A Generalization of Zeckendorf's Theorem via Circumscribed $m$-gons

TL;DR

This paper generalizes Zeckendorf's theorem to a new family of sequences derived from circumscribed m-gons, establishing unique decompositions, Gaussian distribution of summands, geometric decay of gaps, and concentration of the longest gap.

Contribution

It introduces m-gonal sequences with unique decompositions that extend Zeckendorf's theorem, handling cases with initial zero coefficients that previous methods could not address.

Findings

Decompositions are unique for m-gonal sequences.

Number of summands follows a Gaussian distribution.

Gaps between summands decay geometrically.

Abstract

Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$, $F_1=1$ and $F_2=2$. The distribution of the number of summands in such decomposition converges to a Gaussian, the gaps between summands converges to geometric decay, and the distribution of the longest gap is similar to that of the longest run of heads in a biased coin; these results also hold more generally, though for technical reasons previous work needed to assume the coefficients in the recurrence relation are non-negative and the first term is positive. We extend these results by creating an infinite family of integer sequences called the $m$-gonal sequences arising from a geometric construction using circumscribed $m$-gons. They satisfy a recurrence where the first $m+1$ leading terms vanish, and thus cannot be handled by existing techniques. We provide a notion of a legal decomposition, and prove that the decompositions exist and are unique. We then examine the distribution of the number of summands used in the decompositions and prove that it displays Gaussian behavior. There is geometric decay in the distribution of gaps, both for gaps taken from all integers in an interval and almost surely in distribution for the individual gap measures associated to each integer in the interval. We end by proving that the distribution of the longest gap between summands is strongly concentrated about its mean, behaving similarly as in the longest run of heads in tosses of a coin.