The extreme value of local dimension of convolution of the cantor measure

Abstract

Let $\mu$ be the $m-$fold convolution of the standard Cantor measure and $\underline{\alpha}_m$ be the lower extreme value of the local dimension of the measure $\mu$. The values of $\underline{\alpha}_m$ for $m=2,3,4$ were showed in [4] and [5]. In this paper, we show that $$\underline{\alpha}_5=|\frac{\log \big[\frac{2}{3.2^5}\big(\sqrt{145}\cos(\frac{\arccos\frac{427}{59\sqrt{145}}}{3})+5\big)\big]}{\log 3}|\approx 0.972638.$$ This values was estimated by P. Shmerkin in [5], but it has not been proved.