On arithmetic progressions in non-periodic self-affine tilings

On arithmetic progressions in model sets

In this project we show the existence of arbitrary length arithmetic progressions in model sets and Meyer sets in the Euclidean $d$-space. We prove a van der Waerden type theorem for Meyer sets. We show that pure point subsets of Meyer sets with positive density and pure point diffraction contain arithmetic progressions of arbitrary length.

A dichotomy for the stability of arithmetic progressions

Let H stand for the set of homeomorphisms on [0,1]. We prove the following dichotomy for Borel subsets A of [0,1]: either there exists a homeomorphism f in H such that the image f(A) contains no 3-term arithmetic progressions; or, for every f in H, the image f(A) contains arithmetic progressions of arbitrary finite length. In fact, we show that the first alternative holds if and only if the set A is meager (a countable union of nowhere dense sets).

Group automorphisms with prescribed growth of periodic points, and small primes in arithmetic progressions in intervals

We investigate the question of which growth rates are possible for the number of periodic points of a compact group automorphism. Our arguments involve a modification of Linnik's Theorem, concerning small prime numbers in arithmetic progressions which lie in intervals.

Arithmetic progressions and the primes - El Escorial lectures

We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length $k$ in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes.

具有任意长等差数列的数集的豪斯多夫维数

Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions

本文研究了偏商序列严格递增且包含任意长等差数列的无理数的豪斯多夫维数。

关于二进制Morse类词中的长等差数列

On long arithmetic progressions in binary Morse-like words

本文研究了二进制Morse类词中的长等差数列存在性，提出了在Thue-Morse词及广义Thue-Morse词中的相关结果，其论证方法受启发于van der Waerden关于整数有限着色中单色等差数列存在性的证明。