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/(@f^2+1), with @f 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"1H"n+c"2H"n"-"1+...+c"nH"1+1 (1==L), every positive integer can be written uniquely as @?a"iH"i under natural constraints on the a"i@?s, 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@?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 -(21-2@f)/(29+2@f)~-0.551058.

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