Prime-representing functions and Hausdorff dimension

Let \(c \geq 2\) be any fixed real number. Matomäki [4] inverstigated the set of \(A > 1\) such that integer part \( A^{c^k}\) is a prime number for every \(k \in \mathbb{N}\). She proved uncountable, nowhere dense, and has Lebesgue measure 0. In this article, we show Hausdorff dimension 1.