On additive properties of sets defined by the Thue-Morse word

In this paper we study some additive properties of subsets of N : A subset A of N is called ksummable (k a positive integer) if A contains {n∈F xn |F ⊆ {0, . . . , k − 1}} for some k-term sequence of natural numbers x0 < x1 < x2 < · · · < xk−1. We say A ⊆ N is FS big if A is k-summable for each positive integer k. We say is A ⊆ N is infinite FS big if for each positive integer k, A contains {n∈F xn |F ⊂ N ; #F ≤ k} for some infinite sequence of natural numbers x0 < x1 < x2 < · · · . We say A ⊂ N is an IP-set if A contains { ∑ n∈F xn |F ⊂ N ; #F < ∞} for some infinite sequence of natural numbers x0 < x1 < x2 < · · · . By a celebrated result of N. Hindman [3], the property of being an IP-set 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. Recently the authors proved that the property of being FS-big is also partition regular. Let T = 011010011001011010 . . . denote the Thue-Morse word fixed by the morphism 0 7→ 01 and 1 7→ 10. For each factor u of T we consider the set T ∣∣ u ⊆ N of all occurrences of u in T. In this note we characterize the sets T∣∣ u in terms of the additive properties defined above. As a consequence we show that the property of being infinite FS-big is not partition regular.