On the symmetric version of Seaki Theorem and flat densities

Abstract

It is shown that for any $\alpha \in ]\frac{1}{2},1[$ there exists a symmetric probability measure $\sigma$ on the torus such that the Hausdorff dimension of the support of $\sigma$ is $\alpha$ and $\sigma \ast \sigma$ is absolutely continuous with flat continuous Radon-Nikodym derivative. Namely, we obtain a symmetric version of Seaki Theorem but the flat Radon-Nikodym derivative of $\sigma \ast \sigma$ can not be a Lipschitz function.

8 pages. 7 references. In this version, some misprints are corrected and the proof of Lemma 2.1 is added. A scientific comments, questions or remarks are welcome