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 1 G n + · · · + c L G n+1−L with c i positive and some other restrictions. Additionally, a set S ⊂ 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 + 1 d ); in other words, smaller leading digits are more likely to occur. We prove that as n → ∞ 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.
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