Uniform lower bounds for the average length of finite Pierce expansions

Min Woong Ahn

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arXiv · 2610.26594

Uniform lower bounds for the average length of finite Pierce expansions

Min Woong Ahn

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Abstract

For integers $1\leq a\leq b$, the length of the finite Pierce expansion of $a/b$ is the number of steps of the iteration $r\mapsto b\bmod r$ which starts at $a$ and ends at zero. Erdős and Shallit (1991) proved that the average of these lengths over $1\leq a\leq b$ is bounded below by a positive constant times $\log\log b$, and suggested that it is at least of order $\log b$. In this paper, we prove that the average length is at least $\sqrt{2\log b}-3(\log b)^{1/4}-11$ for every $b\geq2$. We also show that, for each $0<c<\sqrt2$, the proportion of numerators $a$ for which the length is less than $c\sqrt{\log b}$ tends to zero exponentially fast in $\sqrt{\log b}$, with an explicit rate. Our approach refines the interval-counting argument of Erdős and Shallit by restricting the product of the initial digits. As applications, we obtain similar bounds for reduced fractions and for numerators in a fixed congruence class.

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BibTeX RIS

Min Woong Ahn. 2026-10-19. Uniform lower bounds for the average length of finite Pierce expansions. https://arxiv.org/abs/2610.26594

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