On digits of Mersenne numbers

Motivated by recently developed interest to the distribution of $q$-arydigits of Mersenne numbers $M_p = 2^p-1$, where $p$ is prime, we estimaterational exponential sums with $M_p$, $p \leq X$, modulo a large power of afixed odd prime $q$. In turn this immediately implies the normality of stringsof $q$-ary digits amongst about $(\log X)^{3/2+o(1)}$ rightmost digits of$M_p$, $p \leq X$. Previous results imply this only for about $(\logX)^{1+o(1)}$ rightmost digits.