Seminar 3.6.14 Carnovale

Title: Gowers Norms and Multiple Recurrence of Sparse Measures on R^d

Speaker: Marc Carnovale, The Ohio State University

Seminar Type:  Ergodic Theory and Probability Seminar

Abstract: It is classical that any positive measure subset of the reals must contain scaled, translated images of any finite configuration of points. Does this still hold for natural classes of ”large” singular sets? A construction of Keleti says that Hausdorff dimension 1 is insufficient to guarantee such a result even for 3-term arithmetic progressions (3APs), while a result of Laba and Pramanik says that the stronger notion of Fourier dimension does yield a result for 3APs, but leaves the case of longer progressions open. Using the notion of intersections of measures from geometric measure theory as a guide, we study a quantity which can be thought of as measuring the multiple recurrence properties under the shift operator of a singular measure, rather than a set, in the torus, and show that it is positive when the measure under question satisfies a certain “higher order Fourier dimension” condition.