Projections of Fractal percolation constructed with inhomogeneous probabilities

TL;DR

This paper studies fractal percolation sets with non-uniform probabilities, showing that in most directions, projections contain intervals and the projected measure is H"older continuous, extending known results from homogeneous cases.

Contribution

It extends the analysis of fractal percolation projections to non-homogeneous probabilities, identifying conditions under which projections retain interval and measure regularity properties.

Findings

Projections contain intervals in all directions except rational or Liouville tangent directions.

The projected natural measure is H"older continuous in most directions.

Non-homogeneous probabilities do not guarantee the same properties as homogeneous cases in all directions.

Abstract

In this paper we consider fractal percolation random Cantor sets $E$ on the plane constructed with non-homogeneous probabilities. We focus on the case when the probabilities are large enough to guarantee that the almost sure dimension of $E$ is greater than $1$. Under this assumption in the case of homogeneous (equal) probabilities it was proved by Rams and the first author that the orthogonal projection of $E$ contains an interval simultaneously in all directions. Moreover, Peres and Rams proved the stronger result that the orthogonal projection of the natural measure on $E$ to every line is absolutely continuous with H\"older-continuous density. We point out that in the case of non-homogeneous probabilities neither of the two previous assertions remain valid. However, we also prove that in the non-homogeneous case every line whose tangent is neither a rational nor a Liouville number is non-exceptional. That is, almost surely for all of these directions the projection of $E$ contains some interval and the projection of the natural measure has H\"older-continuous density.