Prime-representing functions and Hausdorff dimension

TL;DR

This paper proves that the set of real numbers for which the integer parts of their exponential powers are always prime has Hausdorff dimension 1, revealing a complex fractal structure.

Contribution

It establishes that this set, previously known to be uncountable and measure zero, actually has full Hausdorff dimension 1, providing new geometric insight.

Findings

The set has Hausdorff dimension 1.

The set is uncountable and nowhere dense.

The set has Lebesgue measure zero.

Abstract

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 $1$.