A Frostman type lemma for sets with large intersections, and an application to Diophantine approximation

Abstract

We consider classes ${\mathcal{G}}^s([0,1])$ of subsets of $[0,1]$, originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least $s$. We provide a Frostman type lemma to determine if a limsup-set is in such a class. Suppose $E=\mathrm{limsup}{E}_n\subset [0,1]$, and that ${\mu }_n$ are probability measures with support in $E_n$. If there is a constant $C$ such that \[\iint|x-y|^{-s}\, \mathrm{d}μ_n(x)\mathrm{d}μ_n(y)<C\] for all $n$, then under suitable conditions on the limit measure of the sequence $({\mu }_n)$, we prove that the set $E$ is in the class ${\mathcal{G}}^s([0,1])$. As an application we prove that for $\alpha >1$ and almost all $\lambda \in (\frac{1}{2},1)$ the set \[ E_λ(α) = \{\,x\in[0,1] : |x - s_n| < 2^{-αn} \text{infinitely often}\ \}\] where $s_n\in \{(1-\lambda ){\sum }_{k=0}^n{a}_k{\lambda }^k$ and $a_k\in \{0,1\}\}$, belongs to the class ${\mathcal{G}}^s$ for $s\le \frac{1}{\alpha }$. This improves one of our previous results.

1+20 pages; Erratum added

References in corpus (1)

• On the Diophantine properties of lambda-expansions

Cited by in corpus (3)

• A Note on Random Coverings of Tori
• Hausdorff dimension of limsup sets of rectangles in the Heisenberg group
• On Shrinking Targets for Piecewise Expanding Interval Maps