Generalized Hausdorff dimension of irrationals with Lagrange value exactly 3
Carlos Gustavo Moreira, Harold Erazo, Nicolas Angelini
TL;DR
The authors study the generalized Hausdorff measure of the Lagrange-irrational set $k^{-1}(3)$, characterizing precisely when $\mathcal{H}^h(k^{-1}(3))$ is infinite or zero in terms of the gauge function $h$. They prove a sharp dichotomy: if $\limsup_{\varepsilon\to0} \frac{\log h(\varepsilon)}{\log \varepsilon}=0$ then $\mathcal{H}^h(k^{-1}(3))=\infty$, and if the limsup is positive then $\mathcal{H}^h(k^{-1}(3))=0$, with the additional result that the attainable part $A_3$ carries the full measure while the non-attainable part $B_3$ has strictly smaller dimension. The paper further shows that $k^{-1}(3)$ is not $G_\delta$ or $F_\sigma$ but is $F_{\sigma\delta}$, and it connects these size results to properties of Liouville numbers, including non-$\sigma$-finite measures in certain gauge regimes. A Cantor-set construction demonstrates the infinite-measure case, while a zero-measure criterion is derived for a Cantor subset $K_3$, highlighting a rich topological and fractal structure underlying $k^{-1}(3)$ and its subsets.
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 $\left| x-\frac{p}{q} \right|<\frac{1}{(3+\varepsilon)q^2}$ has infinitely many rational solutions $\frac{p}{q}$ for any $\varepsilon<0$ but only finitely many for any $\varepsilon>0$. 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 $σ$--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.