2019/03/19 by García, Ignacio, Hare, Kathryn E., Mendivil, Franklin
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)
paper · doi:10.48550/arxiv.1903.07800
Given a non-negative, decreasing sequence a with sum 1, we consider all the closed subsets of [0,1] such that the lengths of their complementary open intervals are given by the terms of a, the so-called complementary sets. In this paper we determine the almost sure value of the Φ-dimensions of these sets given a natural model of randomness. The Φ-dimensions are intermediate Assouad-like dimensions which include the Assouad and quasi-Assouad dimensions as special cases. The answers depend on the size of Φ, with one size behaving like the Assouad dimension and the other, like the quasi-Assouad dimension.