Differences of random Cantor sets and lower spectral radii

TL;DR

This paper explores when the difference of two independent random Cantor sets contains an interval, linking this property to Hausdorff dimensions and spectral radii, and provides conditions for specific 2-adic cases.

Contribution

It characterizes the interval property of random Cantor sets using lower spectral radii and identifies precise conditions for 2-adic sets where the Palis conjecture fails.

Findings

The Palis conjecture holds when the sum of Hausdorff dimensions exceeds 1.

For 2-adic random Cantor sets, the region where the conjecture fails is explicitly described.

A general spectral radius criterion is established for the interval/no interval property.

Abstract

We investigate the question under which conditions the algebraic difference between two independent random Cantor sets $C_1$ and $C_2$ almost surely contains an interval, and when not. The natural condition is whether the sum $d_1+d_2$ of the Hausdorff dimensions of the sets is smaller (no interval) or larger (an interval) than 1. Palis conjectured that \emph{generically} it should be true that $d_1+d_2>1$ should imply that $C_1-C_2$ contains an interval. We prove that for 2-adic random Cantor sets generated by a vector of probabilities $(p_0,p_1)$ the interior of the region where the Palis conjecture does not hold is given by those $p_0,p_1$ which satisfy $p_0+p_1>\sqrt{2}$ and $p_0p_1(1+p_0^2+p_1^2)<1$. We furthermore prove a general result which characterizes the interval/no interval property in terms of the lower spectral radius of a set of $2\times 2$ matrices.