Benford Behavior of a Higher-Dimensional Fragmentation Process

TL;DR

This paper extends the understanding of Benford's law to higher-dimensional fragmentation processes, showing that volumes in these processes tend to follow Benford's distribution, with conjectures for lower-dimensional cases.

Contribution

It generalizes previous one-dimensional fragmentation results to higher dimensions and proposes conjectures for lower-dimensional volumes under fragmentation.

Findings

Higher-dimensional volumes follow Benford's law in fragmentation processes.

The result extends known one-dimensional cases to any finite dimension.

Conjecture: lower-dimensional volumes also follow Benford's law in unrestricted fragmentation.

Abstract

Nature and our world have a bias! Roughly $30\%$ of the time the number $1$ occurs as the leading digit in many datasets base $10$. This phenomenon is known as Benford's law and it arrises in diverse fields such as the stock market, optimizing computers, street addresses, Fibonacci numbers, and is often used to detect possible fraud. Based on previous work, we know that different forms of a one-dimensional stick fragmentation result in pieces whose lengths follow Benford's Law. We generalize this result and show that this can be extended to any finite-dimensional ``volume''. We further conjecture that even lower-dimensional volumes, under the unrestricted fragmentation process, follow Benford's Law.