arXiv subjects
Publications and source records attributed to William O'Regan.
We present two applications of recent developments in incidence geometry. One is a $δ$-discretised version of a particular `Elekes--Rónyai' expander problem. The second application is an incidence estimate addressing the scenario when both tubes, squares and their shadings satisfy non-concentration assumptions.
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 $4/3$-bound for finite subsets of $\mathbb{R}.$
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 application, we present a result pertaining to the Fourier decay of fractal measures on convex curves.
Utilising recent advances in incidence geometry for balls and tubes, and advances in sum-product theory in the discrete setting, we show that for $0 < s \leq 1/2$ and for any $A \subset \mathbb{R}$ with Hausdorff dimension $s$, either the upper-box dimension of $AA$, or the lower-box dimension of $A+A$ must be at least $29s/23$. We obtain the slightly better bound of $33 s / 26$ when we replace the sum-set with the smoother difference-set.
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.
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 geometric measure theory, harmonic analysis, and homogeneous dynamics. Our main innovation is a short elementary argument that shows that a discretised subset of $\R$ satisfying a weak ``two-ends'' spacing condition is expanded by a polynomial to a set of positive Lebesgue measure.
We use recent advances in the theory of Furstenberg sets to prove new incidence results of Szemerédi--Trotter strength for $δ$-discretized structures with Cartesian product flavor. We use these results to make progress on a number of problems that include energy estimates and Fourier decay of fractal measures supported on curves, as well as various sum-product-like results governed by fractal dimension.
By making use of arithmetic information inequalities, we give a strong quantitative bound for the discretised ring theorem. In particular, we show that if $A \subset [1,2]$ is a $(δ,σ)$-set, with $|A| = δ^{-σ},$ then $A+A$ or $AA$ has $δ$-covering number at least $δ^{-c}|A|$ for any $0 < c < \min\{σ/6, (1-σ)/6\}$ provided that $δ> 0$ is small enough.
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 tubes of arbitrarily small total width. We remark that a more general class of sponge-like sets satisfy this property. For a given $ε> 0$ the proof is able to give an explicit description of the tubes for which the total width is less than $ε.$