# Differences of random Cantor sets and lower spectral radii

### F. Michel Dekking

Delft University of Technology, Netherlands### Bram Kuijvenhoven

ABN AMRO Bank, Amsterdam, Netherlands

## 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 *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 $p_{0}+p_{1}>√2$ and $p_{0}p_{1}(1+p_{0}+p_{1})<1$. We furthermore prove a general result which characterizes the interval/no interval property in terms of the lower spectral radius of a set of $2×2$ matrices.

## Cite this article

F. Michel Dekking, Bram Kuijvenhoven, Differences of random Cantor sets and lower spectral radii. J. Eur. Math. Soc. 13 (2011), no. 3, pp. 733–760

DOI 10.4171/JEMS/266