2021/10/14 by Hirayama, Michihiro, Karagulyan, Davit
#60G57 #FOS: Mathematics #Probability (math.PR) #primary 28A78 #secondary 60D05
paper · doi:10.48550/arxiv.2110.07350
The classical Dvoretzky covering problem asks for conditions on the sequence of lengths \ℓn\n∈ ℕ so that the random intervals In : = (ωn -(ℓn/2), ωn +(ℓn/2)) where ωn is a sequence of i.i.d. uniformly distributed random variable, covers any point on the circle \mathbbT infinitely often. We consider the case when ωn are absolutely continuous with a density function f. When mf=essinf_\mathbbTf>0 and the set Kf of its essential infimum points satisfies dimB Kf<1, where dimB is the upper box-counting dimension, we show that the following condition is necessary and sufficient for \mathbbT to be μf-Dvoretzky covered \limsupn → ∞ ((ℓ1 + … + ℓn)/(ln n))≥ (1)/(mf). Under more restrictive assumptions on \ℓn\ the above result is true if dimH Kf<1. We next show that as long as \ℓn\n∈ ℕ and f satisfy the above condition and |Kf|=0, then a Menshov type result holds, i.e. Dvoretzky covering can be achieved by changing f on a set of arbitrarily small Lebesgue measure. This, however, is not true for the uniform density.