Additive combinatorics generated by uniformly recurrent words

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Date(s) - 16/09/2014
11 h 05 min - 12 h 00 min

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A subset A of the set N of natural numbers is called an IP-set if A contains all finite sums of distinct terms of some infinite sequence of natural numbers. We show how certain families of aperiodic words of low factor complexity may be used to generate a wide assortment of IP-sets having additional nice properties inherited from the rich combinatorial structure of the underlying word. We consider Sturmian words and their extensions to larger alphabets (so-called Arnoux-Rauzy words), as well as words generated by substitution rules. Our methods simultaneously exploit the general theory of combinatorics on words, the arithmetic properties of abstract numeration systems defined by substitution rules, notions from topological dynamics including proximality and equicontinuity, and the beautiful and elegant theory, developed by N. Hindman, D. Strauss and others, linking IP-sets to the algebraic/topological properties of the Stone-Cech compactification of N.

Together with the property of being IP-set, we consider two related additive properties of subsets of positive integers: We say that a subset A of N is finite (resp., infinite) FS-big if for each positive integer k the set A contains finite sums with at most k summands of some k-term (resp., infinite) sequence of natural numbers. By a celebrated result of Hindman (1974), 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. We prove that the collection of all finite FS-big sets is also partition regular. Using the Thue-Morse word we show that the collection of all infinite FS-big sets is not partition regular. The talk is based on joint work with M. Bucci, N. Hindman, L. Q. Zamboni.

References

M. Bucci, S. Puzynina, L. Q. Zamboni, Central sets generated by uniformly recurrent words, To appear in Ergodic Theory and Dynamical Systems. doi:10.1017/etds.2013.69

M. Bucci, N. Hindman, S. Puzynina, L. Q. Zamboni Additive properties of sets defined by the Thue-Morse word. Journal of Combinatorial Theory, Series A, 120 (2013), 1235–1245.

N. Hindman and D. Strauss, Algebra in the Stone-Cech compactification: theory and applications, 2nd edition, Walter de Gruyter & Co., Berlin, 2012.

S. Puzynina and L.Q. Zamboni, Additive properties of sets and substitutive dynamics. To appear in a forthcoming book “Recent Mathematical Developments in Aperiodic Order” edited by J. Kellendonk, D. Lenz and J. Savinien.

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