Badly approximable numbers for sequences of balls

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 $α\in \mathbb{R}^{d}$ is a badly approximable number with respect to $\{B(x_{i},r_{i})\}_{i=1}^{\infty}$ if there exists $κ(α)>0$ and $N(α)\in\mathbb{N}$ such that $α\notin B(x_{i},κ(α)r_{i})$ for all $i\geq N(α)$. 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.

7 pages. After completing this paper the author was made aware of a result due to Cassels. We now know that the work done in this paper is in fact a reasonably straightforward consequence of this result