On the length of Pierce expansions

TL;DR

This paper improves the upper bound on the length of the process defined by iterated modular operations for a positive integer, refining previous bounds with a more precise asymptotic estimate.

Contribution

It provides a tighter upper bound of $O(n^{1/3 - 2/177 + ext{small epsilon}})$ for the process length, advancing the understanding of Pierce expansion behavior.

Findings

Improved the upper bound from $O(n^{1/3+ ext{epsilon}})$ to $O(n^{1/3 - 2/177 + ext{epsilon}})$

Refined the asymptotic analysis of the process length for Pierce expansions

Enhanced the theoretical understanding of the process duration before reaching zero.

Abstract

For a given positive integer $n$, how long can the process $x \mapsto n\text{ }(\text{mod } x)$ last before reaching $0$? We improve Erd\H{o}s and Shallit's upper bound of $O(n^{\frac{1}{3}+\varepsilon})$ to $O(n^{\frac{1}{3}-\frac{2}{177}+\varepsilon})$ for any $\varepsilon > 0$.