$2$-large sets are sets of Bohr recurrence
Ryan Alweiss
TL;DR
The paper proves that if a set S of positive integers fails to be a set of Bohr recurrence for some fixed dimensions (i.e., there exist α1,...,αd with ||sαi||>ε for all s∈S and some i), then S is not 2-large. The authors construct a sophisticated 2-coloring of the integers via a map into a high-dimensional torus and a random coloring of a fundamental domain, proving that any sufficiently long arithmetic progression with differences in S must be bichromatic. The argument combines a detailed torus-based coloring, a line-orbit counting analysis, and divisibility arguments to force monochromatic avoidance, yielding a strong link between Bohr recurrence properties and 2-largeness. The work suggests broader applicability to nilsystems and distal dynamical systems and contributes to the Large Sets conjecture by bridging combinatorial largeness with dynamical recurrence notions.
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 $S$.