Involve: A Journal of Mathematics

  • Involve
  • Volume 10, Number 1 (2017), 125-150.

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

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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 F1 = 1, F2 = 2, and Fn = Fn1 + Fn2 for n 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.

Article information

Source
Involve, Volume 10, Number 1 (2017), 125-150.

Dates
Received: 10 September 2015
Revised: 5 December 2015
Accepted: 13 December 2015
First available in Project Euclid: 22 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1511371087

Digital Object Identifier
doi:10.2140/involve.2017.10.125

Mathematical Reviews number (MathSciNet)
MR3561734

Zentralblatt MATH identifier
1348.11015

Subjects
Primary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 11B05: Density, gaps, topology
Secondary: 65Q30: Recurrence relations 60B10: Convergence of probability measures

Keywords
Zeckendorf decompositions longest gap

Citation

Dorward, Robert; Ford, Pari L.; Fourakis, Eva; Harris, Pamela E.; Miller, Steven J.; Palsson, Eyvindur; Paugh, Hannah. A generalization of Zeckendorf's theorem via circumscribed $m$-gons. Involve 10 (2017), no. 1, 125--150. doi:10.2140/involve.2017.10.125. https://projecteuclid.org/euclid.involve/1511371087


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