Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$\{F_n\}_{n=1}^{\infty}$. Lekkerkerker proved that the average number of
summands for integers in $[F_n, F_{n+1})$ is $n/(\phi^2 + 1)$, with $\phi$ the
golden mean. This has been generalized to the following: given nonnegative
integers $c_1,c_2,...,c_L$ with $c_1,c_L>0$ and recursive sequence
$\{H_n\}_{n=1}^{\infty}$ with $H_1=1$, $H_{n+1}
=c_1H_n+c_2H_{n-1}+...+c_nH_1+1$ $(1\le n< L)$ and
$H_{n+1}=c_1H_n+c_2H_{n-1}+...+c_LH_{n+1-L}$ $(n\geq L)$, every positive
integer can be written uniquely as $\sum a_iH_i$ under natural constraints on
the $a_i$'s, the mean and the variance of the numbers of summands for integers
in $[H_{n}, H_{n+1})$ are of size $n$, and the distribution of the numbers of
summands converges to a Gaussian as $n$ 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 $\pm F_n$'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 $-(21-2\phi)/(29+2\phi) \approx -0.551058$.

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The categories of this paper are: nlin
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The categories of this paper are: comp-gas
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The categories of this paper are: adap-org
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: mtrl-th