• Corpus ID: 51774496

Benford Behavior of Zeckendorf Decompositions

@article{Best2014BenfordBO,
  title={Benford Behavior of Zeckendorf Decompositions},
  author={Andr{\'e} Best and Patrick J. Dynes and Xixi Edelsbrunner and Brian McDonald and Steven J. Miller and Kimsy Tor and Caroline L. Turnage-Butterbaugh and Madeleine Weinstein},
  journal={arXiv: Number Theory},
  year={2014}
}

• André Best, Patrick J. Dynes, +5 authors Madeleine Weinstein
• Published 1 September 2014
• Mathematics
• arXiv: Number Theory

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.