报告题目：From Fibonacci Numbers to Central Limit Type Theorems

报 告 人: Yinghui Wang (Massachusetts Institute of Technology)

时 间:1月12日（星期三）10:00—11:00

地 点：数学楼102报告厅

摘 要：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 $n/(\varphi^2 + 1)$,
with $\varphi$ the golden mean. We prove the following massive
generalization: given nonnegative integers $c_1,c_2,\dots,c_L$ with
$c_1,c_L>0$ and recursive sequence $\{H_n\}_{n=1}^{\infty}$ with
$H_1=1$, $H_{n+1} =c_1H_n+c_2 H_{n-1} + \cdots +c_ _1+1$ $(1\le n< L)$
and $H_{n+1}= c_1H_n+c_2 H_{n-1}+\cdots +c_L H_{n+1-L}$ $(n\geq L)$,
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 were number
theoretic, involving continued fractions, and were limited to results
on existence and, in some cases, the mean. By recasting as a
combinatorial problem and using generating functions and
differentiating identities, we surmount the limitations inherent in
the previous approaches.

Our method generalizes to a multitude of other problems. For example,
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). We prove similar results as above; for instance,
the distribution of the numbers of positive and negative summands
converges to a bivariate normal with computable, negative correlation,
namely $-(21-2\varphi)/(29+2\varphi) \approx -0.551058$.