On a discrete approach to lower bounds in discrepancy theory

Abstract

In this paper, we prove that some renowned lower bounds in discrepancy theory admit a discrete analogue. Namely, we prove that the lower bound of the discrepancy for corners in the unit cube due to Roth holds true also for a suitable finite family of corners. We also prove two analogous results for the discrepancy on the torus with respect to squares and balls.

29 pages, 1 figure. Accepted for publication on Journal of Fourier Analysis and Applications