1. On the size of $k$-fold sum and product sets of integers
- Author
-
Mei-Chu Chang and Jean Bourgain
- Subjects
Discrete mathematics ,Mathematics::Combinatorics ,Conjecture ,Mathematics - Number Theory ,Applied Mathematics ,General Mathematics ,Combinatorics ,Integer ,FOS: Mathematics ,Mathematics - Combinatorics ,11P70 ,Combinatorics (math.CO) ,Number Theory (math.NT) ,Finite set ,Mathematics - 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��s and Szemer��di on sum and product sets., 33 pages, no figures, submitted, J. Amer. Math. Soc. (Proxy submission)
- Published
- 2003