Quantitative additive energy estimates for regular sets and connections to discretized sum-product theorems

Series: 
Analysis Seminar
Wednesday, March 28, 2018 - 1:55pm
1 hour (actually 50 minutes)
Location: 
Skiles 005
,  
UCLA
,  
Organizer: 
We prove new quantitative additive energy estimates for a large class of porous measures which include, for example, all Hausdorff measures of Ahlfors-David subsets of the real line of dimension strictly between 0 and 1. We are able to obtain improved quantitative results over existing additive energy bounds for Ahlfors-David sets by avoiding the use of inverse theorems in additive combinatorics and instead opting for a more direct approach which involves the use of concentration of measure inequalities. We discuss some connections with Bourgain's sum-product theorem.