Inhomogeneous Diophantine approximation in the coprime setting

TL;DR

This paper investigates two conjectures in coprime inhomogeneous Diophantine approximation, proving one and disproving the other, thereby advancing understanding of approximation properties involving coprimality constraints.

Contribution

The paper proves the first conjecture about the limit infimum being zero for almost every gamma and disproves the second conjecture regarding a uniform bound C.

Findings

Proved that for almost every gamma, the limit infimum is zero.

Disproved the existence of a universal constant C for all irrationals and gamma.

Clarified the behavior of coprime inhomogeneous Diophantine approximation limits.

Abstract

Given $n\in N$ and $x,\gamma\in R$, let \begin{equation*} ||\gamma-nx||^\prime=\min\{|\gamma-nx+m|:m\in 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 $\gamma\in R$, \begin{equation*} \liminf_{n\to \infty}n||\gamma -n\alpha||^{\prime}=0 \end{equation*} and that there exists $C>0$, such that for all $\alpha\in R\backslash Q$ and $\gamma\in [0,1)$ , \begin{equation*} \liminf_{n\to \infty}n||\gamma -n\alpha||^{\prime} < C. \end{equation*} We prove the first conjecture and disprove the second one.