Benford Behavior of Generalized Zeckendorf Decompositions

TL;DR

This paper proves that the leading digits of summands in generalized Zeckendorf decompositions follow Benford's law as the number size grows large, extending the law's applicability to these unique number representations.

Contribution

It establishes the almost sure convergence of leading digit distribution in generalized Zeckendorf decompositions to Benford's law, generalizing previous results to broader recurrence sequences.

Findings

Leading digits in decompositions follow Benford's law asymptotically.

Results apply to sequences from positive recurrence relations.

Distribution convergence holds for sets with positive density.

Abstract

We prove connections between Zeckendorf decompositions and Benford's law. Recall that if we define the Fibonacci numbers by $F_1 = 1, F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$, every positive integer can be written uniquely as a sum of non-adjacent elements of this sequence; this is called the Zeckendorf decomposition, and similar unique decompositions exist for sequences arising from recurrence relations of the form $G_{n+1}=c_1G_n+\cdots+c_LG_{n+1-L}$ with $c_i$ positive and some other restrictions. Additionally, a set $S \subset \mathbb{Z}$ is said to satisfy Benford's law base 10 if the density of the elements in $S$ with leading digit $d$ is $\log_{10}{(1+\frac{1}{d})}$; in other words, smaller leading digits are more likely to occur. We prove that as $n\to\infty$ for a randomly selected integer $m$ in $[0, G_{n+1})$ the distribution of the leading digits of the summands in its generalized Zeckendorf decomposition converges to Benford's law almost surely. Our results hold more generally: one obtains similar theorems to those regarding the distribution of leading digits when considering how often values in sets with density are attained in the summands in the decompositions.