  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.
