DIMACS REU 2024

Student:	Abbas Dohadwala
Office:	CoRE 448
School:	Department of Mathematics, Purdue University
Contact:	ad2129@rutgers.edu
Project:	When Fourier Analysis Meets Ergodic Theory and Combinatorics
Mentor:	Mariusz Mirek, Leonidas Daskalakis

Project Description

Ergodic theory is an area of mathematics which studies the statistical properties of dynamical systems. The Birkhoff ergodic theorem is a classical result of ergodic theory, which establishes pointwise convergence almost everywhere for a specific average under certain conditions. In this project, we worked towards proving an ergodic theorem with some similarities, but involving fractional powers of primes. With techniques from analytic number theory and Fourier analysis, we studied exponential sums involving these fractional powers of primes to obtain certain bounds. Using these bounds, we use harmonic analysis tools to prove a sequence of theorems, eventually leading to our desired result.

Research Log

Week 1 (5/27-5/31)

I arrived at the program this week, attended the orientation and got settled. I met with my mentors, as well as Ish and Erik. This week, I reviewed measure theory, working from Mariusz's slides and working on exercises given to us by Leonidas. I learned a large amount of the prerequisite ergodic theory and worked several exercises from various books. Our mentors discussed how we will use tools from harmonic analysis and analytic number theory to prove the desired result concerning pointwise convergence. In addition we were introduced to Tolev's paper on exponential sum estimates. There is an incredible amount to cover before we can begin the problem, so the first half of the program will be dedicated to acquiring the necessary background. Together with Ish, I prepared a slidedeck for the upcoming presentation.

Outside of program activities, I am adapting to the surrounding areas. I went climbing with a couple students, went on a meandering run arround campus (was disappointed by the lack of sidewalks), and went to New York City.

Week 2 (6/3-6/7)

This week, I learned much of the necessary material on Fourier series. I learned some necessary Hilbert space theory and some more important results in ergodic theory. I also worked several exercises in Fourier analysis, ergodic theory, and measure theory. Ish and I met with Leonidas most days so we could better understand the plan for our project. We also did our initial presentation week, and it was very interesting to see what others were doing in their projects.

Week 3 (6/10-6/14)

This week, we made more progress in terms of the necessary material, and we began going through the Tolev-Laporta paper to understand better the exponential sum estimates. I learned the basic theory of maximal operators and learned additional material in Fourier series.

This week, I also went to several talks in the "Beyond the Freshman Horizons Workshop". Many were close to what my project was concerned with and aligned with my general interests, and I showed up for the additive combinatorics, analytic number theory, random matrix theory, integrable systems, and fourier analysis talks.

Week 4 (6/17-6/21)

I made more progress this week in the Tolev Paper, and I began looking through a paper of Laporta-Tolev which many of the estimates were drawn from. I am starting to become more comfortable with exponential sum estimates and the philosophy of Vaughan's identity for the von Mangoldt function. I also learned some more theory of maximal operators and maximal inequalities, finishing the notes by Mariusz on the topic. I also began the notes on Ergodic theory and Oscillations. The theory of oscillation seminorms will be crucial for demonstrating our result holds on square-integrable functions. We met with Leonidas most days to discuss our progress and to learn more about general analysis principles to prove results of this nature, namely maximal inequalities and pointwise convergence results.

Week 5 (6/24-6/28)

This week we made significant progress towards understanding the exponential sum estimates of the minor arc. There is some uneasiness that Ish, Leonidas, and I have about some of the large steps the authors have skipped, and we intend to fill in the gaps. Otherwise, I made significant progress in the oscillations notes.

Week 6 (7/1-7/5)

This week we finished verifying the minor arc estimates from Laporta-Tolev. I finished most of the oscillations notes. Ish and Erik made some progress in the Major arc estimates.

Otherwise, we were visited by the NYC REU program, which was nice. I connected with one of the students, and I am interested in visiting their program soon.

Week 7 (7/8-7/12)

This week Leonidas discovered that we did not have to use a smooth bump function for the maximal and oscillation inequalities and we could instead use uniform estimates of the difference between multipliers to establish the result. We decided we are going to try to prove full oscillations and try to obtain weak-type (1,1) maximal inequalities for our operator.

I also reached out to one of the REU participants at Baruch and organized a little meet-up with some people at the REU in NY. We ate Ukranian food, hung out in a park, ate from a famous snack shop (with questionable hygiene), and went to Strand's Bookstore (where I ran into Dr. Loebl!).

Week 8 (7/15-7/19)

This week was final presentation week. Ish and I worked on the presentations for the first half of the week. Ish and Erik worked on writing up the Major Arc, and I began trying to understand the techniques for the lacunary oscillations. I started writing up the oscillation inequalities. This week, most of our program left, and it was a pretty sad experience, since I had gotten to know many of my fellow participants over the past couple months.

Week 9 (7/21-7/25)

This week is the final week of the program. We spent the week writing up the technical report. I had not had the maximal inequalities and oscillation inequalities in writing, so I got that done this week. We met with Leonidas and Erik to discuss the report and future directions for the project.

Many participants left throughout the week. I played basketball with Nigel and Rhett for a final time. I tried (somewhat successfully) to organize a couple walks with the other participants. I took them on a route that I began to run on almost every day after Justin had shown it to me several weeks ago. Participants slowly departed the program, and it is definitely starting to feel weird. I ended up going to New York alone to visit the NYC Discrete Math REU for a visit by Ingrid Daubechies, a famous applied mathematician/physicist.

Project Summary

References

Under Construction

Acknowledgements

This work was carried out while the author Abbas Dohadwala was a participant in the 2024 DIMACS REU program at Rutgers University, CNS-2150186, supported by the Rutgers Department of Mathematics and NSF grant DMS-2154712.