Inhomogeneous Diophantine approximation in the coprime setting

Abstract

Given $n\in N$ and $x,\gamma \in R$, let \begin{equation*} ||γ-nx||^\prime=\min\{|γ-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||γ-nα||^{\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||γ-nα||^{\prime} < C. \end{equation*} We prove the first conjecture and disprove the second one.

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• Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase
• On shrinking targets for Z^m actions on tori
• Diophantine approximation, Khintchine's theorem, torus geometry and Hausdorff dimension
• Inhomogeneous approximation by coprime integers