<abstract xmlns="http://www.w3.org/1999/xhtml"> Zeckendorf's theorem states that every positive integer can be decomposed uniquely as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gajw between summands for m € [Fn,Fn+1) converge to geometric decay for almost all m as n→ ∞. While similar results are known for many other recurrences, previous work focused on proving Gaussianity for the number of summands or the average gap measure. We derive general conditions, which are easily checked, that yield geometric decay in the individual gap measures of generalized Zerkendorf decompositions attached to many linear recurrence relations. </abstract>

Zeckendorf's theorem states that every positive integer can be decomposed uniquely as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gajw between summands for m € [Fn,Fn+1) converge to geometric decay for almost all m as n→ ∞. While similar results are known for many other recurrences, previous work focused on proving Gaussianity for the number of summands or the average gap measure. We derive general conditions, which are easily checked, that yield geometric decay in the individual gap measures of generalized Zerkendorf decompositions attached to many linear recurrence relations.

Individual Gap Measures from Generalized Zeckendorf Degompositions

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Individual Gap Measures from Generalized Zeckendorf Degompositions"}

Zeckendorf's theorem states that every positive integer can be decomposed uniquely as a sum of nonconsecutive Fibonacci numbers. The distribution of the number...

Individual Gap Measures from Generalized Zeckendorf Degompositions

Article Category: Dedicated to the fifth international conference on Uniform Distribution Theory (UDT 2016) Sopron, Hungary, July 5–8, 2016

Published Online: Jul 22, 2017

Page range: 27 - 36

Received: Aug 01, 2015

Accepted: Dec 14, 2015

DOI: https://doi.org/10.1515/udt-2017-0002

Keywords
Zeckendorf decompositions, individual gap measures, Levy’s Criterion

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Individual Gap Measures from Generalized Zeckendorf Degompositions

Article Category: Dedicated to the fifth international conference on Uniform Distribution Theory (UDT 2016) Sopron, Hungary, July 5–8, 2016

Published Online: Jul 22, 2017

Page range: 27 - 36

Received: Aug 01, 2015

Accepted: Dec 14, 2015

DOI: https://doi.org/10.1515/udt-2017-0002

KeywordsZeckendorf decompositions, individual gap measures, Levy’s Criterion

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Zeckendorf decompositions, individual gap measures, Levy’s Criterion