5 papers
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.
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 flavor.…
Covering sponges with tubes
William O'Regan
The aim of this note is to give a short proof of a result of Pyörälä--Shmerkin--Suomala--Wu; the Sierpiński carpet, and generalisations, are tube-null; they can be covered with tub…