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Ben Green works on additive combinatorics and number theory. With Emmanuel Breuillard and Terence Tao, he proved a structure theorem for approximate groups, extending the study of sets with small product growth beyond commutative groups.^[ref]

He also coauthored the characteristic-two proof of the polynomial Freiman–Ruzsa conjecture with Timothy Gowers, Freddie Manners and Terence Tao. Both projects turn limited growth under addition or multiplication into structural information; the later proof makes entropy a central quantitative tool.^[ref]

Emmanuel Breuillard works on group theory and its connections with geometry and additive structure. With Ben Green and Terence Tao, he proved the structure theorem for approximate groups.^[ref]

An approximate group is a finite symmetric set whose products can be covered by a bounded number of translates of itself. The theorem finds controlled nilpotent structure behind that bounded growth. It lets a local combinatorial restriction on products constrain the larger algebraic organization of the set.^[ref]

Freddie Manners works on additive combinatorics and its connections with number theory, analysis and ergodic theory. His UC San Diego profile also uses the name Frederick Manners.^[ref]

With Timothy Gowers, Ben Green and Terence Tao, he proved the polynomial Freiman–Ruzsa conjecture in characteristic two. The paper connects additive structure with entropy: sets or random variables whose sums grow unusually little must possess substantial algebraic organization. Its entropy formulation makes inequalities about sums part of a structural theorem in additive combinatorics.^[ref]