Finding a Low-dimensional Piece of a Set of Integers

DOI10.1093/IMRN/RNW153zbMATH Open1405.11010arXiv1512.06272OpenAlexW2962862232MaRDI QIDQ4612006

Publication date: 22 January 2019

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

Abstract: We show that a finite set of integers

A s u b s e t e q m a t h b b Z

with

| A + A | l e K | A |

contains a large piece

X s u b s e t e q A

with Freu{i}man dimension

O (l o g K)

, where large means

| A | / | X | l l e x p (O (l o g^{2} K))

. This can be thought of as a major quantitative improvement on Freu{i}man's dimension lemma, or as a "weak" Freu{i}man--Ruzsa theorem with almost polynomial bounds. The methods used, centered around an "additive energy increment strategy", differ from the usual tools in this area and may have further potential. Most of our argument takes place over

m a t h b b F_{2}^{n}

, which is itself curious. There is a possibility that the above bounds could be improved, assuming sufficiently strong results in the spirit of the Polynomial Freu{i}man--Ruzsa Conjecture over finite fields.

Full work available at URL: https://arxiv.org/abs/1512.06272

zbMATH Keywords

additive energy Freĭman dimension Freĭman-Ruzsa conjecture

Mathematics Subject Classification ID

Arithmetic combinatorics; higher degree uniformity (11B30)