On a question of Gowers related to Littlewood's conjecture

Abstract

In a blogpost in 2009, Gowers raised a possible approach to Littlewood's conjecture in Diophantine approximation, leading to a question about the existence of sufficiently many points in the unit cube such that "hyperbolic distance" between any two of them is large. In this note we answer this question by an explicit construction. This shows that this approach to prove Littlewood's conjecture cannot work, unless some further refinements are added.