On the symmetric version of Seaki Theorem and flat densities

TL;DR

This paper constructs symmetric probability measures on the torus with supports of arbitrary Hausdorff dimension between 0.5 and 1, whose convolutions are absolutely continuous with flat, non-Lipschitz Radon-Nikodym derivatives, extending Seaki's theorem.

Contribution

It provides a symmetric version of Seaki's theorem demonstrating the existence of measures with flat convolution densities and arbitrary support dimensions.

Findings

Existence of symmetric measures with specified Hausdorff dimension

Convolution results in absolutely continuous measures with flat densities

Flat densities cannot be Lipschitz functions

Abstract

It is shown that for any $\alpha \in ]\frac12,1[$ there exists a symmetric probability measure $\sigma$ on the torus such that the Hausdorff dimension of the support of $\sigma$ is $\alpha$ and $\sigma*\sigma$ 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 $\sigma*\sigma$ can not be a Lipschitz function.