Speaker
Will O’Regan, University of British Columbia
Abstract
Post the resolution of the Furstenberg set conjecture it is a reasonable question, both in the sense of furthering human enlightenment, and for applications to other mathematical problems, to ask to what extent the non-concentration hypotheses may be relaxed and still obtain non-trivial bounds. To this end, we present a sharp incidence bound for points with a Katz-Tao non-concentration condition and tubes which satisfy a very weak `three-ends’ condition, that is, through each point there are three separated bundles of tubes, with each bundle having roughly the same size. This may be considered to be the weakest condition on the tubes to gain something better than what one may obtain `trivially’. As an application, we present an upper bound on the number of approximate rectangles that may be drawn with vertices in a set of points with some non-concentration. Joint with with Ciprian Demeter.