Fred Tyrrell

Multiplicative subgroups of prime fields are not sumsets.

Let H be a proper multiplicative subgroup of F_p, and suppose that H = A+B for some sets A and B. In joint work with Misha Rudnev, we prove that either one of the summands is a singleton, or |A|=|B|=2 and |H|=4. In particular, no proper multiplicative subgroup of F_p can be written as A+B with |A|,|B| > 2.

In this talk, I will motivate the problem, as part of a broader phenomenon of additive vs multiplicative structure, and give some background and context. Then I will sketch the proof, which builds on the Hanson--Petridis polynomial method and Kalmynin’s subsequent resolution of Sárközy’s conjecture for quadratic residues. Finally, I will state some possible future directions of research.