Benford Behavior of Zeckendorf Decompositions
Abstract
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of nonconsecutive 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2014
 arXiv:
 arXiv:1409.0482
 Bibcode:
 2014arXiv1409.0482B
 Keywords:

 Mathematics  Number Theory;
 11B39;
 11B05;
 60F05 (primary) 11K06;
 65Q30;
 62E20 (secondary)
 EPrint:
 Version 1.0, 12 pages, 1 figure