A note on the hitting probabilities of random covering sets

Abstract

Abstract. Let E = lim sup n→∞ (g n + ξ n ) be the random covering set on the torus T d , where {g n } is a sequence of ball-like sets and ξ n is a sequence of independent random variables uniformly distributed on T d . We prove that E ∩ F = ∅ almost surely whenever F ⊂ T d is an analytic set with Hausdorff dimension, dim H (F ) > d − α, where α is the almost sure Hausdorff dimension of E. Moreover, examples are given to show that the condition on dim H (F ) cannot be replaced by the packing dimension of F .

Hitting probabilities of random covering sets in tori and metric spaces

“…Bugeaud and Durand [4] also recovered these results within the context of Diophantine approximation. Li and Suomala [41] proved the analogue of (2.6) in the torus T d and showed that the assumption dim P F > d − α alone is not enough to guarantee that P(H(F )) > 0.…”

“…The proof of (2.6) given in [41] for the case X = T d can be generalised in a straightforward way to the current setting (replace the dyadic cubes by the generalised dyadic cubes Q throughout). We will not repeat the details.…”

“…Several authors have considered random covering problems in the context of metric spaces (see [36] and the references therein), but their emphasis has been on the case of full covering and on the size of the uncovered set X \ n∈N Z n . To the best of our knowledge, the intersection properties of the random covering sets have been studied only for randomly placed balls on tori [13,14,40,41]. This has motivated us to investigate hitting probabilities of random covering sets on more general metric spaces.…”

On the Intersection of Dynamical Covering Sets with Fractals

“…In general, in Theorem 2.1, P(E ∩ G = ∅) = 1 may not hold if dim H G > s − α is replaced by the weaker condition dim P G > s − α. A counter-example was given by Li and Suomala [26] when {ξ n } is a sequence of independent and uniformly distributed random variables on the circle. Therefore in the result below, we will make use of the following condition on the sequence {ℓ n }:…”

On the hitting probabilities of limsup random fractals

“…Li, Shieh and Xiao [11] investigated the hitting probability of random covering sets in which the use of limsup random fractals is essential. Later Li and Suomala [12] studied the same problem under conditions different from those in [11]. Wang, Wu and Xu [16] considered the dynamical covering problems on the middle-third Cantor set.…”