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

Let {n k } be an increasing lacunary sequence, that is, n k + 1 /n k > 1 + ϵ for some ϵ > 0. In 1987, P. Erdős asked for the chromatic number χ(G) of a graph G with vertex set ℤ, where two integers x, y ∈ ℤ are connected by an edge if and only if their difference |x − y| is in the sequence {n k }. Y. Katznelson found a connection to a Diophantine approximation problem (also due to Erdős): the existence of θ ∈ (0, 1) such that all the multiples n j θ are at a least distance δ(θ) > 0 from ℤ. Katznelson showed that χ(G) ≤ Cϵ −2 |log ϵ|. We apply the Lovasz local lemma to establish that δ(θ) > cϵ |log ϵ| −1 for some θ, which implies that χ(G) < Cϵ −1 |log ϵ|. This is sharp up to the logarithmic factor.