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

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 {∑n∈Fxn|∅≠F⊆{1,2,…,k}} for some k-term sequence of natural numbers 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 A⊆N is infinite FS-big if for each positive integer k, A contains {∑n∈Fxn|∅≠F⊆Nand#F⩽k} for some infinite sequence of natural numbers satisfying uniqueness of finite sums. We say A⊆N is an IP-set if A contains {∑n∈Fxn|∅≠F⊆Nand#F<∞} for some infinite sequence of natural numbers . By the Finite Sums Theorem (Hindman, 1974) [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 also partition regular. Let T=011010011001011010010110011010… denote the Thue–Morse word fixed by the morphism 0↦01 and 1↦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. Using the Thue–Morse word we show that the collection of all infinite FS-big sets is not partition regular.