One of the central problems in combinatorics is to study the structure of “nonexpanding” objects. A well-known example is Freiman-type problem: Given an ambient group G and a subset A of size n, suppose|A+A| < K|A|for some constant K, A should be well structured. It was conjectured by Ruzsa that when G is an Abelian group of finite torsion r, then the size of the affine span of A is bounded by r^CK|A|for some absolute constant C. The case r = 2 is resolved by Green and Tao, and the case when r is prime is resolved by Even-Zohar and Lovett. With Souktik Roy (UIUC), we confirm the conjecture when r is a power of prime. Another problem along this line is to study the structure of non-expanding polynomials, i.e., Elekes–Ronyai type problem. With Souktik Roy and Chieu-Minh Tran (Notre Dame), we obtain a sumproduct phenomenon for pairs of polynomials over characteristic 0 fields. Special cases of this phenomenon encompass some known results, such as Erdős–Szemerédi Theorem, Elekes–Ronyai Theorem, a result by Elekes–Nathanson–Ruzsa, a result by Shen, and a 5/4-bound on Bourgain–Katz–Tao Theorem for very large finite fields by Grosu.

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