On additive properties of sets defined by the Thue-Morse word
Michelangelo Bucci, Neil Hindman, Svetlana Puzynina, Luca Q. Zamboni
aa r X i v : . [ m a t h . C O ] M a r On additive properties of sets defined by the Thue-Morse word
Michelangelo Bucci a , Neil Hindman b,1 , Svetlana Puzynina a,c,2 , Luca Q. Zamboni a,d,3 a Department of Mathematics and Statistics & FUNDIM, University of Turku, Finland. b Department of Mathematics, Howard University, USA. c Sobolev Institute of Mathematics, Russia. d Universit´e de Lyon 1, France.
Abstract
In this paper we study some additive properties of subsets of the set N of positive integers: A subset A of N is called k -summable (where k ∈ N ) if A contains (cid:8) P n ∈ F x n | ∅ 6 = F ⊆ { , , . . . , k } (cid:9) for some k -termsequence of natural numbers h x t i kt =1 satisfying uniqueness of finite sums. We say A ⊆ N is finite FS-big if A is k -summable for each positive integer k . We say is A ⊆ N is infinite FS-big if for each positiveinteger k, A contains { P n ∈ F x n | ∅ 6 = F ⊆ N and F ≤ k } for some infinite sequence of natural numbers h x t i ∞ t =1 satisfying uniqueness of finite sums. We say A ⊆ N is an IP-set if A contains { P n ∈ F x n | ∅ 6 = F ⊆ N and F < ∞} for some infinite sequence of natural numbers h x t i ∞ t =1 . By the Finite Sums Theorem[5], the collection of all IP-sets is partition regular , i.e., if A is an IP-set then for any finite partition of A , one cell of the partition is an IP-set. Here we prove that the collection of all finite FS-big sets is alsopartition regular. Let T = 011010011001011010010110011010 . . . denote the Thue-Morse word fixed bythe morphism and . For each factor u of T we consider the set T (cid:12)(cid:12) u ⊆ N of all occurrences of u in T . In this note we characterize the sets T (cid:12)(cid:12) u in terms of the additive properties defined above. Using theThue-Morse word we show that the collection of all infinite FS-big sets is not partition regular. Keywords:
Partition regularity, additive combinatorics, IP-sets, Thue-Morse infinite word.
1. Introduction
A fundamental result in Ramsey theory, originally due to Issai Schur [12], states that given a finitepartition of the natural numbers N , one cell of the partition contains two points x, y and their sum x + y . Anextension of Schur’s Theorem, which we will call the finite Finite Sums Theorem states that whenever N isfinitely partitioned, there exist arbitrarily large sets of numbers all of whose sums belong to the same elementof the partition. The finite Finite Sums Theorem is an easy consequence of Rado’s Theorem [10]. Given afinite sequence h x t i kt =1 or an infinite sequence h x t i ∞ t =1 in N we say that the sequence satisfies uniqueness offinite sums provided that whenever F and H are finite nonempty subsets of the domain of the the sequenceand P t ∈ F x t = P t ∈ H x t , one must have F = H . For k a positive integer, we say that a subset A of N is k - summable , if A contains all finite sums of distinct terms of some k -term sequence h x n i kn =1 of naturalnumbers satisfying uniqueness of finite sums. We say that A ⊆ N is k ∞ - summable if there exists an infinite Email addresses: [email protected] (Michelangelo Bucci), [email protected] (Neil Hindman), [email protected] (Svetlana Puzynina), [email protected] (Luca Q. Zamboni) Partially supported by the US National Science Foundation under grant DMS-1160566. Partially supported by the Academy of Finland under grant 251371, by RFBR (grant 10-01-00424) and by RF President grantfor young scientists (MK-4075.2012.1). Partially supported by a FiDiPro grant (137991) from the Academy of Finland and by ANR grant SUBTILE.
Preprint submitted to Elsevier June 11, 2018 equence h x n i ∞ n =1 of natural numbers satisfying uniqueness of finite sums such that A contains all sums of atmost k distinct terms of h x n i ∞ n =1 . As a consequence of the (infinite) Finite Sums Theorem, given any finitepartition of N , one element of the partition is k ∞ -summable.In this paper we consider three different families of subsets of N each characterized by an additiveproperty which may be regarded as an extension of the finite Finite Sums Theorem: finite FS-big, infiniteFS-big, and IP-sets. A subset A of N is called finite FS-big if it is k -summable for every positive integer k .A subset A of N is called infinite FS-big if it is k ∞ -summable for every positive integer k . A subset A of N is called an IP-set if A contains all finite sums of distinct terms of some infinite sequence h x n i ∞ n =1 of naturalnumbers.A collection of sets S is said to be partition regular if for each A ∈ S , whenever A is partitioned intofinitely many sets, at least one set of the partition is in S . By the Finite Sums Theorem, the collection of allIP-sets is partition regular. Other examples of partition regular families are sets having positive upper density,and sets having arbitrarily long arithmetic progressions. (This latter fact is an almost immediate consequenceof van der Waerden’s Theorem [15]. Assume A ⊆ N contains arbitrarily long arithmetic progressions. Let k, r ∈ N , and let A = S ri =1 C i . By van der Waerden’s Theorem pick n such that whenever { , , . . . , n } ispartitioned into r classes, one class contains a length k arithmetic progression. Pick a and d in N such that { a + d, a + 2 d, . . . , a + nd } ⊆ A . For i ∈ { , . . . , r } let B i = { t ∈ { , , . . . , n } | a + td ∈ C i } . Pick i , b ,and c such that { b + c, b + 2 c, . . . , b + kc } ⊆ B i . Then { a + bd + cd, a + bd + 2 cd, . . . , a + bd + kcd } ⊆ C i .)We shall show in Section 2 that the collection of all finite FS-big subsets of N is partition regular (seeTheorem 2.3). In contrast, for any fixed value of k , the property of being k -summable or k ∞ -summable isnot partition regular. For example, the set A = { n ∈ N | n mod } is clearly ∞ -summable. On theother hand A = A ∪ A where A = { n ∈ N | n ≡ } and A = { n ∈ N | n ≡ } , andneither A i is -summable. Nevertheless, for each fixed k we could consider the set R ∞ ( k ) = { A ⊆ N | whenever r ∈ N and A = r [ i =0 A i , ∃ ≤ i ≤ r such that A i is k ∞ -summable } Then each R ∞ ( k ) is non-empty. In fact every IP-set belongs to R ∞ ( k ) . It is a difficult open question ofImre Leader’s [3, Question 8.1] whether there is any member of R ∞ (2) which is not an IP-set. In general,the question of determining whether a given subset A ⊆ N is in R ∞ ( k ) or is an IP-set is typically quitedifficult, even if for every A, either A or its complement belongs to R ∞ ( k ) or is an IP-set.In this note we focus on sets A which are defined in terms of the binary expansions of its elements. In thisrespect it is natural to consider the Thue-Morse infinite word T = t t t t . . . ∈ { , } ω where t n is definedas the sum modulo of the digits in the binary expansion of n. The Thue-Morse word is -automatic [2]:In fact t n is computed by feeding the binary expansion of n in the deterministic finite automata depicted inFigure 1. Starting from the initial state labeled , we read the binary expansion of n starting from the mostsignificant digit. Then t n is the corresponding terminal state. For example, the binary representation of is and the path starting at terminates at vertex . Whence t = 1 .
00 1 011
Figure 1: The Thue-Morse automaton
The origins of T go back to the beginning of the last century with the works of the Norwegian mathe-matician Axel Thue [13, 14]. Thue noted that every binary word of length four contains a square, that is twoconsecutive equal blocks XX.
He then asked whether it was possible to find an infinite word on distinct2ymbols which avoided squares. He also asked whether there exists an infinite binary word without cubes,that is with no three consecutive equal blocks. Thue showed that in each case the answer is positive andconstructed this very special infinite word T to produce the desired words. In fact the word T contains nofractional power greater than , i.e., contains no word of the form XXX ′ where X ′ is a prefix of X. Thue’swork originally appeared in an obscure Norwegian journal and for many years remained largely unknownand unappreciated.A few years later in the 1920s, Marston Morse and Gustav Hedlund [8, 9] were pioneering a new branchof mathematics known as Symbolic Dynamics, inspired by the study of various classical dynamical systemsdating back to Newton. The basic idea of symbolic dynamics consists in dividing up the set of possible statesinto a finite number of pieces. By discretizing both space and time, one could model a dynamical system ( X, T ) by a space consisting of infinite words of abstract symbols, each symbol corresponding to a state ofthe system, and a shift operator corresponding to the dynamics. Thus from this point of view, the orbits ofmotion are described as symbolic trajectories or flows . A periodic orbit would give rise to a periodic infiniteword, while an aperiodic orbit would correspond to an aperiodic infinite word.Curiously enough, these foundational works of Morse and Hedlund exhibited strong ties with the earlierwork of Thue. This connection stems through the use of infinite words to describe infinite geodesic curveson a surface of negative curvature. And so, the word T originally defined by Thue to study combinatorialproperties of words was rediscovered in 1921 by Morse [7] in connection with differential geometry. Heproved that every surface of negative curvature having at least two normal segments, admits a continuum ofrecurrent aperiodic geodesics.An alternative definition of the Thue-Morse word which will be useful to us is in terms of the morphism τ : { , } → { , } ∗ given by and . More precisely, iterating τ on the symbol gives
7→ · · · . In general, τ n +1 (0) = τ n (0) τ n (0) where τ n (0) is obtained from τ n (0) by exchanging ’s and ’s. Inparticular, since τ n (0) is a prefix of τ n +1 (0) , the sequence ( τ n (0)) n ≥ tends in the limit to the infinite word T = 0110100110010110100101100110100110010110 . . . For more background and information on the Thue-Morse word we refer the reader to [1] or [2].Let T denote the word obtained from T by exchange of ’s and ’s, i.e., T is the fixed point of the Thue-Morse morphism beginning in . We consider subsets of N defined by the Thue-Morse word via occurrencesof its factors. More precisely, writing T = t t t . . . with t i ∈ { , } , for each factor u of T we set T (cid:12)(cid:12) u = { n ∈ N | t n t n +1 . . . t n + | u |− = u } . In other words, T (cid:12)(cid:12) u denotes the set of all occurrences of u in T . The main result of this note is to obtain a fullcharacterization of each of the sets T (cid:12)(cid:12) u in terms of the three additive properties defined above. We show thatfactors of the Thue-Morse word can be split into three classes: one corresponding to factors u for which T (cid:12)(cid:12) u is an IP-set; these factors are precisely all prefixes of T . The second class consists of all factors u such that T (cid:12)(cid:12) u is infinite FS-big but not an IP-set; this corresponds to all prefixes of T . Finally, for all remaining factors u of T , the set T (cid:12)(cid:12) u is not -summable, and in some cases not even -summable (see Theorem 3.1). We alsoshow that the set T (cid:12)(cid:12) may be partitioned into two cells neither of which is ∞ -summable (see Lemma 3.3).Thus, the collection of all infinite FS-big sets is not partition regular (see Corollary 3.5). As pointed out to usby the referees of this paper, this latter point may be proved independently of the Thue-Morse word (eitherdirectly using the binary representation of digits without reference to T , or via other digital representationsof the integers). Our use of T is merely one of convenience as it provides a uniform framework on which toinvestigate the various additive properties defined above.3e conclude this introduction with some notation that we will be using. We denote the set of all k -summablesubsets of N by Σ k and the set of all finite FS-big subsets of N by Σ . Thus, Σ = T k ≥ Σ k . We denote theset of all k ∞ -summable sets by Σ ∞ k and the set of all infinite FS-big sets by Σ ∞ so that Σ ∞ = T k ≥ Σ ∞ k . Itis immediate that Σ ∞ k +1 ⊆ Σ ∞ k and Σ ∞ k ⊆ Σ k .We have seen that { n ∈ N | n mod } ∈ Σ ∞ \ Σ . More generally, for each k > we have that { n ∈ N | n mod k } ∈ Σ ∞ k − \ Σ k . This follows immediately from the following simple lemma which islikely a well known fact but the authors were unable to find it in the literature. We thus include a proof herefor the sake of completeness.
Lemma 1.1.
Given any set S of k nonnegative integers, some subset L of S sums up to modulo k, i.e., P x ∈ L x ≡ mod k for some L ⊆ S. Proof.
Equivalently, given a k -term sequence h x i i ki =1 in the cyclic group Z k of order k, we will show thatsome subsequence sums up to . To see this, for each ≤ i ≤ k, define sets C i ⊆ Z k recursively as follows: C = { } and for i ≥ , set C i +1 = C i S ( C i + x i +1 ) , in other words C i +1 = { } [ { X i ∈ F x i | F ⊆ { , . . . , i + 1 }} . We claim that ∈ C i + x i +1 for some ≤ i ≤ k − , i.e., some subsequence of h x j i kj =1 sums to . In fact,for each ≤ i ≤ k − , if / ∈ C i + x i +1 then C i +1 > C i (since ∈ C i ) . Thus if for every ≤ i ≤ k − we had that / ∈ C i + x i +1 , then C k ≥ k + C = k + 1 , a contradiction (since C k ⊆ Z k ) . Acknowledgements:
The authors would like to express their gratitude to the two referees of this paper. Inparticular one of the referees suggested a simplification to the proof of item (1) of Theorem 3.1 which wedecided to use, as well as an alternative simple but clever proof that the collection of all infinite FS-big setsis not partition regular which we included as one of two proofs of Corollary 3.5.
2. Finite FS-big sets
In this section we prove that the collection of all finite FS-big sets is partition regular (see Theorem 2.3below). Surprisingly the authors were unable to find a proof of this fact in the existing literature. Thus wetake this opportunity to present two different derivations: Our first proof is a straightforward application ofthe so-called finite Finite Unions Theorem (see Theorem 2.2 below) and is by now quite routine to experts inRamsey theory. Our second proof uses a clever argument suggested to us by Imre Leader which establishesTheorem 2.2 using only the finite Finite Sums Theorem (Theorem 2.4 ) and Ramsey’s Theorem [11].Throughout this section we will use the notation Fin ( A ) for the set of all non-empty finite subsets of A and let F S ( h x t i t ∈ A ) = { P t ∈ F x t | F ∈ Fin ( A ) } .We first observe that we had some choices to make when we defined k -summable. That is, we couldhave defined A to be k -summable if there is a sequence h x t i kt =1 such that F S ( h x t i kt =1 ) ⊆ A ; we could havedefined A to be k -summable if there is an increasing sequence h x t i kt =1 such that F S ( h x t i kt =1 ) ⊆ A ; andwe could have defined A to be k -summable if there is a sequence h x t i kt =1 satisfying uniqueness of finitesums such that F S ( h x t i kt =1 ) ⊆ A . (We actually chose k -summable because it generalizes most naturally toarbitrary semigroups as in Theorem 2.3.) These notions are progressively strictly stronger. For example if k > , { , , . . . , k } is k -summable but not k -summable . And if k > , { , , . . . , k + k } is k -summable but not k -summable . However, for the notion of finite FS-big subsets of N , it does not matter which choicewas made for k -summable. The reason is that for each k there is some m such that if A is an m -summable subset of N , then A is k -summable . Similarly, for each k there is some m such that if A is an m -summable subset of N , then A is k -summable . 4he main key for proving that finite FS-big sets are partition regular is the finite Finite Unions Theorem.The first proof that we will present, and much the simpler of the two, uses a standard compactness argumentand the infinite Finite Unions Theorem. Theorem 2.1 (Infinite Finite Unions Theorem) . Let r ∈ N . If Fin ( N ) = S ri =1 F i , then there exist i ∈{ , , . . . , r } and a sequence h F t i ∞ t =1 in Fin ( N ) such that for each t ∈ N , max F t < min F t +1 and for each H ∈ Fin ( N ) , S t ∈ H F t ∈ F i .Proof. This is actually stated in [5]. A much easier proof is in [6, Corollary 5.17]. It is an immediatecorollary of the (infinite) Finite Sums Theorem, because, given any sequence h x t i ∞ t =1 in N one may choosea sequence h F n i ∞ n =1 in Fin ( N ) such that for each n ∈ N , max F n < min F n +1 and for each n and l in N , if l ≤ P t ∈ F n x t , then l +1 divides P t ∈ F n +1 x t . (That is, the maximum of the binary support of P t ∈ F n x t isless than the minimum of the binary support of P t ∈ F n +1 x t .) Theorem 2.2 (Finite Finite Unions Theorem) . Let r, k ∈ N . There is some m ∈ N such that whenever Fin ( { , , . . . , m } ) = S ri =1 F i , there exist i ∈ { , , . . . , r } and a sequence h F t i kt =1 in Fin ( { , , . . . , m } ) such that for each t ∈ { , , . . . , k − } , if any, max F t < min F t +1 and for each H ∈ Fin ( { , , . . . , k } ) , S t ∈ H F t ∈ F i .Proof. Suppose not. For each m ∈ N pick a function ψ m : Fin ( { , , . . . , m } ) → { , , . . . , r } withthe property that there do not exist i ∈ { , , . . . , r } and a sequence h F t i kt =1 in Fin ( { , , . . . , m } ) suchthat for each t ∈ { , , . . . , k − } , if any, max F t < min F t +1 and for each H ∈ Fin ( { , , . . . , k } ) , ψ m ( S t ∈ H F t ) = i . Define σ m : Fin ( N ) → { , , . . . , r } by σ m ( F ) = ψ m ( F ) if F ⊆ { , , . . . , m } and σ m ( F ) = 1 otherwise.Give { , , . . . , r } the discrete topology and let X = × F ∈ Fin( N ) { , , . . . , r } with the product topology.Then X is compact and h σ m i ∞ m =1 is a sequence in X so pick a cluster point ϕ of h σ m i ∞ m =1 . Pick by Theorem2.1, i ∈ { , , . . . , r } and a sequence h F t i ∞ t =1 in Fin ( N ) such that for each t ∈ N , max F t < min F t +1 andfor each H ∈ Fin ( N ) , ϕ ( S t ∈ H F t ) = i . Let U = (cid:8) µ ∈ X | µ ( F i ) = ϕ ( F i ) for all i ∈ { , , . . . , k } (cid:9) . Then U is a neighborhood of ϕ in X so pick m > max F k such that σ m ∈ U . Then for each H ∈ Fin ( { , , . . . , k } ) , ψ m ( S t ∈ H F t ) = σ m ( S t ∈ H F t ) = ϕ ( S t ∈ H F t ) = i , a contradiction.The definition of FS-big makes sense in an arbitrary semigroup ( S, +) . (Even though we are writingthe semigroup additively, we are not assuming commutativity, so we need to specify that the sums are takenin increasing order of indices.) The reader should be cautioned that an arbitrary semigroup might have nonontrivial sequences satisfying uniqueness of finite sums, in which case Σ = ∅ . However, if S is cancellative,then by [6, Lemma 6.31], any infinite subset of S contains a sequence satisfying uniqueness of finite products. Theorem 2.3.
Let ( S, +) be a semigroup. The collection Σ of all finite FS-big subsets of S is partitionregular.Proof. Suppose A ⊆ S is finite FS-big and A = S ri =1 B i for some r ∈ N . Let k ∈ N . We shall showthat there are some i ∈ { , , . . . , r } and some h x t i kt =1 satisfying uniqueness of finite sums such that F S ( h x t i kt =1 ) ⊆ B i . By the pigeon hole principle, there is thus one i which contains such a set for arbi-trarily large k , and thus for all k .By Theorem 2.2 pick m ∈ N such that whenever Fin ( { , , . . . , m } ) = S ri =1 F i , then there exist i ∈{ , , . . . , r } and a sequence h F t i kt =1 in Fin ( { , , . . . , m } ) such that for each t ∈ { , , . . . , k − } , if any, max F t < min F t +1 and for each H ∈ Fin ( { , , . . . , k } ) , S t ∈ H F t ∈ F i . Since A is finite FS-big wemay pick h y t i mt =1 satisfying uniqueness of finite sums with F S ( h y t i mt =1 ) ⊆ A . For each i ∈ { , , . . . , r } F i = { H ∈ Fin ( { , , . . . , m } ) | P t ∈ H y t ∈ B i } . Pick i ∈ { , , . . . , r } and a sequence h F t i kt =1 in Fin ( { , , . . . , m } ) such that for each t ∈ { , , . . . , k − } , if any, max F t < min F t +1 and for each H ∈ Fin ( { , , . . . , k } ) , S t ∈ H F t ∈ F i . For n ∈ { , , . . . , k } let x n = P t ∈ F n y t . Then since max F t < min F t +1 when t < k , if H ∈ Fin ( { , , . . . , k } ) and K = S n ∈ H F n , then P n ∈ H x n = P t ∈ K y t ∈ B i . Further it is an easy exercise to show that, since h y t i mt =1 satisfies uniqueness of finite sums, so does h x t i kt =1 .Notice that, since we did the above proof for an arbitrary semigroup, it would not be good enough tohave F t ∩ F l = ∅ when t = l . For example, if F = { , } , F = { } , H = { , } , and K = S n ∈ H F n ,then K = { , , } . Thus P n ∈ H x n = y + y + y which need not equal y + y + y = P t ∈ K y t .As we remarked earlier, the finite Finite Sums Theorem, (sometimes called Folkman’s Theorem), hasbeen known, or at least easily knowable, since the proof of Rado’s Theorem [10] was published in 1933. Theorem 2.4 (Finite Finite Sums Theorem) . Let k, r ∈ N . There exists m ∈ N such that whenever { , , . . . , m } = S ri =1 B i , there exist i ∈ { , , . . . , r } and a sequence h x t i kt =1 such that F S ( h x t i kt =1 ) ⊆ B i .Proof. This is an easy consequence of Rado’s Theorem. See [6, Exercise 15.3.1].While it is immediate that Theorem 2.2 implies Theorem 2.4 (by means of the binary support of integers),it is by no means obvious that one can derive Theorem 2.2 from Theorem 2.4. We are grateful to Imre Leaderfor providing an argument which establishes Theorem 2.2 using only Theorem 2.4 and Ramsey’s Theorem[11].
Lemma 2.5.
Let k, r, s ∈ N with k ≤ s . There exists m ∈ N such that whenever A is a set with A = m ,and Fin ( A ) = S ri =1 F i , there exist ϕ : { , , . . . , k } → { , , . . . , r } and B ⊆ A with B = s such thatfor all C ∈ Fin ( B ) , if t = C and t ≤ k , then C ∈ F ϕ ( t ) .Proof. We proceed by induction on k (for all r and all s ≥ k ). For k = 1 the conclusion is an immediateconsequence of the pigeon hole principle.Now assume that the lemma holds for k . Let r, s ∈ N be given with s ≥ k + 1 . By Ramsey’s Theorempick n ∈ N such that if D = n and { C ⊆ D | C = k + 1 } ⊆ S ri =1 F i , then there exist i ∈ { , , . . . , r } and B ⊆ D such that B = s and { C ⊆ B | C = k + 1 } ⊆ F i .Pick m as guaranteed by the induction hypothesis for k , r , and n (with n replacing s ) and let A = m .Assume that Fin ( A ) = S ri =1 F i . Pick ϕ : { , , . . . , k } → { , , . . . , r } and D ⊆ A with D = n such thatfor all C ∈ Fin ( D ) , if t = C and t ≤ k , then C ∈ F ϕ ( t ) . Then { C ⊆ D | C = k + 1 } ⊆ S ri =1 F i , sopick ϕ ( k + 1) ∈ { , , . . . , r } and B ⊆ D such that B = s and { C ⊆ B | C = k + 1 } ⊆ F ϕ ( k + 1) . Second proof of Theorem k, r ∈ N and pick by Theorem 2.4 s ∈ N such that whenever { , , . . . , s } = S ri =1 C i , there exist x , x , . . . , x k and i ∈ { , , . . . , r } such that F S ( h x t i kt =1 ) ⊆ C i .Let k ′ = s and pick m as guaranteed by Lemma 2.5 for r , k ′ , and s . Let Fin ( { , , . . . , m } ) = S ri =1 F i .Pick ϕ : { , , . . . , s } → { , , . . . , r } and B ⊆ { , , . . . , m } with B = s such that for all C ∈ Fin ( B ) , if t = C and t ≤ s , then C ∈ F ϕ ( t ) . Pick x , x , . . . , x k and i ∈ { , , . . . , r } such that ϕ [ F S ( h x t i kt =1 )] = { i } . Pick h F t i kt =1 with max F t < min F t +1 for all t < k such that F t = x t , which one can do since P kt =1 x t ≤ s .
3. Sets defined by the Thue-Morse word
In this section we define a class of subsets of N defined by the occurrences of factors in the Thue-Morseword. Theorem 3.1.
Let u be a factor of the Thue-Morse word T = 011010011001011010 . . . . Then If u is a prefix of T then T (cid:12)(cid:12) u is an IP-set. If u is a prefix of T then T (cid:12)(cid:12) u is infinite FS-big but is not an IP-set. If u is neither a prefix of T nor a prefix of T then T (cid:12)(cid:12) u is not -summable. Moreover T (cid:12)(cid:12) u is -summableif and only u is a prefix of τ n (010) or of τ n (101) for some n ≥ . Before we begin with the proof of Theorem 3.1 we introduce some useful notation: For each positiveinteger n we will denote the binary expansion of n by [ n ] , i.e., if n = r k k + r k − k − + . . . + r with r k = 1 and r i ∈ { , } we write [ n ] = r k r k − . . . r . We define the support of n , denote supp( n ) by supp( n ) = { i ∈ { , , . . . , k } | r i = 1 } . For instance, supp(19) = { , , } . Thus t n = 0 ⇔ n ) is even . Finally, for each length n we denote by pref n T the prefix of T of length n . Proof of Theorem 3.1, part 1.
It follows from the definition of the Thue-Morse word T that if u = u u . . . u k ∈{ , } k is a factor of T , then m ∈ T (cid:12)(cid:12) u if and only if m + j ) ≡ u j +1 mod 2 for each ≤ j ≤ | u | − . Thus, if n > m + | u | − then n +1 + 2 n + m + j ) = 2 + m + j ) for each ≤ j ≤ | u | − from which it follows that n +1 + 2 n + m ∈ T (cid:12)(cid:12) u . Hence if ∈ T (cid:12)(cid:12) u (equivalentlyif u is a prefix of T ) , then there is a sequence of positive integers of the form n + 2 n +1 whose finite sumsare all in T (cid:12)(cid:12) u . Thus, T (cid:12)(cid:12) u is an IP-set. This completes the proof of 1.We will need the following lemma in the proof of 2.: Lemma 3.2.
Let i, j, k and r be positive integers with r odd and j ≤ k − . If [ r ] = r j r j − . . . r then (cid:0) r i (2 k − (cid:1) = k. Proof.
Since (cid:0) r i (2 k − (cid:1) = (cid:0) r (2 k − (cid:1) it suffices to show that (cid:0) r (2 k − (cid:1) = k .We have that [ r k ] = r l r l − . . . r k . Thus [ r k − = r l r l − . . . r k − .Since l ≤ k − we have (cid:0) r (2 k − (cid:1) = r k − − r + 1) = r ) + k − − r ) + 1 = k where the last +1 term comes from the fact that r = 1 since r is odd. Proof of Theorem 3.1, part 2.
We first note that those n ’s for which n ) is odd and [ n ] ends in l correspond to occurrences of pref l T .Let u be a prefix of T and k a positive integer. To prove that T u ∈ Σ ∞ k − consider the sequence h x n i ∞ n =0 of numbers whose binary representation is given by [ x n ] = 110 n + j k − l where j = ⌈ log (2 k − ⌉ and l = ⌈ log | u |⌉ . Consider any r ≤ k − distinct numbers x n i and consider their sum r X i =1 x n i = r X i =1 (2 n i +2 k − l + j + 2 n i +2 k + l + j ) + r (2 k − − l . By Lemma 3.2 it follows that r (2 k − − l ) = 2 k − and hence that P ri =1 x n i ) =2 k − r . As this is an odd number, and [ P ri =1 x n i ] ends in at least l = ⌈ log | u |⌉ many ’s, it followsthat P ri =1 x n i is an occurrence of u .Next we will prove that if u is a prefix of T then T (cid:12)(cid:12) u is not an IP-set. We will make use of the followinglemma: 7 emma 3.3. There exists a partition of the set T (cid:12)(cid:12) into two sets neither of which is in Σ ∞ .Proof. Consider the partition T (cid:12)(cid:12) = A ∪ A defined as follows: Let A be the set of all n ∈ T (cid:12)(cid:12) such that the min (cid:0) supp( n ) (cid:1) is even, and let A be the set of all n ∈ T (cid:12)(cid:12) such that the min (cid:0) supp( n ) (cid:1) is odd. For instance,
25 = 2 + 2 + 2 , and hence the least nonzero digit is in position , so ∈ A . We will show that neither A i is in Σ ∞ . Fix i ∈ { , } and suppose to the contrary that A i is in Σ ∞ , i.e., there is an infinite sequence h x n i ∞ n =1 in A i satisfying uniqueness of finite sums such that for every n = m we have x n + x m ∈ A i .Note first that for each n > we have supp( x n ) ∩ supp( x ) = ∅ . Otherwise x + x n ) would beeven. Therefore, there exists a positive constant M such that min (cid:0) supp( x n ) (cid:1) ≤ M for each n ∈ N . Bythe pigeon hole principle there exists a positive integer r and an infinite subsequence x n , x n , . . . of thesequence h x n i ∞ n =1 such that min (cid:0) supp( x n j ) (cid:1) = r for each j ∈ N . Again by the pigeon hole principle thereexists infinitely many of the x n j whose binary expansions also agree in position r + 1 . Thus there exists n = m such that min (cid:0) supp( x n ) (cid:1) = min (cid:0) supp( x m ) (cid:1) = r and such that r + 1 ∈ supp( x n ) if and only if r + 1 ∈ supp( x m ) . It is readily verified that min (cid:0) supp( x n + x m ) (cid:1) = r + 1 . Hence x n + x m ∈ A − i . Remark.
It is not difficult to see that the sets A and A from the proof of Lemma 3.3 are both finiteFS-big. So they provide examples of sets which are finite FS-big but not ∞ -summable.It follows from the above lemma that T (cid:12)(cid:12) is not an IP-set. In fact, the property of being an IP-set ispartition regular, so for any finite partition of T (cid:12)(cid:12) one element of the partition must be an IP-set and inparticular must be in Σ ∞ . But this contradicts Lemma 3.3. Let u be a prefix of T . Since T (cid:12)(cid:12) u ⊆ T (cid:12)(cid:12) itfollows that T (cid:12)(cid:12) u is not an IP-set. Proof of Theorem 3.1, part 3.
We will make use of the following lemma:
Lemma 3.4.
Let u be a factor of T which is neither a prefix of T nor a prefix of T . Then there exists anonnegative integer k such that one of the two following properties holds: For each x ∈ T (cid:12)(cid:12) u , [ x ] ends in k . For each x ∈ T (cid:12)(cid:12) u either [ x ] ends in k or in k +1 and both cases happen. Furthermore, u is aprefix of τ n ( aba ) for some nonnegative integer n , with a, b distinct letters.Proof. Let u be a factor of T which is neither a prefix of T nor a prefix of T and let { a, b } = { , } . If x isan occurrence of aa , then the x ) and x + 1) have the same parity, and hence [ x ] ends in and the statement is verified for all factors beginning with aa .We can then assume that u begins with ab and we will proceed by induction on | u | . Clearly the shortestsuch u is of the kind aba . Then for each x ∈ T u , the number of ’s in the binary expansion of x and of x + 2 have the same parity. It follows that [ x ] must end in or (and in fact it is easily verified that both arepossible). Thus the result of the lemma is verified with k = 0 .Next suppose | u | = N ≥ and that the claim is true for all factors u of length smaller than N . If u begins in aba , then T (cid:12)(cid:12) u ⊆ T (cid:12)(cid:12) aba and hence, as we have just seen if x ∈ T (cid:12)(cid:12) u we have that [ x ] must end ineither or . Otherwise u must begin in either or . In this case, let v denote the longest prefixof u which is either a prefix of T or of T . Then we can write u = vaλ where v begins in either or , a ∈ { , } and v ∈ { , } ∗ . Since both v and v are factors of T it follows that v = τ ( v ′ ) for some v ′ strictly shorter than v such that v ′ begins in or , and v ′ a is a factor of T which is neither a prefix of T nor of T . By the induction hypothesis we deduce that there exists a k such that for all x ′ ∈ T (cid:12)(cid:12) v ′ a we havethat [ x ′ ] ends in either k or in k +1 . Moreover, since every occurrence of va in T (and hence of u ) isthe image of τ of an occurrence in T of v ′ a it follows that if x ∈ T (cid:12)(cid:12) u then x = 2 x ′ for some x ′ ∈ T (cid:12)(cid:12) v ′ a .Whence [ x ] ends in either k +1 or in k +2 . We have thus proved that if u is neither a prefix of T nor of T and u begins in ab , then there exists a k such that for any x ∈ T (cid:12)(cid:12) u either [ x ] ends in k or in k +1 . If8nly one of these cases occurs, then clearly property 1 holds and we are done. Assume then that both casesoccur, we need to prove that u is a prefix of τ n ( aba ) for some nonnegative integer n (i. e. that we are incase 2). It is not difficult to prove (given the definition of τ ) that every factor of T of length at least eitherappears only in odd positions or only in even positions. Since we are assuming that there exist x, y ∈ T (cid:12)(cid:12) u such that [ x ] ends in k and [ y ] ends in k +1 , it must be k > and u occurs only in even positions.Again from the definition of τ , it is easy to see that if | u | is odd, then there exists a unique letter c such thatevery occurrence of u is followed by c . Hence there exists a unique α ∈ { , , ε } such that | uα | is even and T (cid:12)(cid:12) u = T (cid:12)(cid:12) uα . From the uniformity of τ , since uα is a factor of T of even length which appears only in evenpositions, there exists u ′ shorter than u such that τ ( u ′ ) = uα and T (cid:12)(cid:12) uα = { x, x ∈ T (cid:12)(cid:12) u ′ } . Hence, for each x ∈ T (cid:12)(cid:12) u ′ either [ x ] ends in k − or in k and both cases actually happen, thus, by induction hypothesis, u ′ is a prefix of τ n ( aba ) for some n and u is a thus a prefix of τ n +1 ( aba ) .We are now able to easily prove item 3. of our main theorem. First of all, let us observe that it is readilyverified that { , , } ⊆ T (cid:12)(cid:12) and { , , } ⊆ T (cid:12)(cid:12) and hence { n · , n · , n · } ⊆ T (cid:12)(cid:12) τ n (010) and { n · , n · , n · } ⊆ T (cid:12)(cid:12) τ n (101) which proves that if aba ∈ { , } , then T (cid:12)(cid:12) τ n ( aba ) are -summablefor every nonnegative integer n. Clearly then, if u is a prefix of some τ n ( aba ) , T (cid:12)(cid:12) u is - summable as well.Let u be a factor of T . In case point 1 of the preceding lemma holds, we have that there exists anonnegative integer k such that each x ∈ T (cid:12)(cid:12) u ends k . But then for any x, y ∈ T (cid:12)(cid:12) u , it follows that [ x + y ] ends in k +1 and hence x + y / ∈ T (cid:12)(cid:12) u .Thanks to point 2 of the preceding lemma we have thus proved that T (cid:12)(cid:12) u is -summable if and only if u is a prefix of τ n ( aba ) for some n and a, b distinct letters (considering that prefixes of T and of T are aswell prefixes of τ n ( aba ) ). We are left to prove that if u is neither a prefix T nor a prefix of T , then T (cid:12)(cid:12) u isnot -summable. Of course, the statement is trivial if T (cid:12)(cid:12) u is not -summable, so, as observed before, we canassume that point 2 of Lemma 3.4 holds, that is there exists k such that [ x ] ends in k or k for each x ∈ T u and both cases happen. Consider three points x, y, z ∈ T (cid:12)(cid:12) u . If [ x ] ends in k , then [ x + y ] endsin k or k ; in either way it follows that x + y / ∈ T (cid:12)(cid:12) u . On the other hand if [ x ] and [ y ] both end in k , then [ x + y ] ends in k and hence as above it follows that x + y + z / ∈ T (cid:12)(cid:12) u . It follows that T (cid:12)(cid:12) u isnot -summable, and the statement is complete. Remark.
We proved part 1 of Theorem 3.1 directly using the numeration system, though it actually followsfrom parts 2, 3, and the Finite Sums Theorem [5].As a corollary of Theorem 3.1 2. and Lemma 3.3 we obtain:
Corollary 3.5. Σ ∞ is not partition regular, i.e., there exists a set A ⊆ N which is infinite FS-big and apartition of A = A ∪ A such that neither A i is ∞ -summable. One of the two referees suggested the following alternative proof of Corollary 3.5. For each positiveinteger n, let supp ( n ) denote the support of the ternary expansion of n. Let D = F S ( h n i n ∈ N ) . We notethat for any m, n ∈ D, if m + n ∈ D then supp ( m ) ∩ supp ( n ) = ∅ . Let ( E i ) ∞ i =1 be a partition of N intoinfinite disjoint sets. For each i ∈ N set D i = { n ∈ D : supp ( n ) ⊆ E i and ( n ) ≤ i } and put A = S ∞ i =1 D i . Then clearly A ∈ Σ ∞ . However, let B and B be a partition of N so that no twointegers in N of the form x and x are both in B or both in B . For i ∈ { , } , put A i = { n ∈ A : ( n ) ∈ B i } .
9t is clear we cannot have x, y ∈ A i satisfying x + y ∈ A i and ( x ) = ( y ) . Thus A i is not ∞ -summable.We next derive two additional consequences of Theorem 3.1. For this purpose, we recall some termi-nology which will be needed. Let A be a finite non-empty set, and let A ω denote the set of all right infinitewords ( x n ) n ∈ N with x n ∈ A . We endow A ω with the topology generated by the metric d ( x, y ) = 12 n where n = min { k : x k = y k } whenever x = ( x n ) n ∈ N and y = ( y n ) n ∈ N are two elements of A ω . (This is also the product topology when A has the discrete topology.) Let T : A ω → A ω denote the shift transformation defined by T : ( x n ) n ∈ N ( x n +1 ) n ∈ N . A point x ∈ X is said to be uniformly recurrent in X if for every neighborhood V of x the set { n | T n ( x ) ∈ V } is syndetic, i.e., of bounded gap. Two points x, y ∈ A ω are said to be proximal if for every ǫ > there exists n ∈ N such that d ( T n ( x ) , T n ( y )) < ǫ. Let X be a closed and T -invariant subset of A ω ; the pair ( X, T ) is called a subshift of A ω . A subshift ( X, T ) is said to be minimal whenever X and the empty set are the only T -invariant closed subsets of X .To each x ∈ A ω is associated the subshift (Ω( x ) , T ) where Ω( x ) is the shift orbit closure of x . A point x ∈ A ω is called distal if the only point in Ω( x ) proximal to x is x itself. If x ∈ A ω is uniformly recurrent,then the associated subshift (Ω( x ) , T ) is minimal. And, if (Ω( x ) , T ) is minimal, then every point of Ω( x ) isuniformly recurrent. (For the proofs of the last two assertions see for example [4, Theorems 1.17 and 1.15].)It is well known that the Thue-Morse word is uniformly recurrent. (See for example [8, p. 832].)As an application of Theorem 3.1 we have the following corollary. In the proof of this corollary weuse some facts from [6] about the algebraic structure of the Stone- ˇCech compactification β N of N , thepoints of which are the ultrafilters on N . Given an ultrafilter p ∈ β N and a sequence h x n i ∞ n =1 in a compactHausdorff space X , p - lim n ∈ N x n is the unique point y ∈ X with the property that for every neighborhood U of y , { n ∈ N | x n ∈ U } ∈ p . Corollary 3.6.
The Thue-Morse word T is distal. In particular, for each n ≥ , exactly one of the sets { T n ( T ) (cid:12)(cid:12) , T n ( T ) (cid:12)(cid:12) } is an IP-set.Proof. Suppose x ∈ Ω( T ) is proximal to T . Then, since T is uniformly recurrent, we have by [6, Theorem19.26] that there exists a (minimal) idempotent ultrafilter p ∈ β N with p - lim n ∈ N T n ( T ) = x . Given a prefix u of x , U = { y ∈ A ω | u is a prefix of y } is a neighborhood of x so { n ∈ N | T n ( T ) ∈ U } ∈ p ; that is T (cid:12)(cid:12) u ∈ p .Therefore by [6, Theorem 5.12] T (cid:12)(cid:12) u is an IP-set. By Theorem 3.1 it follows that u is a prefix of T and hence x = T as required. Having established that T is distal, it follows that T n ( T ) is distal for each n ≥ . Finally,let us fix n ≥ , and let a ∈ { , } denote the initial symbol of T n ( T ) . We claim that T n ( T ) (cid:12)(cid:12) a is an IP-setwhile T n ( T ) (cid:12)(cid:12) a is not, where a := 1 − a. Since T n ( T ) is uniformly recurrent, it follows from [6, Theorem19.23] that there exists a idempotent ultrafilter p ∈ β N with p - lim m ∈ N T m ( T n ( T )) = T n ( T ) . Then as above, T n ( T ) (cid:12)(cid:12) a ∈ p so by [6, Theorem 5.12] we have that T n ( T ) (cid:12)(cid:12) a is an IP-set. Now suppose on the other handthat T n ( T ) (cid:12)(cid:12) a is also an IP-set. Then by [6, Theorem 5.12] there exists an idempotent q ∈ β N such that T n ( T ) (cid:12)(cid:12) a ∈ q . We claim that q - lim m ∈ N T m (cid:0) T n ( T ) (cid:1) is proximal to T n ( T ) for which it suffices by [6, Lemma19.22] to show that q - lim r ∈ N T r (cid:0) q - lim m ∈ N T m (cid:0) T n ( T ) (cid:1)(cid:1) = q - lim k ∈ N T k (cid:0) T n ( T ) (cid:1) . To this end10 - lim r ∈ N T r (cid:0) q - lim m ∈ N T m (cid:0) T n ( T ) (cid:1)(cid:1) = q - lim r ∈ N q - lim m ∈ N T r + m (cid:0) T n ( T ) (cid:1) by [6, Theorem 3.49] = ( q + q ) - lim k ∈ N T k (cid:0) T n ( T ) (cid:1) by [6, Theorem 4.5] = q - lim k ∈ N T k (cid:0) T n ( T ) (cid:1) . Since q - lim m ∈ N T m (cid:0) T n ( T ) (cid:1) is proximal to T n ( T ) and T n ( T ) is distal, we have that q - lim m ∈ N T m ( T n ( T )) = T n ( T ) . Thus, T n ( T ) (cid:12)(cid:12) a ∈ q from which it follows that ∅ = T n ( T ) (cid:12)(cid:12) a ∩ T n ( T ) (cid:12)(cid:12) a ∈ q , a contradiction. Corollary 3.7.
Let N be a positive integer and set x = t N t N − . . . t T ∈ Ω( T ) where T = t t t . . . . Consider the partition N = A ∪ A where A = x (cid:12)(cid:12) and A = x (cid:12)(cid:12) . Then A i − n is an IP-set for each i ∈ { , } and ≤ n ≤ N. On the other hand, for each n > N, exactly one of the sets { A − n, A − n } isan IP-set.Proof. For a ∈ { , } we put a = 1 − a. We first note that since a T ∈ Ω( T ) for some a ∈ { , } , by iteratively applying the morphism , we have that both t n t n − . . . t T ∈ Ω( T ) and t n t n − . . . t T ∈ Ω( T ) for each n ≥ . Fix a positive integer N and put x = t N t N − . . . t T and y = t N t N − . . . t T . Then for each ≤ n ≤ N , we have that T n ( x ) and T n ( y ) are proximal and begin indistinct symbols. Whence applying [6, Theorem 19.26 & Theorem 5.12 ] we deduce that A − n = T n ( x ) (cid:12)(cid:12) and A − n = T n ( x ) (cid:12)(cid:12) are both IP-sets for ≤ n ≤ N. On the other hand, applying Corollary 3.6 we seethat for each n > N, exactly one of the sets { A − n, A − n } is an IP-set. References [1] J.-P. Allouche and J. Shallit,
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