An elementary proof that the set of exceptions to the law of large numbers in Pierce expansions has full Hausdorff dimension

TL;DR

This paper provides an elementary, constructive proof that the set of exceptions to the law of large numbers in Pierce expansions has full Hausdorff dimension, highlighting the fractal structure of these exceptional sets.

Contribution

It introduces a new elementary proof method, adapted from Wu's approach for Engel expansions, to analyze the Hausdorff dimension of exception sets in Pierce expansions.

Findings

The set of exceptions has Hausdorff dimension 1.

The proof avoids advanced machinery, using explicit sequences and constructive techniques.

Method can be extended to other number systems like Engel, Lüroth, and Sylvester expansions.

Abstract

The digits of the Pierce expansion satisfy the law of large numbers. It is known that the Hausdorff dimension of the set of exceptions to the law of large numbers is 1. We provide an elementary proof of this fact by adapting Jun Wu's method, which was originally used for Engel expansions. Our approach emphasizes the fractal nature of exceptional sets and avoids advanced machinery, thereby relying instead on explicit sequences and constructive techniques. Furthermore, our method opens the possibility of extending similar analyses to other real number representation systems, such as the Engel, L\"uroth, and Sylvester expansions, thus paving the way for further explorations in metric number theory and fractal geometry.