Hausdorff dimension and uniform exponents in dimension two

Abstract

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 These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent.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