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 ∈ F x n | ∅ ≠ F ⊆ { 1 , 2 , … , k } } for some k-term sequence of natural numbers 〈 x t 〉 t = 1 k 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 ∈ F x n | ∅ ≠ F ⊆ N and # F ⩽ k } for some infinite sequence of natural numbers 〈 x t 〉 t = 1 ∞ satisfying uniqueness of finite sums. We say A ⊆ N is an IP-set if A contains { ∑ n ∈ F x n | ∅ ≠ F ⊆ N and # F < ∞ } for some infinite sequence of natural numbers 〈 x t 〉 t = 1 ∞ . 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.