A generalization of Zeckendorf’s theorem via circumscribed m-gons

Dorward, Robert; Ford, Pari L.; Fourakis, Eva; Harris, Pamela E.; Miller, Steven J.; Palsson, Eyvindur; Paugh, Hannah

doi:10.2140/involve.2017.10.125

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A generalization of Zeckendorf's theorem via circumscribed $m$-gons

Robert Dorward, Pari L. Ford, Eva Fourakis, Pamela E. Harris, Steven J. Miller, Eyvindur Palsson and Hannah Paugh

Vol. 10 (2017), No. 1, 125–150

DOI: 10.2140/involve.2017.10.125

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 $F_{1} = 1$ , $F_{2} = 2$ , and $F_{n} = F_{n - 1} + F_{n - 2}$ for $n \geq 3$ . The distribution of the number of summands in such a decomposition converges to a Gaussian, the gaps between summands converge 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 is needed to assume the coefficients in the recurrence relation are nonnegative 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.

Keywords

Zeckendorf decompositions, longest gap

Mathematical Subject Classification 2010

Primary: 11B39, 11B05

Secondary: 65Q30, 60B10

Milestones

Received: 10 September 2015

Revised: 5 December 2015

Accepted: 13 December 2015

Published: 11 October 2016

Communicated by Stephan Garcia

Authors

Robert Dorward
Department of Mathematics Oberlin College Oberlin, OH 44074 United States

Pari L. Ford
Department of Mathematics and Physics Bethany College Lindsborg, KS 67456 United States

Eva Fourakis
Department of Mathematics and Statistics Williams College Williamstown, MA 01267 United States

Pamela E. Harris
Department of Mathematical Sciences United States Military Academy West Point, NY 10996 United States	Department of Mathematics and Statistics Williams College Williamstown, MA 01267 United States

Steven J. Miller
Department of Mathematics and Statistics Williams College Williamstown, MA 01267 United States

Eyvindur Palsson
Department of Mathematics and Statistics Williams College Williamstown, MA 01267 United States	Department of Mathematics Virginia Tech Blacksburg, VA 24061 United States

Hannah Paugh
Department of Mathematical Sciences United States Military Academy West Point, NY 10996 United States

Vol. 10, No. 1, 2017