Intersections of certain deleted digits sets

TL;DR

This paper studies the intersection properties of deleted digits Cantor sets with their translations, focusing on conditions that determine the Hausdorff dimension of intersections and the structure of the translation set.

Contribution

It characterizes when the set of translations with intersections of a given Hausdorff dimension is dense and when this set forms an interval, based on properties of the digit set.

Findings

The set of translations with a specific Hausdorff dimension intersection can be dense under certain digit set conditions.

Conditions are identified under which the translation set forms an interval.

The paper provides simple characterizations of the structure of the translation set F.

Abstract

We consider some properties of the intersection of deleted digits Cantor sets with their translates. We investigate conditions on the set of digits such that, for any t between zero and the dimension of the deleted digits Cantor set itself, the set of translations such that the intersection has Hausdorff dimension equal to t is dense in the set F of translations such that the intersection is non-empty. We make some simple observations regarding properties of the set F, in particular, we characterize when F is an interval, in terms of conditions on the digit set.