  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.
