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arXiv · 1704.04691

A Fourier analytic approach to inhomogeneous Diophantine approximation

Han Yu
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Abstract

In this paper, we study inhomogeneous Diophantine approximation with rational numbers of reduced form. The central object to study is the set $W(f,θ)$ as follows, \begin{eqnarray*} \left\{x\in [0,1]:\left |x-\frac{m+θ(n)}{n}\right|<\frac{f(n)}{n}\text{ for infinitely many coprime pairs of numbers } m,n\right\}, \end{eqnarray*} where $\{f(n)\}_{n\in\mathbb{N}}$ and $\{θ(n)\}_{n\in\mathbb{N}}$ are sequences of real numbers in $[0,1/2]$. We will completely determine the Hausdorff dimension of $W(f,θ)$ in terms of $f$ and $θ$. As a by-product, we also obtain a new sufficient condition for $W(f,θ)$ to have full Lebesgue measure and this result is closely related to the study of \ds with extra conditions.

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Han Yu. 2018-09-26. A Fourier analytic approach to inhomogeneous Diophantine approximation. https://arxiv.org/abs/1704.04691

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