Colloquium Mathematics, Dr. Jakub Byszewski (Jagiellonian University, Krakow) / Sparse generalized polynomials and nilmanifolds

Title:

Sparse generalized polynomials and nilmanifolds

Abstract:

We investigate generalised polynomials (i.e., polynomial-like expressions involving the use of the floor function) which take the value 0 at all integers except for a set of density 0. By a theorem of Bergelson--Leibman, generalised polynomials can be completely described in terms of dynamics on nilmanifolds.
Our main result is that the set of integers where a sparse generalised polynomial takes non-zero value cannot be combinatorially rich (specifically, cannot contain a translate of an IP set). We study some explicit constructions and show that the characteristic functions of the Fibonacci and Tribonacci numbers are given by generalised polynomials. Finally, we show that any sufficiently sparse {0,1}-valued sequence is given by a generalised polynomial. We apply these results to a question on automatic sequences.

This is joint work with Jakub Konieczny (Jerusalem)

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