Freiman–Ruzsa polynomial conjecture

Establish the Freiman–Ruzsa polynomial conjecture concerning the polynomial quantitative bounds in Freiman-type structure theory.

Background

The paper develops a categorical framework for additive sets and Freiman homomorphisms and concludes by identifying the categorical formulation and proof of Freiman’s structure theorem as an immediate direction for further work. In discussing the need to track quantitative complexity in such a framework, the authors cite the Freiman–Ruzsa polynomial conjecture as a related example of a result whose quantitative structure would need to be handled. The paper does not state the conjecture’s precise mathematical formulation or resolve it.

References

It seems that this would require enriching the world of objects and morphisms with suitably designed quantitative measures so that we can keep track of the complexity (size) of various structures that occur in this theorem and various others, for example, the Freiman-Ruzsa polynomial conjecture.

— A categorical approach to additive combinatorics  (2501.02097 - Blanco et al., 3 Jan 2025) in Section 6, Concluding remarks