Prime-representing functions and Hausdorff dimension

In 2010, Matom\"{a}ki investigated the set of $A>1$ such that the integer part of $A^{c^k}$ is a prime number for every $k\in \mathbb{N}$, where $c\geq 2$ is any fixed real number. She proved that the set is uncountable, nowhere dense, and has Lebesgue measure $0$. In this article, we show that the set has Hausdorff dimension $1$.