Benford Behavior of Zeckendorf Decompositions

TL;DR

This paper proves that the leading digits of Fibonacci numbers in Zeckendorf decompositions follow Benford's law as the numbers grow large, revealing a natural digit distribution pattern in these unique representations.

Contribution

It establishes the almost sure convergence of leading digit distributions in Zeckendorf decompositions to Benford's law, extending the understanding of digit patterns in Fibonacci-based representations.

Findings

Leading digits in Zeckendorf decompositions follow Benford's law asymptotically.

Distribution convergence holds almost surely as numbers grow large.

Results extend to sets with positive density beyond Fibonacci numbers.

Abstract

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law 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 $[0, F_{n+1})$ the distribution of the leading digits of the Fibonacci summands in its Zeckendorf decomposition converge to Benford's law almost surely. Our results hold more generally, and instead of looking at the distribution of leading digits one obtains similar theorems concerning how often values in sets with density are attained.