On Additive Doubling and Energy

Abstract

We discuss some ideas related to the polynomial Freiman–Ruzsa conjecture. We show that there is a universal $\epsilon>0$ so that any subset of an abelian group with subtractive doubling K must be polynomially related to a set with additive energy at least $\frac{1}{K^{1-\epsilon}}$. This means that the main difficulty in proving the polynomial Freiman–Ruzsa conjecture consists of studying sets whose energy is greater than that implied by their doubling. One example is a generalized arithmetic progression of high dimension which cannot occur in the finite characteristic setting.

Locations

• SIAM Journal on Discrete Mathematics - View
• arXiv (Cornell University) - View - PDF