  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.
