On inhomogeneous Diophantine approximation and Hausdorff dimension

TL;DR

This paper investigates the Hausdorff dimension of sets of points in ^n that are well-approximated by a dense subgroup, establishing a precise relationship between the approximation exponent and the dimension.

Contribution

It provides a new formula for the Hausdorff dimension of v-approximable points relative to inhomogeneous Diophantine approximation with dense subgroups.

Findings

Hausdorff dimension of B_v equals 1/v for v  ^n

Dimension result holds for any v   with v  (A)

Extends classical Diophantine approximation results to inhomogeneous dense subgroups

Abstract

Let $\Gamma = Z A +Z^n$ be a dense subgroup with rank $n+1$ in $R^n$ and let $\omega(A)$ denote the exponent of uniform simultaneous rational approximation to the point $A$. We show that for any real number $v\ge \omega(A)$, the Hausdorff dimension of the set $B_v$ of points in $R^n$ which are $v$-approximable with respect to $\Gamma$, is equal to $1/v$.