Ergodic theory meets combinatorial number theory (ETMCNT)
Polonez BIS 3 project
Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, Poland.
Project duration: 01.09.2023 - 31.08.2025
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If we have a sequence starting with 4, 7, 10, 13, what will come next? Questions of this sort are meant to check pattern detection skills; for that reason, they are often used to test mathematical abilities. In this example, each subsequent term is greater by 3 than the previous one; hence the next term will be 16. Sequences of this form, in which the difference between consecutive terms is constant, are called arithmetic progressions.
The search for patterns like arithmetic progressions in sets of numbers is a central task of the subfield of pure mathematics called combinatorial number theory. It is known that large subsets of natural numbers contain plenty of “reasonable” patterns, which includes arithmetic progressions. One of the main objectives of the area is to quantify these statements, either by showing that all sets of a certain size contain a pattern or by estimating the number of patterns in sets of a given size.
By contrast, ergodic theory examines the statistical behaviour of processes over long time, and its origins lie in the study of large systems of particles in statistical mechanics. The systems to study range from relatively simple ones, such as rotations on a circle, to highly complicated flows on surfaces and their higher-dimensional generalisations. If possible, one wants to obtain a definitive form (e.g. space average) for the long-term behaviour of the system (i.e. its time average).
On the surface, there seems to be little connection between these two apparently distant areas, both in terms of the scope of research and available techniques. Yet there is a strong link between them, forged by Furstenberg's ergodic proof of the Szemerédi theorem from combinational number theory. Szemerédi proved that each sufficiently large subset of natural numbers contains arithmetic progressions of arbitrary length; Furstenberg rederived this result by relating it to the problems of multiple recurrence in ergodic theory. The interaction between combinatorial number theory and ergodic theory initiated by Furstenberg has proved enormously fruitful, creating a virtuous cycle that lead to countless new developments in both branches of mathematics. The goal of this project is to tackle a number of open problems at the interface between ergodic theory and combinatorial number theory.
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News
June 2025
Summer is always the time of intense mathematical activity. In my case, this involved:
completing a new paper ("Resolving the joint ergodicity problem for Hardy sequences") jointly with Sebastián Donoso, Andreas Koutsogiannis, Wenbo Sun and Konstantinos Tsinas. As the title suggests, we resolve the longstanding joint ergodicity classification problem for a broad class of Hardy sequences of polynomial growths;
presenting my joint ergodicity results (obtained jointly with Sebastián, Andreas, Wenbo and Kostas) at the "Perspectives in Ergodic Theory" IMPAN conference celebrating Vitaly Bergelson's 70th birthday;
presenting my works with Noah Kravitz and James Leng on polynomial corners and quantitative concatenation at the BIRS workshop "New Trends in Arithmetic Combinatorics and Related Fields" in Granada;
giving a survey talk on ergodic methods in additive combinatorics as a part of the workshop preceding Peter Sarnak's Łojasiewicz Lecture at the Jagiellonian University.
April 2025
I visited my collaborators Nikos Frantzikinakis (University of Crete) and Andreas Koutsogiannis (Aristotle University of Thessaloniki) to discuss various open problems on multiple ergodic averages.
26 February 2025
I spoke about my works with Noah Kravitz and James Leng on polynomial corners and quantitative concatenation at the online analysis seminar at University of Georgia.
21 January 2025
I gave a talk outlining recent progress on the polynomial Szemerédi theorem at the discrete mathematics seminar at Adam Mickiewicz University in Poznań.
October 2024 - January 2025
I was a local coordinator at the Jagiellonian University for the online ISEM 28 program. With two students, Filip Wierzbowski and Ivan Spyrydonov, we were delving into ergodic structure theory and its applications.
November 2024
I gave two introductory lectures to higher order Fourier analysis at the number theory seminar at the Jagiellonian University.
19 October 2024
My paper "Seminorm estimates and joint ergodicity for pairwise independent Hardy sequences", written jointly with Sebastián Donoso, Andreas Koutsogiannis, Wenbo Sun and Konstantinos Tsinas, has finally appeared on arXiv. We are developing a robust structure theory of multiple ergodic averages of commuting transformations along Hardy sequences, and we use to it to obtain new results on convergence, recurrence, and joint ergodicity of such averages. The crux of the paper is the derivation of robust seminorm estimates for the averages.
September 2024
At the 10th Polish Combinatorial Conference in Będlewo, I gave a talk outlining recent progress on the Szemerédi theorem and related questions.
August 2024
I gave a talk about my works with Noah Kravitz and James Leng on polynomial corners and quantitative concatenation at the New Trends in Ergodic Theory conference held at Northwestern University.
11 July 2024
Noah Kravitz, James Leng and I have finally releases two papers on arXiv. In one of them, we prove bounds for sets lacking polynomial corners; among other things, we develop in this paper the first instance of a degree lowering argument based on box norms rather than Gowers norms. In the other paper, we develop a sophisticated multidimensional quantitative concatenation argument that provides a single box norm control on arbitrary multidimensional polynomial progressions.
June 2024
I gave a talk about my upcoming works with Noah Kravitz and James Leng on polynomial corners and quantitative concatenation at the Pointwise Ergodic Theory and Connections II conference held at University of Bristol.
May 2024
I received the Kuratowski Prize for Polish mathematicians under 30.
13-17 May 2024
I had the pleasure of attending the workshop "High-dimensional phenomena in discrete analysis" at the American Institute of Mathematics.
22 April 2024
My paper Multidimensional polynomial patterns over finite fields: bounds, counting estimates and Gowers norm control has been accepted in Advances in Mathematics.
17 April 2024
I gave a colloquium talk at IMPAN titled "The Szemerédi theorem and beyond" in which I outlined recent developments in ergodic theory and additive combinatorics that take their roots in the Szemerédi theorem.
13 March 2024
To commemorate the International Day of Pi taking place on 14 March, I gave an expository talk on the mysteries of pi and the history of numbers in IV Liceum Ogólnokształcące in Kraków.
29 February 2024
I have been interviewed by Radio Kraków on the everyday nature of a mathematician's job, the applications of pure mathematics to real-life problems, and the upsides and downsides of mathematical education.
I was the main guest at a two-hour long discussion hosted by Tarnowski Klub Dyskusyjny. We talked about the nature of pure mathematics, its connection with other sciences, real-life applications, benefits of studying and teaching mathematics, and many other topics. Big thanks to Fundacja Alegoria for hosting the discussion.
Photo: Franciszek Kiełbasa
January-February 2024
I have visited my collaborator Sebastián Donoso at University of Chile, where we worked on our joint project on the joint ergodicity of Hardy sequences. I also gave a talk in the dynamical systems seminar.
07-14.12.2023
I had the pleasure of hosting my two collaborators from Greece, Andreas Koutsogiannis and Konstantinos Tsinas, as a part of collaborative work on the joint ergodicity conjecture for Hardy sequences.
04.12.2023
I gave a talk at the dynamical systems seminar at IMPAN (Warsaw) on my recent works with Nikos Frantzikinakis concerning the limiting behaviour of multiple ergodic averages along polynomials for systems of commuting transformations. The results discussed in this talk are an inspiration a starting point for several research tasks in this Polonez Bis project.
09.11.2023
I gave a talk at the number theory seminar at the Jagiellonian University in which I summarised recent developments in the polynomial Szemerédi theorem. This is one of the main topics of this Polonez BIS project.
This research is part of the project No. 2022/47/P/ST1/00854 within the POLONEZ BIS programme co-funded by the National Science Centre and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 945339.
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