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

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.