$2$-large sets are sets of Bohr recurrence

TL;DR

The paper proves that certain sets defined by diophantine approximation conditions are not large enough to guarantee the existence of arbitrarily long arithmetic progressions within any 2-coloring of natural numbers.

Contribution

It establishes that sets of Bohr recurrence defined by inequalities involving real numbers and a fixed delta are not 2-large, providing new insights into combinatorial number theory.

Findings

Sets of Bohr recurrence are not 2-large.

There exists a 2-coloring avoiding long arithmetic progressions with differences in these sets.

The result connects diophantine approximation with combinatorial coloring properties.

Abstract

Let $\alpha_1, \cdots, \alpha_d$ be real numbers, and let $S$ be the set of integers $s$ so that $||\alpha_i s||_{\mathbb{R}/\mathbb{Z}}>\delta$ for some $i$ and some fixed $\delta>0$. We prove $S$ is not \enquote{$2$-large}, i.e. there is a $2$-coloring of $\mathbb{N}$ that avoids arbitrarily long arithmetic progressions with common differences in $S$.