Problems in combinatorial number theory

Geometric discrepancy;Additive energy, Freiman isomorphism

Amirkhanyan, Gagik M. ; Wick, Brett Mathematics Lacey, Michael T. Lubinsky, Doron Geronimo, Jeff Bilyk, Dmitriy ; Wick, Brett

University:Georgia Institute of Technology

Department:Mathematics

关键词: Geometric discrepancy;    Additive energy, Freiman isomorphism;   

Others  :  https://smartech.gatech.edu/bitstream/1853/51865/1/AMIRKHANYAN-DISSERTATION-2014.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

The dissertation 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 L¹ 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.