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 { F n } n = 1 ∞ . Lekkerkerker (1951–1952) [13] proved the average number of summands for integers in [ F n , F n + 1 ) is n / ( φ 2 + 1 ) , with φ the golden mean. This has been generalized: given non-negative integers c 1 , c 2 , … , c L with c 1 , c L > 0 and recursive sequence { H n } n = 1 ∞ with H 1 = 1 , H n + 1 = c 1 H n + c 2 H n − 1 + ⋯ + c n H 1 + 1 ( 1 ⩽ n < L ) and H n + 1 = c 1 H n + c 2 H n − 1 + ⋯ + c L H n + 1 − L ( n ⩾ L ), every positive integer can be written uniquely as ∑ a i H i under natural constraints on the a i ʼs, the mean and variance of the numbers of summands for integers in [ H n , H n + 1 ) are of size n, and as n → ∞ the distribution of the number of summands converges to a Gaussian. 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 other problems (in the sequel paper (Gaudet et al., preprint [2]) we show how this perspective allows us to determine the distribution of gaps between summands). For example, it is known that every integer can be written uniquely as a sum of the ± F n ʼs, 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 − 2 φ ) / ( 29 + 2 φ ) ≈ − 0.551058 .