New fractal dimensions and some applications to arithmetic patches

Abstract

In this paper, we define new fractal dimensions of a metric space in order to calculate the roughness of a set on a large scale. These fractal dimensions are called upper zeta dimension and lower zeta dimension. The upper zeta dimension is an extension of the zeta dimension introduced by Doty, Gu, Lutz, Elvira, Mayordomo, and Moser. We show that the upper zeta dimension is always a lower bound for the Assouad dimension. Moreover, we apply the upper zeta dimension to the existence of weak arithmetic patches of a given set. Arithmetic patches are higher dimensionalized arithmetic progressions. As a corollary, we get the affirmative solution to a higher dimensional weak analogue of the Erd\H{o}s-Tur\'an conjecture. Here the one dimensional case is proved by Fraser and Yu. As examples, we prove the existence of weak arithmetic patches of the set of all irreducible elements of $\mathbb{Z}[\alpha]$, and the set of all prime numbers of the form $p(p(n))$, where $\alpha$ is an imaginary quadratic integer and $p(n)$ denotes the $n$-th prime number.