On the product of differences of sets in finite fields

Series: 
Combinatorics Seminar
Friday, January 22, 2016 - 16:00
1 hour (actually 50 minutes)
Location: 
Skiles 154
,  
University of Rochester
,  
Organizer: 
We show that there exists an absolute constant c>0 with the following property. Let A be a set in a finite field with q elements.  If |A|>q^{2/3-c}, then the set (A-A)(A-A) consisting of products of pairwise differences of elements of A contains at least q/2 elements. It appears that this is the first instance in the literature where such a conclusion is reached for such type sum-product-in-finite-fileds questions for sets of smaller cardinality than q^{2/3}. Similar questions have been investigated by Hart-Iosevich-Solymosi and Balog.