Abstract
Abstract Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence $\omega =(\omega _n)_{n\geq 1}$ uniformly distributed on the unit circle $\mathbb{T}$ and a sequence $(r_n)_{n\geq 1}$ of positive real numbers with limit $0$. We investigate the size of the random set $$\begin{align*} & {\operatorname{{{\mathcal{U}}}}} (\omega):=\{y\in \mathbb{T}: \ \forall N\gg 1, \ \exists n \leq N, \ \text{s.t.} \ | \omega_n -y | < r_N \}. \end{align*}$$Some sufficient conditions for ${\operatorname{{{\mathcal{U}}}}}(\omega )$ to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that ${\operatorname{{{\mathcal{U}}}}}(\omega )$ is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.
Highlights
Let T = R/Z be the one-dimensional torus
The corresponding uniform approximation problem was quite recently studied by Kim and Liao [10] who proved that the Hausdorff dimension of the set
That for any n there is an event An such that A and An only differ on a set of null probability, and An is measurable with respect to the σ -algebra σ (Xn, Xn+1, . . . )
Summary
In 2003, Bugeaud [2] and, independently, Schmeling and Troubetzkoy [16] proved that for any irrational θ , for any α > 1, the Hausdorff dimension of the set. The corresponding uniform approximation problem was quite recently studied by Kim and Liao [10] who proved that the Hausdorff dimension of the set. When rn decreases to 0 faster, one is interested in the Hausdorff dimension of the set of points that are covered infinitely often by the random intervals. For an i.i.d. random sequence ω = (ωn)n≥1 of uniform distribution and a real positive sequence (rn)n≥1, we want to describe the size (in the sense of Lebesgue measure and Hausdorff dimension) of the random set.
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