The algebraic difference of two random Cantor sets: The Larsson family

TL;DR

This paper proves that for a specific family of random Cantor sets, if the sum of their Hausdorff dimensions exceeds one, then their difference set contains interior points, confirming a conjecture for Larsson's sets.

Contribution

It provides a new, complete proof that the sum of Hausdorff dimensions greater than one guarantees interior points in the difference set of Larsson's random Cantor sets.

Findings

Sum of Hausdorff dimensions > 1 implies interior points in the difference set.

Complete proof for Larsson's random Cantor sets.

Confirms a key conjecture in fractal geometry.

Abstract

In this paper, we consider a family of random Cantor sets on the line and consider the question of whether the condition that the sum of the Hausdorff dimensions is larger than one implies the existence of interior points in the difference set of two independent copies. We give a new and complete proof that this is the case for the random Cantor sets introduced by Per Larsson.