8 papers · 1 filter
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.
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…
Sum-product phenomena for Ahlfors-regular sets
William O'Regan
We utilise the recent work of Orponen to yield a sum-product result for Ahlfors-regular sets. As a corollary, we obtain the fractal analogue of Solymosi's -bound for finite su…
Discretised sum-product theorems by Shannon-type inequalities
András Máthé, William O'Regan
By making use of arithmetic information inequalities, we give a strong quantitative bound for the discretised ring theorem. In particular, we show that if is a $(…