On combinatorial properties of nil-Bohr sets of integers and related problems - Konieczny, J (2017)
This thesis deals with six problems in additive combinatorics and ergodic theory. A brief introduction to this general area and a summary of included results is given in Chapter I. In Chapter II, we consider sets of the form { n ϵ ℕ0 | |p(n) mod 1| ≤ ϵ (n) }, where p is a polynomial and ϵ(n) ≥ 0. We obtain various conditions under which any sufficiently large integer can be represented as a sum of 2 or 3 elements of a given set of this form. In Chapter III, we study the class of weakly mixing sets of integers, and prove that a certain class of polynomial equations can always be solved in such a set. In Chapter IV, we show that any nil--Bohr set contains a certain type of an additive pattern. Combined with earlier results of Host and Kra, his leads to a partial combinatorial characterisation of nil--Bohr sets. In Chapter V, we study the combinatorial properties of generalised polynomials (expressions built from polynomials and the floor function). In contrast with results of Bergelson and Leibman, we show that if the set of integers where a given generalised polynomial takes a non-zero value has asymptotic density 0, then it does not contain any IP set. This leads to a partial characterisation of automatic sequences which are given by generalised polynomial formulas. In Chapter V, we estimate the Gowers norms of the Thue-Morse sequence and the Rudin-Shapiro sequence. This gives some of the simplest deterministic examples of sequences with small Gowers norms of all orders.