Fang, B, Irons, A, Lippelman, E and Miller, Steven J. (2025)

Benford Behavior in Stick Fragmentation Problems

Preprint arXiv:2508.17360 [math.PR]; last accessed November 8, 2025.

ISSN/ISBN: Not available at this time. DOI: Not available at this time.

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Abstract: Benford's law is the statement that in many real-world data sets, the probability of having digit \(d\) in base \(B\), where \(1 \leq d \leq B\), as the first digit is \(\log_{B}\left(\tfrac{d+1}{d}\right)\). We sometimes refer to this as weak Benford behavior, and we say that a data set exhibits strong Benford behavior in base \(B\) if the probability of having significand at most \(s\), where \(s \in [1,B)\), is \(\log_{B}(s)\). We examine Benford behaviors in the stick fragmentation model. Building on the work on the 1-dimensional stick fragmentation model, we employ combinatorial identities on multinomial coefficients to reduce the high-dimensional stick fragmentation model to the 1-dimensional model and provide a necessary and sufficient condition for the lengths of the stick fragments to converge to strong Benford behavior.

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Bibtex:
@misc{, title={Benford Behavior in Stick Fragmentation Problems}, author={Bruce Fang and Ava Irons and Ella Lippelman and Steven J. Miller}, year={2025}, eprint={2508.17360}, archivePrefix={arXiv}, primaryClass={math.PR}, url={https://arxiv.org/abs/2508.17360}, }

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Reference Type: Preprint

Subject Area(s): Number Theory

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