Sets Doubling and Structures in Non-Abelian Additive Combinatorics

This project studies how subsets of symmetry groups grow when their elements are multiplied together. In many familiar settings, combining two sets makes them much larger, but in some cases the resulting product set remains unexpectedly small, revealing hidden algebraic and geometric structure. Understanding this phenomenon is a central problem in additive combinatorics and has strong connections with harmonic analysis, geometry, and number theory. The project will develop new principles for detecting and describing this hidden structure in noncommutative settings. By advancing fundamental knowledge about symmetry, growth, and structure, the project will strengthen core research in the mathematical sciences and deepen links among several major areas of pure mathematics. It will also support the training of graduate students and junior researchers, promote collaboration across fields, and broaden participation through mentoring, workshops, and public mathematical outreach. The project investigates direct and inverse problems for product sets in locally compact nonabelian groups. Its main goals are to obtain sharp measure-doubling estimates and corresponding inverse and stability theorems, with four interconnected directions: rearrangement inequalities and inverse theorems in Lie groups, with applications to endpoint forms of the Kunze–Stein phenomenon; minimal doubling for sets of arbitrary measure in compact semi-simple Lie groups, toward a continuous analogue of Babai’s conjecture; inverse and stability results for the nonabelian Brunn-Minkowski inequality in noncompact locally compact groups; and applications to Elekes-Szabó-type incidence and counting problems. The methods combine additive-combinatorial and probabilistic arguments with harmonic analysis, model-theoretic ultraproduct and limit methods, quotient and fiber-structure analysis, and the structure theory of locally compact and Lie groups. Expected outcomes include new sharp growth inequalities, structural classifications of near-extremizers, and new bridges among additive combinatorics, geometry, harmonic analysis, and model theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2503063 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Yifan Jing | Institution: OHIO STATE UNIVERSITY, THE, COLUMBUS, OH | Award Amount: $149,997 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2503063 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2503063.html