From Fibonacci numbers to central limit type theorems

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers {Fn}n=1. Lekkerkerker proved that the average number of summands for integers in [Fn, Fn+1) is n/(φ 2+1), with φ the golden mean. This has been generalized to the following: given nonnegative integers c1, c2, . . . , cL with c1, cL > 0 and recursive sequence {Hn}n=1 with H1 = 1, Hn+1 = c1Hn + c2Hn−1 + · · · + cnH1 + 1 (1 ≤ n < L) and Hn+1 = c1Hn + c2Hn−1 + · · ·+ cLHn+1−L (n ≥ L), every positive integer can be written uniquely as ∑ aiHi under natural constraints on the ai’s, the mean and the variance of the numbers of summands for integers in [Hn, Hn+1) are of size n, and the distribution of the numbers of summands converges to a Gaussian as n goes to the infinity. Previous approaches used number theory or ergodic theory. We convert the problem to a combinatorial one. In addition to re-deriving these results, our method generalizes to a multitude of other problems (in the sequel paper [BM] we show how this perspective allows us to determine the distribution of gaps between summands in decompositions). For example, it is known that every integer can be written uniquely as a sum of the ±Fn’s, such that every two terms of the same (opposite) sign differ in index by at least 4 (3). The presence of negative summands introduces complications and features not seen in previous problems. We prove that the distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely −(21− 2φ)/(29 + 2φ) ≈ −0.551058.