Additive Energy

Abstract

We introduce the concept of additive energy, which has proved itself indispensable for studying additive problems. We then give two simple applications of the concept. One involves showing that a Sidon set must have large sumset with any other subset. The other is a classical result of Vosper. Recall that, from Kneser’s Theorem, we know |A+B|≥min{|G|, |A|+|B|−1} when $G=\mathbb{Z}$ or G=C _p is cyclic of order p prime. When $G=\mathbb{Z}$, equality can only occur when A and B are arithmetic progressions of common difference or min{|A|, |B|}=1. A result of Vosper shows that, with one minor exception, this also holds for G=C _p .