Title: An elementary proof that the set of exceptions to the law of large numbers in Pierce expansions has full Hausdorff dimension Show Chinese title

Title: 一个关于Pierce展开中大数定律例外集具有全豪斯多夫维数的初等证明

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. Show Chinese abstract

Abstract: Pierce展开的数字满足大数定律。 已知大数定律例外集的Hausdorff维数为1。 我们通过适应Jun Wu最初用于Engel展开的方法，提供了一个简单的证明。 我们的方法强调了例外集的分形性质，并避免使用高级工具，而是依赖于显式序列和构造技术。 此外，我们的方法为将类似的分析扩展到其他实数表示系统（如Engel、Lüroth和Sylvester展开）提供了可能性，从而为度量数论和分形几何的进一步探索铺平了道路。