On combinatorial properties of nil-Bohr sets of integers and related problems

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. Introduction Background Additive combinatorics Basic definitions Group actions Ergodic theory in additive combinatorics IP sets Nilsystems Nil-Bohr sets Filtered groups and polynomial sequences Mal'cev coordinates and generalised polynomials Equidistribution Higher order Fourier analysis Overview Nil–Bohr type sets as bases for the positive integers Introduction Non-bases of order 2 General strategy Quadratic irrationals Badly approximable reals Generic reals Bases and almost bases of order 2 Equidistribution and quantitative rationality Almost bases of order 2 Exceptional values of alpha Threshold for being a basis of order 2 Quadratic irrationals The algorithmic approach Bases of order 3 Set-up Minor arcs Major arcs Main contribution Higher degrees Bases of order 2 Non-bases of order 2 Weakly mixing sets and polynomial equations Introduction Definitions Uniform ergodic theorem Outline and initial reductions Uniform convergence for linear polynomials PET induction Definitions and basic properties Uniform convergence in higher degrees Doubly polynomial averages Initial reductions Polynomial Følner averages Concluding remarks Combinatorial characterisation of nil–Bohr sets of integers Introduction Polynomial maps Main results reformulated Connectivity VIP-systems Host-Kra cube groups Host-Kra cubes and nilmanifolds Sk-sequences IP sets revisited Basic definitions Asymptotic subsequences Stable sequences Stable polynomials Basic results Abelian case Case d=2 Main results Robust version and induction Reduction to an abelian problem A counterexample Model problem Patterns Perturbations Final step Proof of Theorem IV.1.1 Automatic sequences and generalised polynomials Introduction Automatic sequences Density 1 results Polynomial sequences Generalised polynomials Sparse sets Arid sets Proof strategy Comments and applications Sparse generalised polynomials Preliminaries Initial reductions Fractional parts and limits Fractional parts of polynomials Group generated by fractional parts Sparse automatic sets Density of symbols Dichotomy for sparse automatic sets IP rich automatic sets Proof of Theorem V.1.6 Examples Small fractional parts IP rich sequences Very sparse sequences Exponential sequences Automaticity of recursive sequences Exponentially sparse generalised polynomial sets Quadratic Pisot numbers Cubic Pisot numbers Concluding remarks Small fractional parts Exponential sequences Morphic words Regular sequences Uniformity of automatic sequences Introduction Thue-Morse sequence Rudin-Shapiro sequence Closing remarks Continued fractions Basic definitions Ergodic perspective Good rational approximations Ultrafilters and limits Bibliography