5 research outputs found
Estimates of the Discrepancy Function in Exponential Orlicz Spaces
We prove that in all dimensions n at least 3, for every integer N there
exists a distribution of points of cardinality , for which the associated
discrepancy function D_N satisfies the estimate an estimate, of sharp growth
rate in N, in the exponential Orlicz class exp)L^{2/(n+1)}. This has recently
been proved by M.~Skriganov, using random digit shifts of binary digital nets,
building upon the remarkable examples of W.L.~Chen and M.~Skriganov. Our
approach, developed independently, complements that of Skriganov.Comment: 13 page
Problems in combinatorial number theory
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.Ph.D