Uniform Random Covering Problems

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.

Uniform approximation problems of expanding Markov maps

“…By letting monotonically decrease to along a sequence , we get immediately the expected lower bound for all . However, recent progresses in uniform approximation [9, 18, 20, 34] indicate that there is no mass transference principle for uniform approximation set. Therefore, we cannot expect that decreases linearly with respect to as does.…”

“…It is left to show the upper bound of dim H U κ (x) when 1/κ ≤ α max . The proof combines the methods developed in [13, §7] and [20,Theorem 8]. Heuristically, the larger the local dimension of a point is, the less likely it is to be hit.…”

“…x. However, we can conclude that U [20]. In their paper, instead of the orbit (T n x) n≥0 , they investigated the sets of points uniformly approximated by an independent and identically distributed sequence (ω n ) n≥1 .…”

“…(4) For μ φ -a.e.x, However, recent progresses in uniform approximation [9,18,20,34] indicate that there is no mass transference principle for uniform approximation set. Therefore, we cannot expect that dim H U κ (x) decreases linearly with respect to 1/κ as dim H L κ (x) does.…”

“…The main new ingredient of this paper is the difficult upper bound for when . To overcome the difficulty, we fully develop and combine the methods in [13, 20].…”