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arXiv · 1207.3615

Hausdorff dimension of affine random covering sets in torus

Esa Järvenpää·Maarit Järvenpää·Henna Koivusalo·Bing Li·Ville Suomala
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Abstract

We calculate the almost sure Hausdorff dimension of the random covering set $\limsup_{n\to\infty}(g_n + ξ_n)$ in $d$-dimensional torus $\mathbb T^d$, where the sets $g_n\subset\mathbb T^d$ are parallelepipeds, or more generally, linear images of a set with nonempty interior, and $ξ_n\in\mathbb T^d$ are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.

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Esa Järvenpää, Maarit Järvenpää, Henna Koivusalo, Bing Li, Ville Suomala. 2013-05-07. Hausdorff dimension of affine random covering sets in torus. https://arxiv.org/abs/1207.3615

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