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In this paper, we study triple-product-free sets, which are analogous to the widely studied concept of product-free sets. A nonempty subset S of a group G is triple-product-free if $abc \notin S$ for all $a, b, c \in S$. If S is triple-product-free and is not a proper subset of any other triple-product-free set, we say that S is locally maximal. We classify all groups containing a locally maximal triple-product-free set of size 1. We then derive necessary and sufficient conditions for a subset of a group to be locally maximal triple-product-free, and conclude with some observations and comparisons with the situation for standard product-free sets.

In this paper we obtain optimal bounds for the length of the longest arithmetic progression in various kinds of sum-sets. As an application, we derive a sharp estimate for the number of sets $A$ of residues modulo a prime $n$ such that no subsum of $A$ equals $x$ modulo $n$, where $x$ is a fixed residue modulo $n$.

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TRIPLE-PRODUCT-FREE SETS

Long arithmetic progressions in sum-sets and the number x-sum-free sets

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TRIPLE-PRODUCT-FREE SETS

Long arithmetic progressions in sum-sets and the number x-sum-free sets