A DISTRIBUTION ON TRIPLES WITH MAXIMUM ENTROPY MARGINAL
Copyright policy )A DISTRIBUTION ON TRIPLES WITH MAXIMUM ENTROPY MARGINAL
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.
Arithmetic progressions, Combinatorial structures in finite projective spaces, discrete non-negative integer-valued distributions, Croot-Lev-Pach polynomial method, Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.), Arithmetic combinatorics; higher degree uniformity, arithmetic progressions, 11B30 (primary), Additive bases, including sumsets, 05D40 (secondary), tri-colored sum-free sets, QA1-939, FOS: Mathematics, additive combinatorics, Mathematics - Combinatorics, Combinatorics (math.CO), Mathematics
Arithmetic progressions, Combinatorial structures in finite projective spaces, discrete non-negative integer-valued distributions, Croot-Lev-Pach polynomial method, Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.), Arithmetic combinatorics; higher degree uniformity, arithmetic progressions, 11B30 (primary), Additive bases, including sumsets, 05D40 (secondary), tri-colored sum-free sets, QA1-939, FOS: Mathematics, additive combinatorics, Mathematics - Combinatorics, Combinatorics (math.CO), Mathematics
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