Limit theorems and fractal properties of digit gaps in Pierce expansions

Abstract

In this paper, we revisit Shallit's results on the law of large numbers, the central limit theorem, and the law of the iterated logarithm for the digits of Pierce expansions. We extend these limit theorems to the setting of digit gaps in Pierce expansions, showing that digit gaps exhibit the same limit behavior as the digits themselves. However, the fractal properties of digit gaps differ significantly from those of the digits. To capture this difference, we compute the Hausdorff dimension of the exceptional sets associated with the law of large numbers for digit gaps and obtain explicit formulas for these dimensions.