This is an advanced course in analytic number theory and harmonic analysis for the Taught Course Centre network (see "Running Courses" therein). Our main focus will be to introduce the Fourier analytic machinery to cover arithmetic problems that arise in harmonic analysis. The main tool here is the circle method which we commence the course with an introduction of. We then focus on the discrete spherical maximal function, proving its relevant sharp bounds and surveying future directions. We close the course by discussing discrete restriction theory where we introduce the three main methods in the subject.
To sign-up for the TCC course: Note that your university must be a part of the TCC network in order to sign up for the course. If it is, follow the directions here to sign up (it's just sending an email to the right person with the right info).
Format: The lectures are pre-recorded in a seminar that I am running in parallel this semester. You may find them on YouTube. I expect you to watch the lectures ahead of time so that I may use the TCC course time to answer questions and clarify any salient points.
If you are unable to sign-up for the TCC course, you are welcome to watch the lectures on YouTube and email me with any questions. I will answer questions as I can, but if I do not have enough time, precedence goes to the students registered for the course.
I expect that the reader is familiar with - or will learn independently in parallel:
The essentials of analysis as exposed in, say Stein and Shakarchi's Lecture Series in Analysis, especially the books on Fourier Analysis, Complex Analysis and Real Analysis. In particular, I expect the participants to be comfortable with Fourier transforms and series as well as Poisson summation.
I will assume less knowledge from analytic number theory, but a good foundation in the subject will be useful. For instance, I will expect you to be familiar with the fundamentals of p-adic numbers.
There are three parts to the course:
An introduction to the circle method and Waring's problem (Lectures 1-4)
The discrete spherical averages and their maximal functions (Lectures 5-10)
The fundamentals of discrete restriction theory (Lectures 11-15)
Following an overview of this course (Lecture 0), the lectures are as follows.
Sums of squares I: The divisor bound for sums of squares and the delta function as an oscillatory integral.
Sums of squares II: The Farey decomposition, Poisson summation, and the major arcs vs minor arcs paradigm.
Sums of squares III: The Weyl’s bounds, the principle of stationary phase, and completing the singular integral and the singular series.
Sums of squares IV: Interpreting the singular integral as a volume and the singular series as a product.
Sums of squares V: p-adic singular integrals as volumes and the completion of the proof.
Euclidean Maximal functions I: The Hardy--Littlewood maximal function and its weak (1,1) bound.
Euclidean Maximal functions II: L^p bounds for Hardy--Littlewood maximal function and applications.
Euclidean Maximal functions III: Measures on hypersurfaces.
Euclidean Maximal functions IV: Decay of the Fourier transform of the spherical measure.
Euclidean Maximal functions V: Littlewood--Paley theory.
Euclidean Maximal functions VI: The spherical maximal function.
The arithmetic spherical maximal function I: Introduction
The arithmetic spherical maximal function II: Sampling/Transference lemmas.
The arithmetic spherical maximal function III: The circle method decomposition of the spherical measure, Part 1.
The arithmetic spherical maximal function III: The circle method decomposition of the spherical measure, Part 2.
The arithmetic spherical maximal function IV: The low-high method and the proof of Magyar—Stein—Wainger’s theorem.
*The arithmetic spherical maximal function V: Application to ergodic theory.
*Survey of related recent work and open problems.
Cut for time :(
The basics of discrete restriction theory
The even moment method for discrete restriction
The circle method approach to discrete restriction
Decoupling for the parabola.
Efficient congruencing for the parabola.
Stein & Shakarchi’s Fourier Analysis (for background)
Davenport’s Analytic Methods for Diophantine Equations and Inequalities
Bourgain’s Pointwise Ergodic Theorems for Arithmetic Sets
Magyar—Stein—Wainger’s Discrete Analogues in Harmonic Analysis: Spherical Averages
Ionescu’s An Endpoint Estimate for the Discrete Spherical Maximal Function
Magyar’s Diophantine Equations and Ergodic Theorems
Bourgain’s Fourier Transform Restriction Phenomena for Certain Lattice Subsets ... Part 1: Schroedinger Equations
Bourgain—Demeter’s A Study Guide for the \ell^2 Decoupling Theorem
Wooley’s Nested Efficient Congruencing
Heath-Brown’s The Cubic Case of Vinogradov’s Mean Value Theorem
Pierce’s The Vinogradov Mean Value Theorem
There are forthcoming books by Ben Krause and by Lillian Pierce. I expect that these will both be excellent sources available in the near future.