An elementary proof that the set of exceptions to the law of large numbers in Pierce expansions has full Hausdorff dimension
Min Woong Ahn
TL;DR
The paper proves that the exceptional set where the law of large numbers fails for Pierce expansions has full Hausdorff dimension by presenting an elementary, constructive proof inspired by Wu's method for Engel expansions. It defines a carefully crafted embedding from a positive-measure set into the exceptional set and uses Hölder control to transfer dimension, showing the lower bound can be made arbitrarily close to 1 as a parameter grows, hence dim$_H$ = 1. The strategy leverages Pierce-specific digit-growth properties and avoids advanced symbolic dynamics or measure-theoretic machinery, offering an accessible perspective and potential extensions to Engel, Lüroth, and Sylvester representations. This work deepens the fractal-geometry viewpoint in metric number theory and paves the way for further explorations of exceptional sets in real-number representations.
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, Lüroth, and Sylvester expansions, thus paving the way for further explorations in metric number theory and fractal geometry.