IMSc Webinar

Sum-free sets, counting sets with small sumset and hypergraph container lemma

Gyan Prakash

HRI

A subset $A$ of an abelian group $G$ is said to be sum-free if there is no solution of the equation $x+y = z$ with $x, y, z \in A.$ In an earlier work along with R. Balasubramanian and D.S. Ramana, we had obtained an upper as well as lower bound for the number of sum-free subsets in finite abelian group of type III, of a given exponent, in terms of the number of sets with small sumset, that is the number of $A$ with $|A| = k_1$ and $|A+A| \leq k_2.$ Ben Green had obtained an upper bound for the number of sets with small sumset in a finite group $G$, when either $G$ is cyclic or is a vector space over a finite field and we had refined his methods to obtain bounds in a general finite abelian group. These bounds, though improved the known upper bound for the number of sum-free subsets, are inadequate to give the correct order for the number of sum-free sets. In 2020, Marcelo Campos, using the method of hypergraph containers developed by Balogh, Morris, Samotij and Saxton, obtained an upper bound for the number of sets with small sumset. His bounds are better than our bounds for certain ranges of $k_1$ and $k_2.$ In this talk we shall briefly explain the relation between sum-free sets and sets with small sumset and give an exposition of the method of hypergraph containers and the result of Campos.
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