DIMENSIONS OF RANDOM COVERING SETS IN RIEMANN MANIFOLDS

• 成果类型: Article
• 署名作者: Feng, De-Jun; Jarvenpaa, Esa; Jarvenpaa, Maarit; Suomala, Ville
• 署名单位: Chinese University of Hong Kong; University of Oulu
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1210
• 发表日期: 2018
• 页码: 1542-1596
• 关键词: large intersection; circle; intervals; numbers; point

摘要

Let M, N and K be d-dimensional Riemann manifolds. Assume that A := (A(n))(n is an element of N) is a sequence of Lebesgue measurable subsets of M satisfying a necessary density condition and x := (xn)(n is an element of N) is a sequence of independent random variables, which are distributed on K according to a measure, which is not purely singular with respect to the Riemann volume. We give a formula for the almost sure value of the Hausdorff dimension of random covering sets E(x, A) := lim sup(n ->infinity) A(n)(x(n)) subset of N. Here, A(n)(x(n)) is a diffeomorphic image of An depending on x(n). We also verify that the packing dimensions of E(x, A) equal d almost surely.