Badly approximable numbers for sequences of balls

TL;DR

This paper extends classical results in Diophantine approximation by showing that the set of badly approximable numbers with respect to general sequences of Euclidean balls also has Lebesgue measure zero, providing a new proof of this fact.

Contribution

It generalizes the measure-zero property of badly approximable numbers to broader sequences of balls and offers a new proof of the classical result.

Findings

The set of badly approximable numbers with respect to general sequences of balls has Lebesgue measure zero.

Under natural conditions, the measure-zero property holds for these generalized sets.

The approach provides a new proof of the classical measure-zero result for badly approximable numbers.

Abstract

It is a classical result from Diophantine approximation that the set of badly approximable numbers has Lebesgue measure zero. In this paper we generalise this result to more general sequences of balls. Given a countable set of closed $d$-dimensional Euclidean balls $\{B(x_{i},r_{i})\}_{i=1}^{\infty},$ we say that $\alpha\in \mathbb{R}^{d}$ is a badly approximable number with respect to $\{B(x_{i},r_{i})\}_{i=1}^{\infty}$ if there exists $\kappa(\alpha)>0$ and $N(\alpha)\in\mathbb{N}$ such that $\alpha\notin B(x_{i},\kappa(\alpha)r_{i})$ for all $i\geq N(\alpha)$. Under natural conditions on the set of balls, we prove that the set of badly approximable numbers with respect to $\{B(x_{i},r_{i})\}_{i=1}^{\infty}$ has Lebesgue measure zero. Moreover, our approach yields a new proof that the set of badly approximable numbers has Lebesgue measure zero.