  Zeckendorf's theorem states that every positive integer can be written
uniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initial
terms $F_1 = 1, F_2 = 2$. Previous work proved that as $n \to \infty$ the
distribution of the number of summands in the Zeckendorf decompositions of $m
\in [F_n, F_{n+1})$, appropriately normalized, converges to the standard
normal. The proofs crucially used the fact that all integers in $[F_n,
F_{n+1})$ share the same potential summands and hold for more general positive
linear recurrence sequences $\{G_n\}$.
  We generalize these results to subintervals of $[G_n, G_{n+1})$ as $n \to
\infty$ for certain sequences. The analysis is significantly more involved here
as different integers have different sets of potential summands. Explicitly,
fix an integer sequence $\alpha(n) \to \infty$. As $n \to \infty$, for almost
all $m \in [G_n, G_{n+1})$ the distribution of the number of summands in the
generalized Zeckendorf decompositions of integers in the subintervals $[m, m +
G_{\alpha(n)})$, 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 $[0, G_{\alpha(n)})$ to
obtain the result, since the summands are known to have Gaussian behavior in
the latter interval.
