Inhomogeneous Diophantine approximation in the coprime setting

Abstract

Given n∈N and x,γ∈R, let||γ−nx||′=min⁡{|γ−nx+m|:m∈Z,gcd⁡(n,m)=1}, Two conjectures in the coprime inhomogeneous Diophantine approximation state that for any irrational number α and almost every γ∈R,lim infn→∞n||γ−nα||′=0 and that there exists C>0, such that for all α∈R\\Q and γ∈[0,1),lim infn→∞n||γ−nα||′<C. We prove the first conjecture and disprove the second one.