Zeckendorf’s theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers Fn, with initial terms F1 = 1, F2 = 2. Previous work proved that as n ! 1 the distribution of the number of summands in the Zeckendorf decompositions of m 2 [Fn, Fn+1), appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in [Fn, Fn+1) share the same potential summands, and hold for more general positivelinear recurrence sequences {Gn}.We generalize these results to subintervals of [Gn, Gn+1) as n ! 1 for certainsequences. The analysis is significantly more involved here as di?erent integers have di?erent sets of potential summands. Explicitly, fix an integer sequence ?(n) ! 1.As n ! 1, for almost all m 2 [Gn, Gn+1) the distribution of the number of summands in the generalized Zeckendorf decompositions of integers in the subintervals[m, m + G?(n)), appropriately normalized, converges to the standard normal. Theproof follows by showing that, with probability tending to 1, m has at least oneappropriately located large gap between indices in its decomposition. We then usea correspondence between this interval and [0, G?(n)) to obtain the result, since thesummands are known to have Gaussian behavior in the latter interval.
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