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Projections of Fractal percolation constructed with inhomogeneous probabilities

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 Hölder-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 Hölder-continuous density.