39 research outputs found
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
. Lekkerkerker proved that the average number of
summands for integers in is , with the
golden mean. This has been generalized to the following: given nonnegative
integers with and recursive sequence
with , and
, every positive
integer can be written uniquely as under natural constraints on
the 's, the mean and the variance of the numbers of summands for integers
in are of size , and the distribution of the numbers of
summands converges to a Gaussian as 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 \cite{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 '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 .Comment: This is a companion paper to Kologlu, Kopp, Miller and Wang's On the
number of summands in Zeckendorf decompositions. Version 2.0 (mostly
correcting missing references to previous literature