Sieve Methods Beyond Commutativity: Unifying Non-Abelian Growth and Additive Problems

I. Shkredov’s 2021 survey highlights the underexplored potential of non-commutative group actions in additive number theory. While the “affine sieve” and growth results in non-abelian groups have found some applications, there is a dearth of systematic sieve constructions in these settings. This research would develop new sieve frameworks that operate in non-abelian group environments, possibly using growth and expansion properties to control and count additive structures. Such a synthesis could resolve conflicts where classical (commutative) sieves fail, particularly in questions involving sum-product phenomena and incidence geometry. This project not only extends classical sieve theory but also addresses the “reconfigure the conceptual framework” and “generate theories from conflict” heuristics by merging disparate algebraic and additive perspectives.

References:

1. Non-commutative methods in additive combinatorics and number theory. I. Shkredov (2021). Russian Mathematical Surveys.
2. Non-commutative methods in additive combinatorics and number theory. I. Shkredov (2021). Russian Mathematical Surveys.
3. Non-commutative methods in additive combinatorics and number theory. I. Shkredov (2021). Russian Mathematical Surveys.
4. Non-commutative methods in additive combinatorics and number theory. I. Shkredov (2021). Russian Mathematical Surveys.