Exceptional sets to Shallit's law of leap years in Pierce expansions
Min Woong Ahn
TL;DR
This work investigates Shallit's law of leap years, which generalizes Julian and Gregorian rules via an intercalation sequence based on Pierce expansion digits. The authors show that the set of exceptions to the law is not only dense in $[0,1]$ but also has full Hausdorff dimension, and that its intersection with any non-empty open set preserves this maximal dimension. By leveraging Pierce-expansion coding, fundamental intervals, and dimension-preserving transformations, they establish a broader framework: for subsets $A(\alpha)$ defined by digit-growth rates and the associated $S(\alpha)$-sets capturing limsup/liminf behavior, the sets are dense with $\dim_H=1$ on open sets, while the exceptional set $[0,1]\setminus S(1)$ likewise attains full dimension. These results reveal a rich fractal structure of typical versus exceptional calendar-steady-state behavior under the Pierce-based intercalation rule, highlighting substantial sets of points where the long-term leap-year prediction deviates from the naive expectation. The findings extend Sha93/Sha94 insights and provide a dimension-theoretic perspective on calendar alignment using number-theoretic digit expansions.
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.