1,730 research outputs found
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
. Lekkerkerker proved that the average number of
summands for integers in is , with the
golden mean. This has been generalized to the following: given nonnegative
integers with and recursive sequence
with , and
, every positive
integer can be written uniquely as under natural constraints on
the 's, the mean and the variance of the numbers of summands for integers
in are of size , and the distribution of the numbers of
summands converges to a Gaussian as 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 '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 .Comment: This is a companion paper to Kologlu, Kopp, Miller and Wang's On the
number of summands in Zeckendorf decompositions. Version 2.0 (mostly
correcting missing references to previous literature
Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals
Zeckendorf's theorem states that every positive integer can be written
uniquely as a sum of non-consecutive Fibonacci numbers , with initial
terms . We consider the distribution of the number of
summands involved in such decompositions. Previous work proved that as the distribution of the number of summands in the Zeckendorf
decompositions of , appropriately normalized, converges
to the standard normal. The proofs crucially used the fact that all integers in
share the same potential summands.
We generalize these results to subintervals of as ; the analysis is significantly more involved here as different integers
have different sets of potential summands. Explicitly, fix an integer sequence
. As , for almost all the distribution of the number of summands in the Zeckendorf
decompositions of integers in the subintervals ,
appropriately normalized, converges to the standard normal. The proof follows
by showing that, with probability tending to , has at least one
appropriately located large gap between indices in its decomposition. We then
use a correspondence between this interval and 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
A Probabilistic Approach to Generalized Zeckendorf Decompositions
Generalized Zeckendorf decompositions are expansions of integers as sums of
elements of solutions to recurrence relations. The simplest cases are base-
expansions, and the standard Zeckendorf decomposition uses the Fibonacci
sequence. The expansions are finite sequences of nonnegative integer
coefficients (satisfying certain technical conditions to guarantee uniqueness
of the decomposition) and which can be viewed as analogs of sequences of
variable-length words made from some fixed alphabet. In this paper we present a
new approach and construction for uniform measures on expansions, identifying
them as the distribution of a Markov chain conditioned not to hit a set. This
gives a unified approach that allows us to easily recover results on the
expansions from analogous results for Markov chains, and in this paper we focus
on laws of large numbers, central limit theorems for sums of digits, and
statements on gaps (zeros) in expansions. We expect the approach to prove
useful in other similar contexts.Comment: Version 1.0, 25 pages. Keywords: Zeckendorf decompositions, positive
linear recurrence relations, distribution of gaps, longest gap, Markov
processe
Benford Behavior of Zeckendorf Decompositions
A beautiful theorem of Zeckendorf states that every integer can be written
uniquely as the sum of non-consecutive Fibonacci numbers . A set is said to satisfy Benford's law if
the density of the elements in with leading digit is
; in other words, smaller leading digits are more
likely to occur. We prove that, as , for a randomly selected
integer in the distribution of the leading digits of the
Fibonacci summands in its Zeckendorf decomposition converge to Benford's law
almost surely. Our results hold more generally, and instead of looking at the
distribution of leading digits one obtains similar theorems concerning how
often values in sets with density are attained.Comment: Version 1.0, 12 pages, 1 figur
- β¦