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A DISTRIBUTION ON TRIPLES WITH MAXIMUM ENTROPY MARGINAL

Part of: Sequences and sets Extremal combinatorics

Published online by Cambridge University Press:  09 December 2019

SERGEY NORIN
SERGEY NORIN*
Affiliation:
Department of Mathematics and Statistics, McGill University, Montréal, Canada; sergey.norin@mcgill.ca

Abstract

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We construct an $S_{3}$-symmetric probability distribution on $\{(a,b,c)\in \mathbb{Z}_{{\geqslant}0}^{3}\,:\,a+b+c=n\}$ such that its marginal achieves the maximum entropy among all probability distributions on $\{0,1,\ldots ,n\}$ with mean $n/3$. Existence of such a distribution verifies a conjecture of Kleinberg et al. [‘The growth rate of tri-colored sum-free sets’, Discrete Anal. (2018), Paper No. 12, arXiv:1607.00047v1], which is motivated by the study of sum-free sets.

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Secondary: 05D40: Probabilistic methods
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e46
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2019

References

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