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On the length of Pierce expansions

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arxiv 2211.08374 v1 pith:4QJGCSBD submitted 2022-11-15 math.NT math.CO

Zachary Chase , Mayank Pandey This is my paper · ORCID
classification math.NTmath.CO
keywords fracvarepsilontextbeforeboundexpansionsgivenimprove
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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 $\varepsilon > 0$.

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  1. Bounds for sets of remainders

    math.NT 2025-08 conditional novelty 6.0 of 10

    The paper proves a rigorous asymptotic with explicit constant for sequence A283190, bounds consecutive differences by O(log log n), and establishes lower and upper bounds for iterated remainder sets.

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