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papers

Publications (13)

math.CA2025

Prescribed projections and efficient coverings by curves in the plane

Alan Chang, Alex McDonald, Krystal Taylor

Davies efficient covering theorem states that an arbitrary measurable set $W$ in the plane can be covered by full lines so that the measure of the union of the lines has the same m…

math.PR2023

A random line intersects $S^{2}$ in two probabilistically independent locations

Dmitriy Bilyk, Alan Chang, Otte HeinÃ¤vaara +2

We consider random lines in $R^{3}$ (random with respect to the kinematic measure) and how they intersect $S^{2}$ . It is known that the entry point and the exit point…

math.CA2015

The Whitney extension theorem in high dimensions

Alan Chang

We prove a variant of the standard Whitney extension theorem for $C^{m} (R^{n})$ , in which the norm of the extension operator has polynomial growth in $n$ for fixed $m$ …

math.NT2013

Newman's conjecture in various settings

Julio Andrade, Alan Chang, Steven J. Miller

De Bruijn and Newman introduced a deformation of the Riemann zeta function $I^¶ (s)$ , and found a real constant $I^$ which encodes the movement of the zeros of $I^¶ (s)$ under the defo…

math.CA2024

Nikodym sets and maximal functions associated with spheres

Alan Chang, Georgios Dosidis, Jongchon Kim

We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp $L^{p}$ -estimates for Nikodym maximal functions associated with spheres. As…

math.MG2025

Random zero sets with local growth guarantees

Alan Chang, Assaf Naor, Kevin Ren

We prove that if $(M, d)$ is an $n$ -point metric space that embeds quasisymmetrically into a Hilbert space, then for every $I ¨ > 0$ there is a random subset $Z$ of…

math.CA2022

Structure of sets with nearly maximal Favard length

Alan Chang, Damian DÄ browski, Tuomas Orponen +1

Let $E \subset B (1) \subset R^{2}$ be an $H^{1}$ measurable set with $H^{1} (E) < \infty$ , and let $L \subset R^{2}$ be a line segment with $\mat…

math.CA2021

Decoupling for fractal subsets of the parabola

Alan Chang, Jaume de Dios Pont, Rachel Greenfeld +3

We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2} L^{p}$ decoupling for a fractal subset on the parabola ${(t, t^{2}) : 0 \leq t \leq 1}$ to stu…

math.CA2019

Analytic capacity and projections

Alan Chang, Xavier Tolsa

In this paper we study the connection between the analytic capacity of a set and the size of its orthogonal projections. More precisely, we prove that if $E \subset C$ is co…

math.CA2026

Sharp Favard length of random Cantor sets

Alan Chang, Pablo Shmerkin, Ville Suomala

We show that for a large class of planar $1$ -dimensional random fractals $S$ , the Favard length $Fav (S (r))$ of the neighborhood $S (r)$ is comparable to $\log^{-1}(1/…

math.NT2014

Newman's conjecture, zeros of the L-functions, function fields

Alan Chang, David Mehrle, Steven J. Miller +3

De Bruijn and Newman introduced a deformation of the completed Riemann zeta function $I^¶$ , and proved there is a real constant $I^$ which encodes the movement of the nontrivial zer…

math.MG2018

Small unions of affine subspaces and skeletons via Baire category

Our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of the $k$ -skeleton of a fixed polytope centered at the points of a given set. For m…

math.CA2020

The Kakeya needle problem and the existence of Besicovitch and Nikodym sets for rectifiable sets

Alan Chang, Marianna CsÃ¶rnyei

We solve the Kakeya needle problem and construct a Besicovitch and a Nikodym set for rectifiable sets.