Two Erdos problems on lacunary sequences: Chromatic number and Diophantine approximation

TL;DR

This paper investigates Erdos's problems on lacunary sequences, establishing bounds on the chromatic number of certain graphs and related Diophantine approximation properties, using probabilistic methods to improve existing bounds.

Contribution

The authors apply the Lovász local lemma to derive new bounds on the Diophantine approximation problem and the chromatic number, improving previous results by Katznelson.

Findings

Chromatic number of the graph is at most proportional to r^{-1} times log r.

Existence of x with all multiples n_j x at least c r / log r away from integers.

Bounds are sharp up to logarithmic factors.

Abstract

Let ${n_k}$ be an increasing lacunary sequence, i.e., $n_{k+1}/n_k>1+r$ for some $r>0$. In 1987, P. Erdos asked for the chromatic number of a graph $G$ on the integers, where two integers $a,b$ are connected by an edge iff their difference $|a-b|$ is in the sequence ${n_k}$. Y. Katznelson found a connection to a Diophantine approximation problem (also due to Erdos): the existence of $x$ in $(0,1)$ such that all the multiples $n_j x$ are at least distance $\delta(x)>0$ from the set of integers. Katznelson bounded the chromatic number of $G$ by $Cr^{-2}|\log r|$. We apply the Lov\'asz local lemma to establish that $\delta(x)>cr|\log r|^{-1}$ for some $x$, which implies that the chromatic number of $G$ is at most $Cr^{-1} |\log r|$. This is sharp up to the logarithmic factor.