Exceptional sets to Shallit's law of leap years in Pierce expansions

TL;DR

This paper investigates the exceptional sets related to Shallit's law of leap years in Pierce expansions, showing these sets are dense and have full Hausdorff dimension, revealing complex fractal structures in the distribution of leap years.

Contribution

It demonstrates that the set of exceptions to Shallit's law is dense and has full Hausdorff dimension, extending understanding of leap year distribution in Pierce expansions.

Findings

Exceptional sets are dense in [0,1]

Exceptional sets have full Hausdorff dimension

Intersections with open sets preserve Hausdorff dimension

Abstract

In his 1994 work, Shallit introduced a rule for determining leap years that generalizes both the historically used Julian calendar and the contemporary Gregorian calendar. This rule depends on a so-called intercalation sequence. According to what we term Shallit's law of leap years, almost every point of the interval $[0,1]$ with respect to Lebesgue measure has the same limsup and liminf, respectively, of a quotient defined in terms of the number of leap years determined by the rule using the Pierce expansion digit sequence as an intercalation sequence. In this paper, we show that the set of exceptions to this law is dense and has full Hausdorff dimension in $[0,1]$, and that the exceptional set intersected with any non-empty open subset of $[0,1]$ has full Hausdorff dimension in $[0,1]$. As a more general result, we establish that for certain subsets of $[0,1]$ concerning the limiting behavior of Pierce expansion digits, intersecting with a non-empty open subset of $[0,1]$ preserves the Hausdorff dimension.