Certain results on uniform circle random covering problems

TL;DR

This work extends the uniform random circle covering paradigm to general subsets $A$ of the circle by analyzing when $A$ is contained in the liminf of unions of random arcs with radii $r_n$. The authors introduce a dimension-based threshold $\delta = \liminf_{n\to\infty}(n\ell_n)/\ln n$ and prove a dichotomy: if $\delta < \dim_H(A)$ then $A$ is almost surely not fully covered, with the uncovered portion retaining at least $\dim_H(A)-\delta$ dimensionality; conversely, under a suitable summability condition on $\ell_n$, $A$ is almost surely contained in the uniform random covering. The analysis blends Dvoretzky covering techniques with fractal geometry, yielding explicit thresholds in the common parametrization $\ell_n = c\frac{\ln n}{n}$ and connecting covering properties to both Hausdorff and box-counting dimensions. The results illuminate when uniform random coverings can (and cannot) realize full coverage, and highlight stability limits (e.g., no perturbation-based recovery for $A=\mathbb{T}$) within this probabilistic geometric framework.

Abstract

In this note we extend a theorem from [13] about uniform circle random coverings

Theorems & Definitions (9)

• Theorem 1
• Corollary 1
• Corollary 2
• proof : Proof of Theorem \ref{['M1']}-1)
• Lemma 1.1
• proof
• Theorem 1.2
• proof
• proof : Proof of Corollary \ref{['cor:covering_fail']}