Problems in Combinatorial Number Theory.

Dissertation Defense
Thursday, March 27, 2014 - 12:00
1 hour (actually 50 minutes)
Skiles 006
Georgia Tech
The talk consists of two parts.The first part is devoted to results in Discrepancy Theory. We consider geometric discrepancy in higher dimensions (d > 2) and obtain estimates in Exponential Orlicz Spaces.  We establish a series of dichotomy-type results for the discrepancy function which state that if the $L^1$ norm of the discrepancy function is too small (smaller than the conjectural bound), then the discrepancy function has to be very large in some other function space.The second part of the thesis is devoted to results in Additive Combinatorics. For a set with small doubling an order-preserving Freiman 2-isomorphism is constructed which maps the set to a dense subset of an interval. We also present several applications.