported by a grant from IPM (No. 99030417).The authors thank Alex Kruckman for bringing the Theorem of Friedman-Miller 5.3(a) to their attention.
As it turns out, Cohen and random reals belong to essentially every partition regularfamily considered in classical Ramsey theory. Specifically, we prove the following:
Theorem 1.1.
Let V ⊆ W be models of set theory, and let r ∈ W be a Cohen (or random)real over V . (1) Suppose W = V [ r ] . In W , r belongs to every partition regular family that is definableover V . (2) In general, r belongs to every partition regular family that is lightface Σ ( a ) or Π ( a ) -definable, where a is a real number in V . We will provide an example showing that the conclusion of (1) does not hold under thehypotheses of (2); in other words, some further restriction on the niceness of the partitionregular families is really needed for (2).Then we consider Mahler’s classification of the reals. The set of real numbers splits intoalgebraic and transcendental numbers. Kurt Mahler in 1932 partitioned the real numbersinto four classes, called
A, S, T, and U . Class A consists of algebraic numbers, while classes S, T and U contain transcendental numbers. We determine the situation of Cohen andrandom reals by proving the following: Theorem 1.2. ( [3] ) ( a ) Assume r is a Cohen real over some (possibly countable) transitive model. Then r isin the class U (and indeed it is a Liouville number). ( b ) Assume r is a random real over some (possibly countable) transitive model. Then r is in the class S . We show also that Cohen or random reals are wild in the sense of o-minimality, by provingthe following.
Theorem 1.3.
Assume r is a Cohen (a random) real over V . Then r defines Z in thesense that the set of integers Z is definable in the structure ( R V [ r ] , + , · , <, , , r ) , where r isconsidered as a unary predicate. We also show that the above phenomenon is not true for all generic reals by providing anexample of a generic real r which does not define Z . OMBINATORIAL AND NUMBER-THEORETIC PROPERTIES OF GENERIC REALS 3
Finally we consider the situation of adding many Cohen or random reals and extend aclassical result by proving the following:
Theorem 1.4.
Let κ be an infinite cardinal. Then, forcing with R ( κ ) × R ( κ ) adds a genericfilter for C ( κ ) , where R ( κ ) and C ( κ ) are the forcing notions for adding κ -many random realsand adding κ -many Cohen reals respectively. Cohen and random forcing
In this section, we briefly review the definition and basic properties of the Cohen andrandom forcing notions.2.1.
Cohen forcing.
The Cohen forcing, denoted C , is defined as C = { p : p is a function n → n ∈ ω } , ordered by reverse inclusion. Suppose G is C -generic over V and let F = S p ∈ G p. Then F is a function from ω into 2 and if we set r = { n ∈ ω : F ( n ) = 1 } , then V [ G ] = V [ r ] . Thereal r is called the Cohen real. This real has a canonical C -name ˙ r , determined by k ˙ r ( n ) = 1 k = X { p ∈ C : p ( n ) = 1 } . The next lemma is a well-known characterization of Cohen reals which we apply repeatedly.
Lemma 2.1.
A real r is Cohen over V if and only if it does not belong to any meager Borelset coded in V . In general, given a non-empty set I , the forcing notion C ( I ), the Cohen forcing for adding | I | -many Cohen reals is defined by C ( I ) = { p : I × ω → | p | < ℵ } ,which is ordered by reverse inclusion. Lemma 2.2. C ( I ) is c.c.c. Assume G is C ( I )-generic over V , and set F = S G : I × ω → . For each i ∈ I define c i : ω → c i ( n ) = F ( i, n ) . Then for each i ∈ I, c i ∈ ω is a new real and for i = j in I, c i = c j . Furthermore, V [ G ] = V [ h c i : i ∈ I i ] . By κ -Cohen reals over V , we mean asequence h c i : i < κ i which is C ( κ )-generic over V . W. BRIAN AND M. GOLSHANI
Random forcing.
There are several equivalent ways to present the random forcing.Here, we consider the following presentation. Consider the product measure space 2 ω withthe standard product measure µ on it. Let B denote the class of Borel subsets of 2 ω .Note that sets of the form [[ s ]] = { x ∈ ω : x ↾ n = s } , where s : n → ω . For Borel sets
S, T ∈ B set S ∼ T ⇐⇒ S △ T is null,where S △ T denotes the symmetric difference of S and T . The relation ∼ is easily seen tobe an equivalence relation on B . Then R , the random forcing, is defined as R = B / ∼ . Thus elements of R are equivalence classes [ B ] of Borel sets modulo null sets. The orderrelation is defined by [ S ] ≤ [ T ] ⇐⇒ µ ( S \ T ) = 0 . Let ˙ r be an R -name for a real such that k ˙ r ( n ) = 1 k = [ { [[ s ]] : s ( n ) = 1 } . This real is called the random real, and this name is its canonical name.The next lemma gives a characterization of random reals analogous to Lemma 2.1.
Lemma 2.3.
A real r is random over V if and only if it does not belong to any null Borelset coded in V . We think of 2 ω as the end space of the tree 2 <ω . If T is any subtree of 2 <ω , then T defines a closed subset of 2 ω , namely[[ T ]] = \ n ∈ ω [ { [[ s ]] : s ∈ T and dom( s ) = n } . OMBINATORIAL AND NUMBER-THEORETIC PROPERTIES OF GENERIC REALS 5
Conversely, if C is a closed subset of 2 ω , then there is a subtree T of 2 <ω with [[ T ]] = C ,namely T = (cid:8) s ∈ <ω : [[ s ]] ∩ C = ∅ (cid:9) . Lemma 2.4.
Let C ⊆ ω be closed and non-null, and let T be the subtree of <ω with C = [[ T ]] . Then [ C ] (cid:13) ˙ r ↾ n ∈ T for every n ∈ N . As in the case of Cohen forcing, we can generalize the above construction to add manyrandom reals. Suppose I is a non-empty set and consider the product measure space 2 I × ω with the standard product measure µ I on it. Let B ( I ) denote the class of Borel subsets of2 I × ω . Recall that B ( I ) is the σ -algebra generated by the basic open sets[[ s ]] = { x ∈ I × ω : x ⊇ s } , where s ∈ C ( I ). Also µ I ([[ s ]]) = 2 −| s | .For Borel sets S, T ∈ B ( I ) define S ∼ T as above. Then R ( I ), the forcing for adding | I | -many random reals, is defined as R ( I ) = B ( I ) / ∼ . The following fact is standard.
Lemma 2.5. R ( I ) is c.c.c. for any I . Using the above lemma, we can easily show that R ( I ) is in fact a complete Booleanalgebra. Given any i ∈ I , let ˙ r i be an R ( I )-name for a real such that k ˙ r i ( n ) = 1 k = [ { [[ s ]] : s ( i, n ) = 1 } . Given an R ( I )-generic filter G over V , and i ∈ I , set r i = r ∼ i [ G ] . Then each r i ∈ ω is a newreal and for i = j in I, r i = r j . Furthermore, V [ G ] = V [ h r i : i ∈ I i ] . By κ -random reals over V , we mean a sequence h r i : i < κ i which is R ( κ )-generic over V . W. BRIAN AND M. GOLSHANI Largeness properties of Cohen and random reals
In this section we prove Theorem 1.1. The proof of part (1) capitalizes on the existence ofcertain automorphisms of the Cohen and random forcing notions (and then (2) follows from(1) by absoluteness). We begin by establishing the existence of the relevant automorphisms.
Lemma 3.1.
Let P denote either the forcing for adding a Cohen real or a random real,and let ˙ r denote the canonical name for the generic real. Then, for any p ∈ P , there is anautomorphism π : P → P and some q ≤ p such that • π ( q ) = q . • q (cid:13) ˙ r ∩ π ( ˙ r ) is finite. • For some finite set F , q (cid:13) ˙ r ∪ π ( ˙ r ) ∪ F = ω .Proof. For the Cohen forcing the proof is straightforward. Let p ∈ C and let π : C → C bethe map defined by • π ( s )( n ) = s ( n ) for all n ∈ dom( p ) • π ( s )( n ) = 1 − s ( n ) for all n / ∈ dom( p ).Let q = p . It is clear that π ( q ) = q , and if ˙ r is the canonical name for the generic real, thenfor all n / ∈ dom( q ), (cid:13) P ˙ r ( n ) = 1 ⇔ π ( ˙ r )( n ) = 0which in particular implies that q (cid:13) ˙ r ∩ π ( ˙ r ) ⊆ dom( q ) and r ∪ π ( ˙ r ) ∪ dom( q ) = ω .For the random forcing, let p = [ B ] ∈ R . By the Lebesgue Density Theorem (see Exercise17.9 in [11]), there is some s ∈ <ω such that µ ( B ∩ [[ s ]]) > µ ([[ s ]]) . Define a map γ : 2 ω → ω as follows: γ ( x )( n ) = x ( n ) if n ∈ dom( s )1 − x ( n ) otherwise.Observe that γ is a homeomorphism from 2 ω to itself; in particular, it maps Borel sets toBorel sets. Also observe that µ ( γ ([[ t ]])) = µ ([[ t ]]) for every basic open set [[ t ]]. It follows that OMBINATORIAL AND NUMBER-THEORETIC PROPERTIES OF GENERIC REALS 7 γ is measure-preserving: i.e., µ ( A ) = µ ( γ ( A )) for every measurable set A . Also notice that γ ([[ s ]]) = [[ s ]].Let C = ([[ s ]] ∩ B ) ∩ γ ([[ s ]] ∩ B ). By the observations at the end of the last paragraph, C is Borel and µ ( C ) ≥ µ ([[ s ]] ∩ B ) − µ ([[ s ]] − B ) , and this is positive by our choice of s . Thus [ C ] ∈ R , and clearly [ C ] ≤ [ B ]. Let q = [ C ].Define π : 2 ω → ω by π ( x ) = γ ( x ) if x ∈ Cx if x / ∈ C. First, note that π maps Borel sets to Borel sets. Second, π is measure-preserving: µ ( π ( A )) = µ ( π ( A ∩ C ) ∪ ( A − C )) = µ ( γ ( A ∩ C ))+ µ ( A − C ) = µ ( A ∩ C )+ µ ( A − C ) = µ ( A ) . Third, π is a bijection. Fourth, π is order-preserving on R .Therefore π induces an automorphism π of R , namely π ([ A ]) = [ π ( A )]. Let us checkthat it has the required properties. Because γ ◦ γ is the identity map, it follows from ourdefinition of C that γ ( C ) = C . Therefore π ( C ) = C and π ( q ) = q . Furthermore, if T is thesubtree of 2 <ω with [[ T ]] = C , we have q (cid:13) ˙ r ↾ n ∈ T. If t ∈ T and s ⊆ t , then π ([[ t ]]) = γ ([[ t ]]) = [[ u ]], where u ( n ) = t ( n ) if n ∈ dom( σ )1 − t ( n ) otherwise.Thus, for all n / ∈ dom( s ), we have q (cid:13) ˙ r ( n ) = π ( ˙ r )( n ) . It follows that q (cid:13) ˙ r ∩ π ( ˙ r ) ⊆ dom( s ) and ˙ r ∪ π ( ˙ r ) ∪ dom( s ) = ω, which completes the proof. (cid:3) W. BRIAN AND M. GOLSHANI
Proof of Theorem 1.1.
Let F be a partition regular semifilter in W that is definable over V .In other words, there is a formula ϕ ( X, a ), with a ∈ V , such that (in W ) if X ⊆ N then X ∈ F ⇔ ϕ ( X, a )Assume W = V [ r ] for some Cohen (random) real r . Let P denote the Cohen (random)forcing, so that W is a P -generic extension of V , and let ˙ r denote the canonical name forthe generic real. Let G be the generic filter on P with ˙ r G = r .Let ˙ F be the canonical P -name for the semifilter F : k ˙ x ∈ ˙ Fk = k ˙ x ⊆ N k ∧ k ϕ ( ˙ x, a ) k . There is some p ∈ G such that p (cid:13) ˙ F is a partition regular family(in fact, one may argue using automorphisms of P that if p (cid:13) ( ˙ F is a partition regular family),then (cid:13) P ( ˙ F is a partition regular semifilter). We do not need this, though, and content our-selves to work below p .)Let us suppose r / ∈ F and aim for a contradiction. If r / ∈ F , there is some p ′ ≤ p , with p ′ ∈ G , such that p ′ (cid:13) ˙ r / ∈ ˙ F or, equivalently, p ′ (cid:13) ¬ ϕ ( ˙ r, a ) . Applying our lemma, let π be an automorphism of P and q ≤ p ′ such that • π ( q ) = q . • q (cid:13) ˙ r ∩ π ( ˙ r ) is finite. • q (cid:13) ˙ r ∪ π ( ˙ r ) ∪ F = ω for some finite set F .Because q ≤ p ′ , q (cid:13) ˙ F is a partition regular family and ¬ ϕ ( ˙ r, a ) . By well-known properties of automorphisms, we have π ( q ) = q (cid:13) ˙ F is a partition regular family and ¬ ϕ ( π ( ˙ r ) , a ) . OMBINATORIAL AND NUMBER-THEORETIC PROPERTIES OF GENERIC REALS 9
Let G ′ be a P -generic filter containing q (note that q is not necessarily in G ). Workingin W ′ = V [ G ′ ], we will obtain a contradiction. By our choice of π and the fact that q ∈ G ′ , we have ω = ( ˙ r ) G ′ ∪ ( π ( ˙ r )) G ′ ∪ F for some finite set F . Recalling our definition ofpartition regularity, one of ( ˙ r ) G ′ , ( π ( ˙ r )) G ′ , and F should be in F ′ = ( ˙ F ) G ′ . Our definitionof “semifilter” requires that F ′ contain only infinite sets, so we must have either ( ˙ r ) G ′ or( π ( ˙ r )) G ′ in F ′ . Because q ∈ G ′ and q (cid:13) ¬ ϕ ( ˙ r, a ), we have ( ˙ r ) G ′ / ∈ F ′ . Because q ∈ G ′ and q (cid:13) ¬ ϕ ( π ( ˙ r ) , a ), we have ( π ( ˙ r )) G ′ / ∈ F ′ . This contradiction establishes that r ∈ F andfinishes the proof of (1).For (2), let r be Cohen (random) over V , and let ϕ ( x, a ) be a Σ ( a ) or Π ( a ) formuladefining F in W (where a is a real number in V ). By Shoenfield’s Absoluteness Theorem(which implies that such formulas are absolute), if X ∈ V [ r ] then W | = ϕ ( X, a ) if andonly if V [ r ] | = ϕ ( X, a ). Therefore ϕ still defines a partition regular family in V [ r ]. By (1), V [ r ] | = ϕ ( r, a ). By Shoenfield’s Absoluteness Theorem again, W | = ϕ ( r, a ), and it followsthat r ∈ F . (cid:3) Corollary 3.2.
Let M ⊆ V be any transitive model of (enough of ) ZFC. If r is a Cohen orrandom real over M , then (1) r is large. (2) r contains arbitrary long arithmetic progressions. (3) r contains arbitrary long geometric progressions. (4) r contains infinitely many solutions to every partition regular Diophantine equation. (5) r has positive upper density. (6) r is piecewise syndetic, i.e., there are arbitrarily long intervals of N where the gapsin r are bounded by some constant b . (7) r satisfies the conclusion of Hindman’s Theorem: i.e., there is some infinite A ⊆ r such that, for every finite F ⊆ A , P F ∈ r . (8) r is a ∆ -set: i.e., r contains { s j − s i : i, j ∈ N , i < j } for some infinite sequence h s i : i ∈ N i of natural numbers. (9) r is central: i.e., there is a minimal idempotent ultrafilter containing r . Proof.
It is not difficult to check that the various families described in (1) - (8) are allanalytic (they are, at worst, G δ , G δ , G δ , F σ , F σ , G δσ , Σ , and Σ , respectively). For (9),observe that, though the central sets are often defined as those belonging to some minimalidempotent ultrafilter, there are alternative definitions (see, e.g., Theorem 14.25 in [8]) thatmake it clear that the family of central sets is at worst Σ . (cid:3) Next, let us consider an example showing that the conclusions of (1) do not hold for allextensions W of V ; in particular, some restriction on the definition of F is required.Given V , let r be a Cohen real over V , and let W be the generic extension of V [ r ] obtainedby using a (set-sized) Easton forcing to get n ∈ r ⇔ ℵ n = ℵ n +1 . In W , we may define a partition regular semifilter F by the parameter-free formula X ∈ F ⇔ X ⊆ N and ( ∀ m ∈ N )( ∃ n ∈ X ) n > m ∧ ℵ n = ℵ n +1 . In other words, F is the family of all sets that have infinite intersection with the complementof r . Clearly F is partition regular, but r / ∈ F .In fact, one can show that Theorem 1.1(2) can not be improved. To see this, assume V = L [ r ] , where r is a Cohen or a random real over L . Then there is a generic extension W of L [ r ] in which there exists a ∆ -real s which codes r . Then the above argument showsthat there exists a partition regular semifilter F ∈ W , definable in W from the parameter s (hence ∆ ), such that r / ∈ F .To end this section, let us recall that there is a duality between filters and partitionregular semifilters. Lemma 3.3. If P is a filter then F = { X ⊆ N : X ∩ B = ∅ for all B ∈ P } . is a partition regular semifilter. This can be done by selectively destroying the stationarity of certain canonical stationary subsets of ω in L and coding these stationary kills by reals in a nice way. See [4] and [9] for similar arguments anddetails. OMBINATORIAL AND NUMBER-THEORETIC PROPERTIES OF GENERIC REALS 11
The family F is called the dual of P (in fact, partition regular families are sometimesreferred to in the literature as filterduals ). Although we do not need to use the fact, letus point out that families can also be “dualized” as in the lemma above, and a family ispartition regular if and only if its dual is a filter.Theorem 1.1 admits a dual version: Theorem 3.4.
Let V ⊆ W be models of set theory, and let r be a Cohen (random) realover V . (1) Suppose W = V [ r ] . In W , r belongs to no filter that is definable over V . (2) In general, r belongs to no filter that is lightface Π ( a ) -definable, where a is a realnumber in V .Proof. Let P be a filter in W , and let F denote its dual. Clearly, F is definable over V ifand only if P is. Also, if P is lightface Π ( a )-definable, where a is a real number in V , then F is lightface Π ( a )-definable as well.By Theorem 1.1, N − r (which is also a Cohen/random real) belongs to F under thehypotheses of either (1) or (2). Of course, if N − r belongs to F then r / ∈ P . (cid:3) Because the definition of a family from its dual requires a universal quantifier, it is possiblethat the dual of a Σ ( a )-definable filter is neither Σ ( a ) nor Π ( a ). This is why we havedropped the Σ ( a ) part from (2) in the dual version. However, we will point out that onemay go through the proof of Theorem 1.1, dualizing everything, to obtain a stronger versionof Theorem 3.4 in which Σ ( a ) filters are allowed. We leave these details as an exercise.4. Mahler’s classification of Cohen and random reals
In this section we consider Mahler’s classification of the reals. The definition of theseclasses draws and extends the idea of a Liouville number. Below we review the definition ofthese classes and refer to [2] for further details. The results presented here are obtained in[3], where similar results are proved for some other generic reals.For a polynomial P ( X ) ∈ Z [ X ], let H ( P ) denote the height of P and deg( P ) denote thedegree of P . Given positive integer n and real numbers ξ and H ≥
1, define w n ( ξ, H ) = min {| P ( ξ ) | : P ( X ) ∈ Z [ X ] , H ( P ) ≤ H, deg( P ) ≤ n, P ( ξ ) = 0 } . Then set w n ( ξ ) = lim sup H →∞ − log w n ( ξ, H )log H and w ( ξ ) = lim sup n →∞ w n ( ξ ) n . In other words, w n ( ξ ) is the supremum of the real numbers w for which there exist infinitelymany integer polynomials P ( X ) of degree at most n satisfying0 < | P ( ξ ) | < H ( P ) − w . Mahler’s classes
A, S, T and U are defined as follows. Let ξ be a real number. Then • ξ is an A -number if w ( ξ ) = 0 . • ξ is an S -number if 0 < w ( ξ ) < ∞ . • ξ is a T -number if w ( ξ ) = ∞ and w n ( ξ ) < ∞ for any n ≥ . • ξ is a U -number if w ( ξ ) = ∞ and w n ( ξ ) = ∞ for some n onwards.It is known that A -numbers are exactly the class of algebraic numbers. There is anotherclassification of reals known as Koksma’s classification, where instead of looking at theapproximation of 0 by integer polynomials evaluated at the real number ξ , the approximationof ξ by algebraic numbers is considered.For a real number α let deg( α ) = deg( P ) and H ( α ) = H ( P ) , where P ( X ) ∈ Z [ X ] is theminimal polynomial of α . For a positive integer n and real numbers ξ and H ≥
1, define w ∗ n ( ξ, H ) = min {| ξ − α | : α real algebraic deg( α ) ≤ n, H ( α ) ≤ H, α = ξ } . Then set w ∗ n ( ξ ) = lim sup H →∞ − log( Hw ∗ n ( ξ, H ))log H and w ∗ ( ξ ) = lim sup n →∞ w ∗ n ( ξ ) n . In other words, w ∗ n ( ξ ) is the supremum of the real numbers w for which there exist infinitelymany real algebraic numbers α of degree at most n satisfying0 < | ξ − α | < H ( α ) − w − . Koksma’s classes A ∗ , S ∗ , T ∗ and U ∗ are defined as follows. Let ξ be a real number. Then OMBINATORIAL AND NUMBER-THEORETIC PROPERTIES OF GENERIC REALS 13 • ξ is an A ∗ -number if w ∗ ( ξ ) = 0 . • ξ is an S ∗ -number if 0 < w ∗ ( ξ ) < ∞ . • ξ is a T ∗ -number if w ∗ ( ξ ) = ∞ and w ∗ n ( ξ ) < ∞ for any n ≥ . • ξ is a U ∗ -number if w ∗ ( ξ ) = ∞ and w ∗ n ( ξ ) = ∞ for some n onwards.It is well-know that the classifications of Mahler and of Koksma coincide, in the sense thatfor any real number ξ , ξ is an A (resp. S, T or U ) number ⇐⇒ ξ is an A ∗ (resp. S ∗ , T ∗ or U ∗ ) number.We now prove the following. Theorem 4.1. ( [3] ) ( a ) Assume r is a Cohen real over any transitive model M . Then r is in the class U . ( b ) Assume r is a random real over any transitive model M . Then r is in the class S . For the proof of Theorem 4.1( b ), we will need the following result of Sprindzuk. Lemma 4.2. ( Sprindzuk 1965 ) There exists a G δ set A of measure zero which contains alltranscendental numbers ξ with w ∗ ( ξ ) > . Proof.
We present a proof for completeness. Set S = { ξ : ξ is transcendental and w ∗ ( ξ ) > } . Let ǫ > n, H ≥ X n,H = { α : α is real algebraic, deg( α ) ≤ n and H ( α ) ≤ H } . For each n let i n be sufficiently large such that ∞ X H = i n n (2 H + 1) n +1 ( 2 H n +3 ) < ǫ n +1 . Set A ǫ = ∞ [ n =1 ∞ [ H = i n [ y ∈ A n,H ( y − H n +3 , y + 1 H n +3 ) . Then it is easily seen that m ( A ǫ ) = ∞ X n =1 ∞ X H = i n X y ∈ A n,H ( 2 H n +3 ) ≤ ∞ X n =1 ǫ n +1 < ǫ. We now show that A ǫ contains all real numbers ξ with w ∗ ( ξ ) > . Given such a ξ, let n belarge enough such that w ∗ n ( ξ ) n > /n. Then choose
H > i n such that − log( Hw ∗ n ( ξ, H ))log H > n (1 + 2 /n ) . Let α ∈ X n,H be such that w ∗ n ( ξ, H ) = | ξ − α | . It follows that | ξ − α | < H n (1+2 /n )+1 = 1 H n +3 , and hence ξ ∈ A ǫ . It follows that m ( S ) ≤ m ( A ǫ ) < ǫ. Finally let A = T Proof of Theorem 4.1 ( a ) . Assume r is a Cohen real. Let n ≥ . By the proof of Theorem1.1(2), we can find m > n such that r ∩ { m, . . . , m } = ∅ . Let α = P i 0, and so q = [ S ] iswell-defined and it extends p . Clearly, q (cid:13) “ ˙ r is in the class S ”.The result follows. (cid:3) OMBINATORIAL AND NUMBER-THEORETIC PROPERTIES OF GENERIC REALS 15 Recall from number theory that a Liouville number is a real number r with the propertythat, for every positive integer n , there exist infinitely many pairs of integers ( p, q ) with q > < | r − pq | < q n .One can easily show that a real number r is a Liouville number if and only if w ∗ ( r ) = ∞ . Itfollows from the proof of Theorem 4.1 that each Cohen reals is a Liouville number. Let usclose this section with the following corollary of the above result. Corollary 4.3. (a) In the generic extension by C , the set { r ∈ R : r is Cohen genericover V } has measure zero. (b) The set { r ∈ R : r is a Liouville number } contains a perfect set.Proof. (a) It follows from Lemma 4.2 that almost all reals are S -numbers. As Cohen realsare U -numbers, so { r ∈ R : r is Cohen generic over V } must have measure zero. (b) Let θ = (2 ℵ ) + and let M be a countable elementary submodel of H ( θ ) . Let P ⊆ R bea perfect set of reals such that each r ∈ P is Cohen generic over M . Let r ∈ P . By Theorem4.1(a), M [ r ] | =“ r is a Liouville number”, and hence by absoluteness, r is a Liouville number.Thus { r ∈ R : r is a Liouville number } ⊇ T, and the result follows. (cid:3) Addition of Cohen or random reals defines Z An infinite structure ( M, <, . . . ) which is totally ordered by < is called an o-minimalstructure if and only if every definable subset X of M (with parameters from M ) is a finiteunion of intervals and points. It is clear that the set Z of integers is not definable in ano-minimal structure.It follows from Tarski’s elimination of quantifiers that the structure R = ( R , + , · , <, , Z is not definable in it. An important problem is to seewhich extensions of R remain o-minimal or at least retain the property that Z , the set of Indeed, by a result of Kasch and Volkmann [10] the set of T -numbers and the set of U -numbers haveHausdorff dimension zero, in particular the set { r ∈ R : r is Cohen generic over V } has Hausdorff dimensionzero. integers, is not definable in them. There is a vast range of results in this direction; see forexample [5], [7], [12], [13] , [16] and [17].Recall that a subset C ⊆ R is called tame if Z is not definable in the structure ( R , C )and it is called wild otherwise.We relate the above phenomenon to forcing and show that the addition of Cohen orrandom reals to the structure R , when considered as unary predicates, defines Z whichcompletes the proof of Theorem 1.3. It follows that the Cohen and random reals are wildin the above sense. Remark 5.1. It is known that o-minimality is closed under expansions by constants. Itfollows that if we consider Cohen and random reals as constants (or any other generic real),then the structure ( R V [ r ] , + , · , <, , , r ) remains o-minimal and hence it does not define Z . Accordingly, in what follows we interpret a Cohen or random real as a subset of ω , i.e.,as a unary predicate in R . This interpretation works in the obvious way: r ∈ ω is identifiedwith the subset of ω for which it is the characteristic function.First we prove Theorem 1.3 for Cohen reals. Thus assume r is a Cohen real over V and in V [ r ] , consider the structure ( R V [ r ] , + , · , <, , , r ). Then it is easily seen, by densityarguments, that Z = { x ∈ R V [ r ] : ∃ a, b ∈ r such that x = a − b } , and hence Z is definable in ( R V [ r ] , + , · , <, , , r ).For a random real, the theorem follows from the following easy lemma. Lemma 5.2. Almost all reals define Z in the sense that for almost all reals r ∈ ω , the set Z is definable in ( R , + , · , <, , , r ) .Proof. For any m ∈ ω, the set A m = { r ∈ ω : ∃ a, b ∈ ω such that r ( a ) = r ( b ) = 1 and m = a − b } is of measure one, and hence so is the set A = T m ∈ ω A m . But any r ∈ A defines Z , as Z = { x ∈ R : ∃ a, b ∈ r such that x = a − b } . (cid:3) OMBINATORIAL AND NUMBER-THEORETIC PROPERTIES OF GENERIC REALS 17 To complement these observations, we now define a forcing notion which adds a genericreal r that does not define Z . Let P be the set of all finite functions p : n → p ( k ) = 0 if k < n is not a power of 2 (i.e., it is not of the form 2 i for some i ), orderedby reverse inclusion. Let G be P -generic over V and let r be the generic real added by G ,namely r = { k < ω : ∃ p ∈ G such that p ( k ) = 1 } . Note that r ⊆ Z = { n : n ∈ Z } . In order to show that Z is not definable in ( R V [ r ] , + , · , <, , , r ), we need the following theorem. Theorem 5.3. (a) (Friedman and Miller [6] ) Let R be an o-minimal expansion of ( R , <, +) and E ⊆ R be such that for every n ∈ N and definable function f : R n → R , the image f ( E n ) has no interior. If every subset of R definable in ( R , E ) has interior or is nowheredense, then every subset of R definable in ( R , E ) ♯ has interior or is nowhere dense.The same holds true with “nowhere dense” replaced by any of “null”, “countable”,“a finite union of discrete sets” or “discrete”. (b) (Van den Dries [16] ) Every subset of R definable in ( R , <, + , ., , , Z ) is the unionof an open set and finitely many discrete sets. It follows from the above theorem that Z is not definable in ( R V [ r ] , + , · , <, , , Z ) ♯ , andhence it is not definable in ( R V [ r ] , + , · , <, , , r ) either.We note that some (but not all) Silver-generic reals r have the property that r ⊆ Z . (Thisproperty is forced by a condition in the Silver poset, namely the partial function mapping ω \ Z to 0.) Similarly, some (but not all) Sacks-generic reals r have the property that r ⊆ Z .(This is forced by a condition in the Sacks poset, { r ∈ ω : r ( k ) = 0 whenever k / ∈ Z } .)Consequently, at least some Silver-generic and Sacks-generic reals have the property that( R V [ r ] , + , · , <, , , r ) does not define Z . Question 5.4. If r is an arbitrary Silver-generic real over V , does ( R V [ r ] , + , · , <, , , r ) define Z ? What about Sacks-generic reals? Other minimal reals? ( R , E ) ♯ is the structure obtained by adding to R predicates picking out every subset of every cartesianpower E k of E . Recall that Silver reals and Sacks reals are two primary examples of minimal reals: i.e.,reals r generic over V such that if s ∈ V [ r ] then either s ∈ V or V [ s ] = V [ r ]. Question 5.5. Assume V [ r ] is a generic extension of V by Cohen (res. random) forcingand (in V [ r ] ) set C = { s ∈ R : s is Cohen (res. random) generic over V } .Does the structure ( R V [ r ] , + , · , <, , , C ) define Z ? . Remark 5.6. It is evident that the structure ( R V [ r ] , + , · , <, , , C ) is not o-minimal, as theset C is not a finite union of intervals and points. Adding many random reals may add many Cohen reals It is a well-know fact that forcing with R × R adds a Cohen real; in fact, if r , r are theadded random reals, then c = r + r is Cohen [1]. This in turn implies all reals c + a, where a ∈ R V , are Cohen, and so, we have continuum many Cohen reals over V . However, thesequence h c + a : a ∈ R V i fails to be C ((2 ℵ ) V )-generic over V . In fact, there is no sequence h c i : i < ω i ∈ V [ r , r ] of Cohen reals which is C ( ω )-generic over V .The usual proof of the above fact is based on the characterization of Cohen reals givenby Lemma 2.1. In this section, we give a proof of Theorem 1.4 which is direct and avoidsany characterization of Cohen reals. As in the classical case, the Steinhaus’s lemma playsan essential role in our proof.A famous theorem of Steinhaus [14] from 1920 asserts that if A, B ⊆ R n are measurablesets with positive Lebesgue measure, then A + B has an interior point; see also [15]. Here,we need a version of Steinhaus theorem for the space 2 κ × ω .For S, T ⊆ κ × ω , set S + T = { x + y : x ∈ S and y ∈ T } , where x + y : κ × ω → x + y )( α, n ) = x ( α, n ) + y ( α, n ) (mod 2) . Note that the above addition operation is continuous. Lemma 6.1. Suppose S ⊆ κ × ω is Borel and non-null. Then S − S contains an open setaround the zero function . OMBINATORIAL AND NUMBER-THEORETIC PROPERTIES OF GENERIC REALS 19 Proof. We follow [15]. Set µ = µ κ be the product measure on 2 κ × ω . As S is Borel andnon-null, there is a compact subset of S of positive µ -measure, so may suppose that S itselfis compact. Let U ⊇ S be an open set with µ ( U ) < · µ ( S ) . By continuity of addition, wecan find an open set V containing the zero function 0 such that V + S ⊆ U. We show that V ⊆ S − S . Thus suppose x ∈ V. Then ( x + S ) ∩ S = ∅ , as otherwise wewill have ( x + S ) ∪ S ⊆ U, and hence µ ( U ) ≥ · µ ( S ) , which is in contradiction with ourchoice of U . Thus let y , y ∈ S be such that x + y = y . Then x = y − y ∈ S − S asrequired. (cid:3) Similarly, we have the following: Lemma 6.2. Suppose S, T ⊆ κ × ω are Borel and non-null. Then S + T contains an openset. Suppose S, T ⊆ κ × ω are Borel and non-null. It follows from Lemma 6.2 that for some p ∈ C ( κ ) , [[ p ]] ⊆ S + T. Thus, by continuity of the addition, we can find x ∈ S and y ∈ T such that: • ( x + y ) ↾ dom( p ) = p. • The sets S ∩ [[ x ↾ dom( p )]] and T ∩ [[ y ↾ dom( p )]] are Borel and non-null.We now complete the proof of Theorem 1.4. Thus force with R ( κ ) × R ( κ ) and let G × H begeneric over V . Let hh r α : α < κ i , h s α : α < κ ii be the sequence of random reals added by G × H. For α < κ set c α = r α + s α . The following completes the proof: Lemma 6.3. The sequence h c α : α < κ i is a sequence of κ -Cohen reals over V .Proof. It suffices to prove the following:For every ([ S ] , [ T ]) ∈ R ( κ ) × R ( κ ), and every open dense subset D ∈ V ( ∗ ) of C ( κ ), there is ([ ¯ S ] , [ ¯ T ]) ≤ ([ S ] , [ T ]) such that ([ ¯ S ] , [ ¯ T ]) k− “ h c ∼ α : α ∈ κ i extends some element of D ”.Thus fix ([ S ] , [ T ]) ∈ R ( κ ) × R ( κ ) and D ∈ V as above, where S, T ⊆ κ × ω are Borel andnon-null. By Lemma 6.2 and the remarks after it, we can find p ∈ C ( κ ) and ( x, y ) ∈ S × T such that: (1) [[ p ]] ⊆ S + T. (2) ( x + y ) ↾ dom( p ) = p. (3) The sets S ∩ [[ x ↾ dom( p )]] and T ∩ [[ y ↾ dom( p )]] are Borel and non-null.Now let q ∈ D be such that([ S ∩ [[ x ↾ dom( p )]]] , [ T ∩ [[ y ↾ dom( p )]]]) (cid:13) “ q ≤ C ( κ ) p ”.Using continuity of the addition and further application of Lemma 6.2 and the remarks afterit, we can find x ′ , y ′ such that:(4) x ′ ∈ S ∩ [[ x ↾ dom( p )]] and y ′ ∈ T ∩ [[ y ↾ dom( p )]] . (5) ( x ′ + y ′ ) ↾ dom( q ) = q. (6) The sets S ∩ [[ x ′ ↾ dom( q )]] and T ∩ [[ y ′ ↾ dom( q )]] are Borel and non-null.It is now clear that([ S ∩ [[ x ′ ↾ dom( q )]]] , [ T ∩ [[ y ′ ↾ dom( q )]]]) (cid:13) “ h c ∼ α : α ∈ κ i extends q ”.The result follows. (cid:3) References [1] Bartoszynski, Tomek; Judah, Haim, Set theory. On the structure of the real line. A K Peters, Ltd.,Wellesley, MA, 1995. xii+546 pp. ISBN: 1-56881-044-X.[2] Bugeaud, Yann. Approximation by algebraic numbers. Cambridge Tracts in Mathematics, 160. 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