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Best, A, Dynes, P, Edelsbrunner, X, McDonald, B, Miller, SJ, Tor, K, Turnage-Butterbaugh, C and Weinstein, M (2014)

Benford Behavior of Zeckendorf Decompositions

Fibonacci Quarterly 52(5), pp. 35Ė46.

ISSN/ISBN: Not available at this time. DOI: Not available at this time.



Abstract: A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers {Fi}∞i=1. A set S ⊂ Z is said to satisfy Benfordís law if the density of the elements in S with leading digit d is log (1 + 1 ). 10 d We prove that, as n → ∞, for a randomly selected integer m in [0,Fn+1) the distribution of the leading digits of the Fibonacci summands in its Zeckendorf decomposition converge to Benfordís law almost surely. Our results hold more generally; instead of looking at the distribution of leading digits of summands in Zeckendorf decompositions, one obtains simi- lar theorems concerning how often values in sets with positive density inside the Fibonacci numbers are attained in these decompositions.


Bibtex:
@article {, AUTHOR = {Best, Andrew and Dynes, Patrick and Edelsbrunner, Xixi and Mcdonald, Brian and Miller, Steven and Tor, Kimsy and Turnage-Butterbaugh, Caroline and Weinstein, Madeleine}, TITLE = {Benford Behavior of Zeckendorf Decompositions}, JOURNAL = {Fibonacci Quarterly}, YEAR = {2014}, VOLUME = {52}, NUMBER = {5}, PAGES = {35--46}, DOI = {}, URL = {https://www.fq.math.ca/Papers1/52-5/Best-Benford.pdf},


Reference Type: Journal Article

Subject Area(s): Number Theory