Laboratory of Mathematical Logic at PDMI

Пятница, 9 ноября, ауд. 203. Начало в 18:00.

Докладчик: Светлана Пузынина (Турку).

Тема: IP-sets defined by words of low factor complexity.

Abstract

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 x_n, n\in N, 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 higher alphabets (so-called Arnoux-Rauzy words), as well as words generated by substitution rules including the famous Thue-Morse word. 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. Using the key notion of p-lim_n, regarded as a mapping from words to words, we apply ideas from combinatorics on words in the framework of ultrafilters.

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 FS-big if for each positive integer k the set A contains finite sums with at most k summands of some k-term sequence of natural numbers x_1 The talk is based on joint work with M. Bucci, N. Hindman, L. Q. Zamboni.

References

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

M. Bucci, S. Puzynina and L.Q. Zamboni, Words, Numerations and the Stone-Cech compactification of N. To appear in a forthcoming book ``Recent Mathematical Developments in Aperiodic Order" edited by J. Kellendonk, D. Lenz and J. Savinien.

M. Bucci, S. Puzynina, L. Q. Zamboni, Central Sets Defined by Words of Low Factor Complexity. Submitted.

M. Bucci, N. Hindman, S. Puzynina, L. Q. Zamboni Additive properties of sets deined by the Thue-Morse word. Submitted. Preliminary version reported at JM2012.

Доклад состоится на русском языке.