$2$-large sets are sets of Bohr recurrence

Abstract

Let ${\alpha }_1,\dots ,{\alpha }_d$ be real numbers, fix $\delta >0$, and let $S$ be the set of positive integers $s$ such that $\Vert {\alpha }_{i}s{\Vert }_{\mathbb{R}/\mathbb{Z}}>\delta$ for some $1\le i\le d$. We prove that $S$ is not \enquote{$2$-large}, i.e. there is a $2$-coloring of $\mathbb{N}$ that avoids arbitrarily long monochromatic arithmetic progressions with common differences in $S$. The case $d=1$ was previously known; here we prove the result in every finite dimension.

arXiv comment7 pages, 1 figure, comments welcome!