Hausdorff dimension and uniform exponents in dimension two

TL;DR

This paper determines the Hausdorff dimension of certain singular vectors in two dimensions based on their uniform exponents, revealing precise values and bounds that improve previous estimates and connect to the set of all singular vectors.

Contribution

It provides exact formulas and bounds for the Hausdorff dimension of sets of two-dimensional vectors with given uniform exponents, advancing understanding in Diophantine approximation.

Findings

Hausdorff dimension equals 2(1 - μ) for μ ≥ √2/2

Dimension exceeds 2(1 - μ) for μ < √2/2 and is bounded above by a specific function

Dimension approaches 4/3 as μ approaches 1/2

Abstract

In this paper we prove the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent $\mu$ $\in$ (1/2, 1) is 2(1 -- $\mu$) when $\mu$ $\ge$ $\sqrt$ 2/2, whereas for $\mu$ \textless{} $\sqrt$ 2/2 it is greater than 2(1 -- $\mu$) and at most (3 -- 2$\mu$)(1 -- $\mu$)/(1 + $\mu$ + $\mu$ 2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when $\mu$ tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. We also prove a lower bound on the packing dimension that is strictly greater than the Hausdorff dimension for $\mu$ $\ge$ 0.565. .. .