$2$-large sets are sets of Bohr recurrence

Let ${\alpha }_1,\cdots ,{\alpha }_d$ be real numbers, and let $S$ be the set of integers $s$ so that $\mathrm{\mid }\mathrm{\mid }{\alpha }_{i}s\mathrm{\mid }{\mathrm{\mid }}_{\mathbb{R}\mathrm{/}\mathbb{Z}}>\delta$ 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 $S$.