Gauss-Benford Rules and Their Harmonic Peers

Julio Rives

doi:doi:10.11648/j.ml.20241003.11

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Gauss-Benford Rules and Their Harmonic Peers

Julio Rives^*

Published in Mathematics Letters (Volume 10, Issue 3)

Received: 5 November 2024 Accepted: 25 November 2024 Published: 18 December 2024

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Abstract

We research the relationship between the probability functions of the twofold hyperbolic universe, consisting of the logarithmic (real) and harmonic (rational) worlds. Within the logarithmic realm, we study the connection between the Gauss- Kuzmin distribution and Newcomb-Benford law and prove that they are fundamentally equivalent; the former corresponds to the probability decrements of the latter, i.e., log(2,1+1/(k(k+2))) is the difference between the function log(2,1+1/n) evaluated at n=k and n=k+1, where k is the index of a coefficient of a real number’s regular continued fraction expansion and n is a positive numeral written in positional notation. Thus, the binary Newcomb-Benford probability of n=1 is the sum of all the Gauss-Kuzmin masses, of n=2, is the sum of all the Gauss-Kuzmin masses minus the Gauss-Kuzmin mass at k=1, and of n=m, is the sum of all the Gauss-Kuzmin masses minus the sum of the first m-1 ones. Besides, the extrapolation of the Gauss-Kuzmin measure outside the unit interval subsumes Newcomb-Benford’s cumulative distribution function. These findings lead to the Gauss-Benford law, which specifies that the occurrence possibility of a positive real number represented in positional notation is log(2,1+x) (i.e., the Gauss-Kuzmin measure) if x is within the unit interval and log(2,1+1/x) if x is outside. Geometrically, these possibilities indicate proximity to 1. The map between both domain partitions is a sheer inversion, the unique conformal transformation that fixes one and respects the minimum information principle. Moreover, we introduce the Gauss-Benford measure, log(1+1/x,1+y), as the probability of a random variable accumulated between x-1 and x, with density 1/((1+y)ln(1+1/x)), where 0≤x-1

Published in	Mathematics Letters (Volume 10, Issue 3)
DOI	10.11648/j.ml.20241003.11
Page(s)	24-35
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Gauss-Kuzmin Distribution, Regular Continued Fraction, Positional Notation, Newcomb-Benford Law, Gauss-Benford Law, Smallness, Harmonic Scale, Logarithmic Scale

References

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[8]	Eduard Wirsing. On the Theorem of Gauss-Kusmin-Lvy and a Frobenius-type Theorem for Function Spaces. Acta Arithmetica, 24(5): 507-528, 1974.
[9]	Michelle He, Brian Huang, Aadeesh Shastry, and Mirilla Zhu. Benford’s Law in Linear Recurrences and Continued Fractions. University of Washington, August 2017.
[10]	Linas Vepstas. Entropy of Continued Fractions (Gauss- Kuzmin Entropy). Unpublished, 2014.
[11]	Nelson M. Blachman. The Continued Fraction as an Information Source (Corresp.). IEEE Transactions on Information Theory, 30(4): 671-674, July 1984.
[12]	Dan L. (Mathoverflow). Gauss-Kuzmin Theorem (continued fractions) - Why Is important? MathOverflow, November 2011.
[13]	Doug Hensley. Metric Diophantine Approximation and Probability. New York Journal of Mathematics, 4: 249- 257, July 1998.
[14]	Viviane Baladi and Brigitte Valle. Euclidean Algorithms Are Gaussian. Journal of Number Theory, 110(2): 331- 386, February 2005.
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[18]	George J. Klir and Bo Yuan. Fuzzy Sets and Fuzzy Logic - Theory and Applications. Prentice Hall PTR, 1995.
[19]	John James Milne. An Elementary Treatise on Cross- ratio Geometry: With Historical Notes. The University Press, 1911.
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[22]	Linas Vepstas. The Gauss-Kuzmin-Wirsing Operator. Unpublished, 2012. Revised 7 July 2020.
[23]	Julio Rives. The Code Underneath. (Submitted to) Axioms, pages 1-60, 2024.

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APA Style

Rives, J. (2024). Gauss-Benford Rules and Their Harmonic Peers. Mathematics Letters, 10(3), 24-35. https://doi.org/10.11648/j.ml.20241003.11

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ACS Style

Rives, J. Gauss-Benford Rules and Their Harmonic Peers. Math. Lett. 2024, 10(3), 24-35. doi: 10.11648/j.ml.20241003.11

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Rives J. Gauss-Benford Rules and Their Harmonic Peers. Math Lett. 2024;10(3):24-35. doi: 10.11648/j.ml.20241003.11

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@article{10.11648/j.ml.20241003.11,
  author = {Julio Rives},
  title = {Gauss-Benford Rules and Their Harmonic Peers},
  journal = {Mathematics Letters},
  volume = {10},
  number = {3},
  pages = {24-35},
  doi = {10.11648/j.ml.20241003.11},
  url = {https://doi.org/10.11648/j.ml.20241003.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20241003.11},
  abstract = {We research the relationship between the probability functions of the twofold hyperbolic universe, consisting of the logarithmic (real) and harmonic (rational) worlds. Within the logarithmic realm, we study the connection between the Gauss- Kuzmin distribution and Newcomb-Benford law and prove that they are fundamentally equivalent; the former corresponds to the probability decrements of the latter, i.e., log(2,1+1/(k(k+2))) is the difference between the function log(2,1+1/n) evaluated at n=k and n=k+1, where k is the index of a coefficient of a real number’s regular continued fraction expansion and n is a positive numeral written in positional notation. Thus, the binary Newcomb-Benford probability of n=1 is the sum of all the Gauss-Kuzmin masses, of n=2, is the sum of all the Gauss-Kuzmin masses minus the Gauss-Kuzmin mass at k=1, and of n=m, is the sum of all the Gauss-Kuzmin masses minus the sum of the first m-1 ones. Besides, the extrapolation of the Gauss-Kuzmin measure outside the unit interval subsumes Newcomb-Benford’s cumulative distribution function. These findings lead to the Gauss-Benford law, which specifies that the occurrence possibility of a positive real number represented in positional notation is log(2,1+x) (i.e., the Gauss-Kuzmin measure) if x is within the unit interval and log(2,1+1/x) if x is outside. Geometrically, these possibilities indicate proximity to 1. The map between both domain partitions is a sheer inversion, the unique conformal transformation that fixes one and respects the minimum information principle. Moreover, we introduce the Gauss-Benford measure, log(1+1/x,1+y), as the probability of a random variable accumulated between x-1 and x, with density 1/((1+y)ln(1+1/x)), where 0≤x-1
											
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TY  - JOUR
T1  - Gauss-Benford Rules and Their Harmonic Peers
AU  - Julio Rives
Y1  - 2024/12/18
PY  - 2024
N1  - https://doi.org/10.11648/j.ml.20241003.11
DO  - 10.11648/j.ml.20241003.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 24
EP  - 35
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20241003.11
AB  - We research the relationship between the probability functions of the twofold hyperbolic universe, consisting of the logarithmic (real) and harmonic (rational) worlds. Within the logarithmic realm, we study the connection between the Gauss- Kuzmin distribution and Newcomb-Benford law and prove that they are fundamentally equivalent; the former corresponds to the probability decrements of the latter, i.e., log(2,1+1/(k(k+2))) is the difference between the function log(2,1+1/n) evaluated at n=k and n=k+1, where k is the index of a coefficient of a real number’s regular continued fraction expansion and n is a positive numeral written in positional notation. Thus, the binary Newcomb-Benford probability of n=1 is the sum of all the Gauss-Kuzmin masses, of n=2, is the sum of all the Gauss-Kuzmin masses minus the Gauss-Kuzmin mass at k=1, and of n=m, is the sum of all the Gauss-Kuzmin masses minus the sum of the first m-1 ones. Besides, the extrapolation of the Gauss-Kuzmin measure outside the unit interval subsumes Newcomb-Benford’s cumulative distribution function. These findings lead to the Gauss-Benford law, which specifies that the occurrence possibility of a positive real number represented in positional notation is log(2,1+x) (i.e., the Gauss-Kuzmin measure) if x is within the unit interval and log(2,1+1/x) if x is outside. Geometrically, these possibilities indicate proximity to 1. The map between both domain partitions is a sheer inversion, the unique conformal transformation that fixes one and respects the minimum information principle. Moreover, we introduce the Gauss-Benford measure, log(1+1/x,1+y), as the probability of a random variable accumulated between x-1 and x, with density 1/((1+y)ln(1+1/x)), where 0≤x-1
											
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Author Information

Julio Rives

Department of Artificial Intelligence, Universidad Nacional de Educacion a Distancia (UNED), Madrid, Spain

Contact Email

http://orcid.org/0000-0003-3980-5829

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APA Style

Rives, J. (2024). Gauss-Benford Rules and Their Harmonic Peers. Mathematics Letters, 10(3), 24-35. https://doi.org/10.11648/j.ml.20241003.11

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ACS Style

Rives, J. Gauss-Benford Rules and Their Harmonic Peers. Math. Lett. 2024, 10(3), 24-35. doi: 10.11648/j.ml.20241003.11

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AMA Style

Rives J. Gauss-Benford Rules and Their Harmonic Peers. Math Lett. 2024;10(3):24-35. doi: 10.11648/j.ml.20241003.11

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@article{10.11648/j.ml.20241003.11,
  author = {Julio Rives},
  title = {Gauss-Benford Rules and Their Harmonic Peers},
  journal = {Mathematics Letters},
  volume = {10},
  number = {3},
  pages = {24-35},
  doi = {10.11648/j.ml.20241003.11},
  url = {https://doi.org/10.11648/j.ml.20241003.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20241003.11},
  abstract = {We research the relationship between the probability functions of the twofold hyperbolic universe, consisting of the logarithmic (real) and harmonic (rational) worlds. Within the logarithmic realm, we study the connection between the Gauss- Kuzmin distribution and Newcomb-Benford law and prove that they are fundamentally equivalent; the former corresponds to the probability decrements of the latter, i.e., log(2,1+1/(k(k+2))) is the difference between the function log(2,1+1/n) evaluated at n=k and n=k+1, where k is the index of a coefficient of a real number’s regular continued fraction expansion and n is a positive numeral written in positional notation. Thus, the binary Newcomb-Benford probability of n=1 is the sum of all the Gauss-Kuzmin masses, of n=2, is the sum of all the Gauss-Kuzmin masses minus the Gauss-Kuzmin mass at k=1, and of n=m, is the sum of all the Gauss-Kuzmin masses minus the sum of the first m-1 ones. Besides, the extrapolation of the Gauss-Kuzmin measure outside the unit interval subsumes Newcomb-Benford’s cumulative distribution function. These findings lead to the Gauss-Benford law, which specifies that the occurrence possibility of a positive real number represented in positional notation is log(2,1+x) (i.e., the Gauss-Kuzmin measure) if x is within the unit interval and log(2,1+1/x) if x is outside. Geometrically, these possibilities indicate proximity to 1. The map between both domain partitions is a sheer inversion, the unique conformal transformation that fixes one and respects the minimum information principle. Moreover, we introduce the Gauss-Benford measure, log(1+1/x,1+y), as the probability of a random variable accumulated between x-1 and x, with density 1/((1+y)ln(1+1/x)), where 0≤x-1
									
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TY  - JOUR
T1  - Gauss-Benford Rules and Their Harmonic Peers
AU  - Julio Rives
Y1  - 2024/12/18
PY  - 2024
N1  - https://doi.org/10.11648/j.ml.20241003.11
DO  - 10.11648/j.ml.20241003.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 24
EP  - 35
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20241003.11
AB  - We research the relationship between the probability functions of the twofold hyperbolic universe, consisting of the logarithmic (real) and harmonic (rational) worlds. Within the logarithmic realm, we study the connection between the Gauss- Kuzmin distribution and Newcomb-Benford law and prove that they are fundamentally equivalent; the former corresponds to the probability decrements of the latter, i.e., log(2,1+1/(k(k+2))) is the difference between the function log(2,1+1/n) evaluated at n=k and n=k+1, where k is the index of a coefficient of a real number’s regular continued fraction expansion and n is a positive numeral written in positional notation. Thus, the binary Newcomb-Benford probability of n=1 is the sum of all the Gauss-Kuzmin masses, of n=2, is the sum of all the Gauss-Kuzmin masses minus the Gauss-Kuzmin mass at k=1, and of n=m, is the sum of all the Gauss-Kuzmin masses minus the sum of the first m-1 ones. Besides, the extrapolation of the Gauss-Kuzmin measure outside the unit interval subsumes Newcomb-Benford’s cumulative distribution function. These findings lead to the Gauss-Benford law, which specifies that the occurrence possibility of a positive real number represented in positional notation is log(2,1+x) (i.e., the Gauss-Kuzmin measure) if x is within the unit interval and log(2,1+1/x) if x is outside. Geometrically, these possibilities indicate proximity to 1. The map between both domain partitions is a sheer inversion, the unique conformal transformation that fixes one and respects the minimum information principle. Moreover, we introduce the Gauss-Benford measure, log(1+1/x,1+y), as the probability of a random variable accumulated between x-1 and x, with density 1/((1+y)ln(1+1/x)), where 0≤x-1
									
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