Zeckendorf's theorem states that every positive integer can be written
uniquely as a sum of non-consecutive Fibonacci numbers Fnβ, with initial
terms F1β=1,F2β=2. We consider the distribution of the number of
summands involved in such decompositions. Previous work proved that as nββ the distribution of the number of summands in the Zeckendorf
decompositions of mβ[Fnβ,Fn+1β), appropriately normalized, converges
to the standard normal. The proofs crucially used the fact that all integers in
[Fnβ,Fn+1β) share the same potential summands.
We generalize these results to subintervals of [Fnβ,Fn+1β) as nββ; the analysis is significantly more involved here as different integers
have different sets of potential summands. Explicitly, fix an integer sequence
Ξ±(n)ββ. As nββ, for almost all mβ[Fnβ,Fn+1β) the distribution of the number of summands in the Zeckendorf
decompositions of integers in the subintervals [m,m+FΞ±(n)β),
appropriately normalized, converges to the standard normal. The proof follows
by showing that, with probability tending to 1, m has at least one
appropriately located large gap between indices in its decomposition. We then
use a correspondence between this interval and [0,FΞ±(n)β) to obtain
the result, since the summands are known to have Gaussian behavior in the
latter interval. % We also prove the same result for more general linear
recurrences.Comment: Version 1.0, 8 page