Large values of the additive energy in R^d and Z^d

TL;DR

This paper establishes optimal bounds linking large additive energy in subsets of R^d and Z^d to their containment in small-rank generalized arithmetic progressions, advancing additive combinatorics theory.

Contribution

It proves the optimal bound for the rank of generalized arithmetic progressions containing large energy subsets, refining previous results in additive combinatorics.

Findings

Large additive energy implies containment in small-rank progressions

Optimal bounds for the rank of progressions are established

The proof involves bounds on additive energy in R^d and Z^d

Abstract

Combining Freiman's theorem with Balog-Szemeredi-Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In this paper, we prove the above statement with the optimal bound for the rank of the progression. The proof strategy involves studying upper bounds for additive energy of subsets of R^d and Z^d.