Inhomogeneous Diophantine approximation in the coprime setting

Given $nin N$ and $x,gammain R$, let \begin{equation*} ||gamma-nx||^prime=min{|gamma-nx+m|:min Z, gcd (n,m)=1}, end{equation*} %where $(n,m)$ is the largest common divisor of $n$ and $m$. Two conjectures in the coprime inhomogeneous Diophantine approximation state that for any irrational number $alpha$ and almost every $gammain R$, \begin{equation*} liminf_{n o infty}n||gamma -nalpha||^{prime}=0 end{equation*} and that there exists $C>0$, such that for all $alphain R\backslash Q$ and $gammain [0,1)$ , \begin{equation*} liminf_{n o infty}n||gamma -nalpha||^{prime} < C. end{equation*} We prove the first conjecture and disprove the second one.

Inhomogeneous Diophantine approximation in the coprime setting Institutional Repository Document