Generalized Hausdorff dimension of irrationals with Lagrange value exactly 3

Abstract

We study the generalized Hausdorff dimension of some natural subsets of $k^{-1}(3)$, where $k^{-1}(3)$ consists of the real numbers $x$ for which $\left|x-\frac{p}{q}\right|<\frac{1}{(3+\epsilon )q^2}$ has infinitely many rational solutions $\frac{p}{q}$ for any $\epsilon <0$ but only finitely many for any $\epsilon >0$. It is well known that $k^{-1}(3)$ is an uncountable set with Hausdorff dimension zero. Given any dimension function $h$, we determine the exact "cut point" at which the generalized Hausdorff dimension ${\mathcal{H}}^h(k^{-1}(3))$ drops from infinity to zero. In particular we show that such a measure is always zero or not $\sigma$--finite, and, as an application, we can classify topologically $k^{-1}(3)$. Moreover, we show that the subset of attainable elements of $k^{-1}(3)$ has the same generalized Hausdorff dimension as $k^{-1}(3)$, but the subset of non--attainable elements of $k^{-1}(3)$ has a "strictly smaller" generalized Hausdorff dimension.

arXiv commentThis revised 28-page version refines our previous result for $K_3$ and establishes a more precise generalized Hausdorff dimension for the non--attainable elements of $k^{-1}(3)$