Alex Iosevich
Introductions to central ideas in harmonic analysis, geometric measure theory, and additive combinatorics, written to make the main questions and methods accessible.
Featured expository article
This article introduces one of the central open problems in geometric measure theory: how large a compact subset of Euclidean space must be before its set of pairwise distances necessarily has positive Lebesgue measure. It explains the conjecture's relationship with the Erdős distinct-distances problem and the role of Fourier analysis in the subject.
Alex Iosevich, Notices of the American Mathematical Society 66 (2019), no. 4, 552–555.
Short introductions
Self-contained accounts of several classical results and the Fourier-analytic ideas behind them.
A short proof, using the Stein–Tomas restriction theorem, that a set of Hausdorff dimension greater than (d+1)/2 has a distance set of positive Lebesgue measure. This perspective became influential in later work on the problem.
Read the note (PDF) →A nearly self-contained proof that a domain tiles Euclidean space by translations along a lattice precisely when the dual lattice gives an orthogonal basis of exponentials for its L2 space. The argument essentially rebuilds the needed Poisson summation formula.
Read the note (PDF) →A simple, self-contained proof that every subset of the positive integers with positive upper density contains a three-term arithmetic progression.
Read the note (PDF) →Five lectures
This series on the Kakeya problem, the restriction conjecture, and related questions was delivered in the Probability Learning Seminar at the University of Missouri–Columbia during the fall semester of 2000. The subject has advanced considerably since then, but the lectures remain useful introductions to several fundamental techniques and historical ideas.
For research papers, books, and a complete publication list, please see my curriculum vitae.