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

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:120

Article

Bucci, Michelangelo^1,2  Hindman, Neil³  Puzynina, Svetlana^1,2  Zamboni, Luca Q.^1,2,4 

[1] Univ Turku, Dept Math & Stat, SF-20500 Turku, Finland

[2] Univ Turku, FUNDIM, SF-20500 Turku, Finland

[3] Howard Univ, Dept Math, Washington, DC 20059 USA

[4] Univ Lyon 1, F-69622 Villeurbanne, France

关键词: Partition regularity;    Additive combinatorics;    IP-sets;    Thue-Morse infinite word;   

DOI  :  10.1016/j.jcta.2013.03.005

来源: Elsevier

【 摘 要 】

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 is an element of N) if A contains {Sigma(n is an element of F) x(n) vertical bar theta not equal F subset of {1, 2, . . . , k}} for some k-term sequence of natural numbers < x(t)>(k)(t=1) satisfying uniqueness of finite sums. We say A subset of N is finite FS-big if A is k-summable for each positive integer k. We say A subset of N is infinite FS-big if for each positive integer k, A contains {Sigma(n is an element of F) x(n) vertical bar theta not equal F subset of N and #F <= k} for some infinite sequence of natural numbers < x(t)>(infinity)(t=1) satisfying uniqueness of finite sums. We say A subset of N is an IP-set if A contains {Sigma(n is an element of F) x(n) vertical bar theta not equal F subset of N and #F < infinity) for some infinite sequence of natural numbers < x(t)>(infinity)(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 bar right arrow 01 and 1 bar right arrow 10. For each factor u of T we consider the set T vertical bar(u) subset of N of all occurrences of u in T. In this note we characterize the sets T vertical bar(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. (C) 2013 Elsevier Inc. All rights reserved.

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