On the size of $k$-fold sum and product sets of integers

Jean Bourgain; Mei-Chu Chang

On the size of $k$-fold sum and product sets of integers

Combinatorics 2007-05-23 v1 Number Theory

Authors: Jean Bourgain , Mei-Chu Chang

Abstract

We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$ , such that for every finite set $A$ of integers with cardinality $|A| > 1$ , we have either $|A + ... + A| \geq |A|^b$ or $|A \cdot ... \cdot A| \geq |A|^b$ where $A + ... + A$ and $A \cdot ... \cdot A$ are the collections of $k$ -fold sums and products of elements of $A$ respectively. This is progress towards a conjecture of Erd\"os and Szemer\'edi on sum and product sets.

Keywords

sum sets product sets finite sets of integers erdos-szemeredi conjecture k-fold sums k-fold products

Cite

@article{arxiv.math/0309055,
  title  = {On the size of $k$-fold sum and product sets of integers},
  author = {Jean Bourgain and Mei-Chu Chang},
  journal= {arXiv preprint arXiv:math/0309055},
  year   = {2007}
}

Comments

33 pages, no figures, submitted, J. Amer. Math. Soc. (Proxy submission)

On the size of $k$-fold sum and product sets of integers

Abstract

Keywords

Cite

Comments

Related papers