9 papers
Two-ends Furstenberg inequality for transversal families and applications to Fourier decay
William O'Regan, Shukun Wu, Guangzeng Yi
We generalise the recent two-ends Furstenberg inequality due to Wang and the second author from lines to a family of transversal curves, and give a much simplified proof. As an app…
A note on the sum-product problem for fractal sets
Adam Cushman, William O'Regan
Utilising recent advances in incidence geometry for balls and tubes, and advances in sum-product theory in the discrete setting, we show that for and for any $A \s…
Bourgain's projection theorem over normed division algebras
William O'Regan
We give a simple and self-contained proof of an extension of a projection theorem of Bourgain over the reals to division algebras over local fields of zero characteristic.
New discretised polynomial expander and incidence estimates
Ciprian Demeter, William O'Regan
We present two applications of recent developments in incidence geometry. One is a -discretised version of a particular `Elekes--Rónyai' expander problem. The second applicati…
Simple proofs of discretised projection theorems
William O'Regan, Pablo Shmerkin, Hong Wang
We give a simple, short and self-contained presentation of Bourgain's discretised projection theorem from 2010, which is a fundamental tool in many recent breakthroughs in geometri…
Incidence estimates for quasi-product sets and applications
Ciprian Demeter, William O'Regan
We use recent advances in the theory of Furstenberg sets to prove new incidence results of Szemerédi--Trotter strength for -discretized structures with Cartesian product flavo…