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.
Author(s): Jakub Konieczny
Series: PhD thesis
Publisher: University of Oxford
Year: 2017
Language: English
Pages: 231
City: Oxford
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
