##### Department of Mathematics,

University of California San Diego

****************************

### Math 269 - Combinatorics

## Jozsef Solymosi

#### University of British Columbia

## Sum-product estimates for sets of numbers and reals

##### Abstract:

An old conjecture of Erd\H os and Szemer\'edi states that if $A$ is a finite set of integers then the sum-set or the product-set should be large. The sum-set of $A$ is defined as $A+A=\{a+b | a,b \in A\}$ and the product set is $A\cdot A=\{ab | a,b \in A\}.$ Erd\H os and Szemer\'edi conjectured that the sum-set or the product set is almost quadratic in the size of $A,$ i.e. $\max (|A+A|,|A\cdot A|)\geq c|A|^{2-\delta}$ for any positive $\delta$. I proved earlier that $\max (|A+A|,|A\cdot A|)\geq c|A|^{14/11}/\log{|A|},$ for any finite set of complex numbers, $A.$ In this talk we improve the bound further for sets of real numbers.

Host: Fan Chung Graham

### May 13, 2008

### 4:00 PM

### AP&M 7321

****************************