$2$-large sets are sets of Bohr recurrence

Abstract

Let $\hat{I}{\pm }_1,\cdots ,\hat{I}{\pm }_d$ be real numbers, and let $S$ be the set of integers $s$ so that $||\hat{I}{\pm }_{i}s|{|}_{\mathbb{R}/\mathbb{Z}}>\hat{I}´$ for some $i$ and some fixed $\hat{I}´>0$. 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 $S$.

6 pages, 1 figure, comments welcome!