$2$-large sets are sets of Bohr recurrence

Abstract

Let $α_1, \cdots, α_d$ be real numbers, and let $S$ be the set of integers $s$ so that $||α_i s||_{\mathbb{R}/\mathbb{Z}}>δ$ for some $i$ and some fixed $δ>0$. 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 $S$.