Almost sure Assouad-like dimensions of complementary sets

GARCÍA, IGNACIO; HARE, KATHRYN; MENDIVIL, FRANKLIN

MATHEMATISCHE ZEITSCHRIFT

SPRINGER

Lugar: Berlin; Año: 2020

0025-5874

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. These are the so-called complementary sets, or rearrangements of the Cantor set, constructed from a. In this paper we determine the almost sure value of the Φ-dimension 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 almost surely like the Assouad dimensions of the associated Cantor set and the other, like the quasi-Assouad dimensions. These results are new even for the Assouad dimensions.