Benford Behavior of Generalized Zeckendorf Decompositions

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 nonadjacent 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 $n\rightarrow \infty$ 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.