Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-12-01T04:32:28.241Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Blasiak, J., Church, T., Cohn, H., Grochow, J. A., Naslund, E., Sawin, W. F. and Umans, C., ‘On cap sets and the group-theoretic approach to matrix multiplication’, Discrete Anal. (2017), Paper No. 3, 27 pp.Google Scholar
Croot, E., Lev, V. F. and Pach, P. P., ‘Progression-free sets in ℤ4 n are exponentially small’, Ann. of Math. (2) 185(1) (2017), 331–337.Google Scholar
Ellenberg, J. S. and Gijswijt, D., ‘On large subsets of 𝔽q n with no three-term arithmetic progression’, Ann. of Math. (2) 185(1) (2017), 339–343.Google Scholar
Fox, J. and Lovász, L. M., ‘A tight bound for Green’s arithmetic triangle removal lemma in vector spaces’, Adv. Math. 321 (2017), 287–297.Google Scholar
Kleinberg, R., Sawin, W. and Speyer, D. E., ‘The growth rate of tri-colored sum-free sets’, Discrete Anal. (2018), Paper No. 12, arXiv:1607.00047v1, 10 pp.Google Scholar
Pebody, L., ‘Proof of a conjecture of Kleinberg–Sawin–Speyer’, Discrete Anal. (2018), Paper No. 13, 7 pp.Google Scholar