A local Benford Law for a class of arithmetic sequences

TL;DR

This paper investigates the local digit distribution properties of various arithmetic sequences, establishing a maximal order of local Benford distribution and revealing that many sequences known to follow Benford's Law do not exhibit strong local properties.

Contribution

It introduces the concept of local Benford distribution of order k and determines the maximal k for a broad class of sequences, refining understanding of Benford's Law at the local digit level.

Findings

Most sequences satisfying Benford's Law have poor local distribution properties.

The paper establishes the maximal local Benford order for many arithmetic sequences.

Sequences like ^n, ^{n^d}, and factorials follow the maximal local Benford Law.

Abstract

It is well-known that sequences such as the Fibonacci numbers and the factorials satisfy Benford's Law, that is, leading digits in these sequences occur with frequencies given by $P(d)=\log_{10}(1+1/d)$, $d=1,2,\dots,9$. In this paper, we investigate leading digit distributions of arithmetic sequences from a local point of view. We call a sequence locally Benford distributed of order $k$ if, roughly speaking, $k$-tuples of consecutive leading digits behave like $k$ independent Benford-distributed digits. This notion refines that of a Benford distributed sequence, and it provides a way to quantify the extent to which the Benford distribution persists at the local level. Surprisingly, most sequences known to satisfy Benford's Law have rather poor local distribution properties. In our main result we establish, for a large class of arithmetic sequences, a "best-possible" local Benford Law, that is, we determine the maximal value $k$ such that the sequence is locally Benford distributed of order $k$. The result applies, in particular, to sequences of the form $\{a^n\}$, $\{a^{n^d}\}$, and $\{n^{\beta} a^{n^\alpha}\}$, as well as the sequence of factorials $\{n!\}$ and similar iterated product sequences.