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arXiv · 1302.0954

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

Tomas Persson·Henry W. J. Reeve
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Abstract

We consider classes $\mathscr{G}^s ([0,1])$ 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 $E = \limsup E_n \subset [0,1]$, and that $μ_n$ 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) 1$ and almost all $λ\in (\frac{1}{2},1)$ the set \[ E_λ(α) = \{\,x\in[0,1] : |x - s_n| < 2^{-αn} \text{infinitely often}\ \}\] where $s_n \in \{\,(1-λ)\sum_{k=0}^na_kλ^k$ and $a_k\in\{0,1\}\,\}$, belongs to the class $\mathscr{G}^s$ for $s \leq \frac{1}α$. This improves one of our previous results.

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Tomas Persson, Henry W. J. Reeve. 2017-09-11. A Frostman type lemma for sets with large intersections, and an application to Diophantine approximation. https://doi.org/10.1017/s0013091514000066

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