Dvoretzky covering problem for general measures
Roope Anttila, Markus Myllyoja
TL;DR
This work resolves the Dvoretzky covering problem for random coverings driven by general Borel measures on the line by introducing a generalized capacity $\mathrm{Cap}_{\mu,\underline{r}}$ and a refined energy framework, proving that an analytic set $A$ is almost surely covered by $E_{\underline{r}}$ iff $\mathrm{Cap}_{\mu,\underline{r}}(A\cap X_{\mu,\underline{r}})=0$. The authors overcome nonuniformity of general measures with the Jankov-von Neumann uniformisation to obtain a deterministic, full-measure set for each measure, and then lift this to the whole analytic set. They also derive a polynomial-radius critical-exponent result in $\mathbb{R}^d$ and apply the main theorem to natural measures on strongly separated self-conformal sets, revealing a multifractal dependence of the critical constant via the spectrum $f(\alpha)$. In the Cantor-set case, they provide explicit bounds for the critical constant and numerically sharpen these using average densities and $n$-step Bernoulli measures, illustrating the intricate role of multifractality. Overall, the paper advances the understanding of random covering phenomena beyond Lebesgue-type measures and demonstrates the power of capacity-based and descriptive-set-theoretic techniques in geometric probability.
Abstract
We study the Dvoretzky covering problem for random covering sets driven by general Borel probability measures. As our main result, we solve the problem of covering analytic sets by random covering sets generated by arbitrary Borel probability measures on the real line. Prior to this work, a complete solution was not known for any singular measure. Our solution is potential theoretic and involves a generalisation of a notion of capacity in the work of Kahane, who solved the problem of covering compact sets in the classical setting where the random covering process is driven by the Lebesgue measure on the unit circle. One of our key innovations is a simple but powerful application of the Jankov-von Neumann uniformisation theorem, which we believe to have interest outside of this work. In addition, we determine the critical exponent for the covering problem for polynomially decreasing sequences $(cn^{-t})_n$ for random covering sets driven by Borel probability measures on $\mathbb{R}^d$. At exactly the critical exponent, the covering property generally depends on the constant $c>0$, and as an application of our main result, we determine the critical constant for random covering sets driven by natural measures on strongly separated self-conformal sets on the line. The critical constant depends on the multifractal structure of the average densities of the measure, and the result is new even for the simplest case of the Hausdorff measure on the Cantor set.