Convergence for nonconventional ergodic theorems

Jan Fornal
University of Bristol

The programme is funded by Professor Krause’s New Investigator Grant on Pointwise Ergodic Theory, and he is also my supervisor during my stay in Bristol. My interests span the pointwise ergodic theory, analytic number theory and additive combinatorics. Previously, I studied at the University of Bonn (West Germany) and Jagiellonian University in Kraków (Poland).

Background

The first theorem that describes convergence behaviour of pointwise averages was obtained by George David Birkhoff (1931). Its statement can be presented as:

Theorem 1. Suppose that (𝑋, 𝜇, 𝑇) is a measure-preserving system and 𝑓 ∈ 𝐿¹ (𝑋). Then for almost all points 𝑥 in 𝑋 ¹, one has that:

exists. On top of that, when transformation is ergodic, then for almost all points 𝑥 ∈ 𝑋 limit is equal to:

Birkhoff was motivated to establish this theorem from studying dynamical systems arising from differential equations on a smooth manifold. Later on, mathematicians discovered many applications of that statement. For instance, Furstenberg (1977), Katznelson and Ornstein used that to prove Szemerédi theorem in ergodic theory language:

Theorem 2. Let (𝑋, μ, 𝑇 ) be a measure-preserving system. Then, for every non-negative ⨍ ∈ 𝐿^∞ (𝑋 ) with positive spatial average, the limit:

converge for almost every 𝑥∈ 𝑋.

In the penultimate section of his 1989 work, Bourgain suggested obtaining the same result for a sequence of times where 𝑃 ∈ ℝ [𝑥]. One of the my objectives during graduate studies will be to establish that convergence.

Another example of a nonconventional ergodic theorem with a single test function was obtained by Wierdl in 1988:

Theorem 4. Let (𝑋, μ, 𝑇 ) be a measure-preserving system and 𝑓 ∈ 𝐿^𝑝 ( 𝑋 ) for some 𝑝 > 1. Then the averages:

converge for almost every 𝑥∈ 𝑋.

Number theorists investigated many arithmetic configurations on prime numbers. For instance, Euler, in the middle of 18^th century, showed that every prime 𝑝 ≡ 1 (mod4) can be written as 𝑥² + 𝑦². Later on, Weber (1882) made heavy use of tools from algebraic number theory to prove that for any positive integer 𝑢 there are infinitely many primes of the form 𝑥² + 𝑢 𝑦². Denoting the set of these primes as ℙ , then ℙ also has good averaging properties in the sense that:

Theorem 5. Let ( 𝑋, 𝜇, 𝑇 ) be a measure-preserving system and 𝑓∈ 𝐿^𝑝 (𝑋) for some 𝑝 > 1. Then the averages:

converge for almost every 𝑥 ∈ 𝑋.

This is the first result that I’ve obtained under the support of a Professor Krause Grant.

Pointwise convergence of bilinear ergodic averages

The aforementioned Szemerédi theorem 2 invited ergodic theory specialists to investigate different versions of multiple ergodic averages. Contrary to the grant topic proposal, the norm convergence for many of these averages is understood well; perhaps Furstenberg and Weiss (1996) and Walsh (2012) may serve as good references in that area. Meanwhile, pointwise convergence, in most cases, remains mysterious. The first significant result on this topic was realised by Jean Bourgain (1989):

Theorem 6. Suppose that (𝑋, μ, 𝑇 ) is a dynamical system and 𝑇₁ , 𝑇₂ are powers of 𝑇. Then for 𝑓1, 𝑓2 ∈ 𝐿^∞ (𝑋 )

converges almost surely.

That was the only result for more than 30 years, until Krause, Mirek and Tao (2022) managed to obtain the seemingly simplest polynomial bilinear result.:

Theorem 7. Suppose that ( 𝑋, 𝜇, 𝑇 ) is a dynamical system, 𝑃 ∈ ℤ[𝑥] is polynomial of degree at least 2 and 𝑝₁, 𝑝₂, ∈ (1, ∞] are exponents satisfying . Sample two functions 𝑓₁, 𝑓₂ within respectively L^P1 (X) and L^P2 (X). Then:

converges almost surely.

The proof of that result relies on the work of Peluse and Prendiville (Peluse and Prendiville, 2022) within the additive combinatorics. In the future, together with two collaborators supported by the same grant (i.e. Dr Mousavi and Dr Sun) we intend to prove another result of that flavour:

Theorem 8. Suppose that (𝑋, μ, 𝑇) is a dynamical system, 𝛼, 𝛽 ∈ ℤ and 𝑝1, 𝑝2 ∈ (1, ∞] are exponents satisfying . Sample two functions 𝑓₁, 𝑓₂ within respectively and . Then:

converges almost surely.

As far as I know, this theorem remains resistant to modern tools like double recurrence method exploited in (Krause, Mirek and Tao, 2022) or (Krause et al., 2024). We believe, however that modern methods from analytic number theory (different approximations of von Mangoldt function in the Gowers norm or additive combinatorics (various inverse theorems like those in [Peluse, 2020; Leng, Sah and Sawhney, 2024]) together with simplified approach to 6 will be sufficient for the actual proof of 8.

Additive combinatorics

I am lucky enough to start my cooperation with Dr Sean Prendiville from Lancaster University. He is an author of the result regarding quantitative bound for sets avoiding polynomial patterns in [𝑁] (see Peluse and Prendiville[2022]).

Theorem 9. There exists 𝑐 > 0 such that any subset of {1, … , 𝑁}of density at least (log log 𝑁)^–^𝑐 contains a nontrivial progression of the form 𝑥, 𝑥 + 𝑦, 𝑥 + 𝑦².

My objective is to obtain different quantitative bounds of that flavour in two different settings (i.e. either for subsets of ℤ or subsets of finite characteristic Dedekind domains 𝔽𝑞[𝑇]). One of most technical propositions in (Peluse and Prendiville, 2022) shows that trilinear correlations evaluated on respectively 𝑥, 𝑥 + 𝑦, 𝑥 + 𝑦² can be locally controlled by Gowers norms. Recent progress in that topic shows that more sophisticated progressions (like those described in Peluse, Sah and Sawhney [2023], Kuca [2019] or Matthiesen, Teräväinen and Wang [2024] ) can also be controlled by different norms that originate from Gowers norm. For instance, in the paper (Kravitz, Kuca and Leng, 2019), authors established that patterns of form (𝑥, 𝑦), (𝑥 + 𝑃(𝑧), 𝑦), (𝑥, 𝑦 + 𝑃(𝑧))(where 𝑃 ∈ ℤ[𝑥] is polynomial with an integer root of multiplicity 1) can be controlled by Peluse-Gowers box norms with small boxes. It seems that current understanding of these norms is rather unsatisfactory and scholars from different areas look forward to further rise of this theory. I wish I could be one of the people that manage to accomplish a bit in that regard.

References

Birkhoff, G.D. (1931) ‘Proof of the Ergodic Theorem’, Proceedings of the National Academy of Sciences of the United States of America, 17(12), pp. 656–660. doi: 10.1073/pnas.17.2.656.

Bourgain, J. (1989) ‘Pointwise ergodic theorems for arithmetic sets’, Publications Mathématiques de l’IHÉS, 69, pp. 5–45. With an appendix by Furstenberg, H., Katznelson, Y., and Ornstein, D.S.

Bourgain, J. (1990) ‘Double recurrence and almost sure convergence’, Journal für die Reine und Angewandte Mathematik, 404, pp. 140–161. Available at: http://eudml.org/doc/153200.

Furstenberg, H. (1977) ‘Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions’, Journal d’Analyse Mathématique, 31, pp. 204–256.

Furstenberg, H. and Weiss, B. (1996) ‘A mean ergodic theorem for nonlinear averages of the form 1/N Σf(Tⁿx)g(Tⁿ²x)’, Convergence in Ergodic Theory and Probability (Columbus, OH, 1993), Ohio State University Mathematical Research Institute Publications., 5, pp. 193–227. doi: 10.1515/9783110889383.193.

Green, B. and Tao, T. (2008) ‘The primes contain arbitrarily long arithmetic progressions’, Annals of Mathematics, 167(2), pp. 481–547.

Host, B. and Kra, B. (2005) ‘Nonconventional ergodic averages and nilmanifolds’, Annals of Mathematics, 161(1), pp. 397–488.

Krause, B., Mirek, M. and Tao, T. (2022) ‘Pointwise ergodic theorems for non-conventional bilinear polynomial averages’, Annals of Mathematics, 195(3), pp. 997–1109.

Krause, B., Mousavi, H., Tao, T. and Teräväinen, J. (2024) ‘Pointwise convergence of bilinear polynomial averages over the primes’, arXiv preprint, arXiv:2409.10510. Available at: https://arxiv.org/ abs/2409.10510 .

Kravitz, N., Kuca, B. and Leng, J. (2024) ‘Corners with polynomial side length’, arXiv preprint, arXiv:2407.08637. Available at: https://arxiv.org/abs/2407.08637.

Kuca, B. (2019) ‘Further bounds in the polynomial Szemerédi theorem over finite fields’, arXiv preprint, arXiv:1907.08446. Available at: https://arxiv.org/abs/1907.08446.

Leng, J., Sah, A. and Sawhney, M. (2024) ‘Quasipolynomial bounds on the inverse theorem for the Gowers norm’, arXiv preprint, arXiv:2402.17994. Available at: https://arxiv.org/abs/2402.17994.

Matthiesen, L., Teräväinen, J. and Wang, M. (2024) ‘Quantitative asymptotics for polynomial patterns in the primes’, arXiv preprint, arXiv:2405.12190. Available at: https://arxiv.org/abs/2405.12190. Peluse, S. (2020) ‘Bounds for sets with no polynomial progressions’, Forum of Mathematics, Pi, 8, e16. doi: 10.1017/fmp.2020.11.

Peluse, S. and Prendiville, S. (2022) ‘A polylogarithmic bound in the nonlinear Roth theorem’, International Mathematics Research Notices, (8), pp. 5658–5684.

Peluse, S., Sah, A. and Sawhney, M. (2023) ‘Effective bounds for Roth’s theorem with shifted square common difference’, arXiv preprint, arXiv:2309.08359. Available at: https://arxiv.org/abs/2309.08359.

Walsh, M. (2012) ‘Norm convergence of nilpotent ergodic averages’, Annals of Mathematics, 175(3), pp. 1667–1688. doi: 10.4007/annals.2012.175.3.15.

Weber, H. (1882) ‘Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist’, Mathematische Annalen, 20, pp. 301–329. doi: 10.1007/BF01443599.

Wierdl, M. (1988) ‘Pointwise ergodic theorem along the prime numbers’, Israel Journal of Mathematics, 64(3), pp. 315–336. doi: 10.1007/BF02882425.

PROJECT SUMMARY

Given an arbitrary measure preserving system we show that the multilinear ergodic averages sampled along an arbitrary number of sequences coming from a Hardy field converge pointwise almost everywhere. We aim to prove this for as wide a class of Hardy field functions as possible. To do so, we establish a long variational inequality along lacunary sequences which implies a maximal inequality, norm convergence, and pointwise convergence.

By a transference argument it suffices prove this long variational inequality in the case that the measure preserving system is the integers. This reduction allows us to use tools from discrete harmonic analysis, additive combinatorics, and analytic number theory. We then give applications in areas such as upcrossings, equidistribution, and combinatorics.

PROJECT LEAD

Jan Fornal begun his PhD programme at the University of Bristol in September 2024. The program is funded by Professor Krause’s New Investigator Grant on Pointwise Ergodic Theory.

Jan’s interests span the pointwise ergodic theory, analytic number theory and additive combinatorics.

Previously Jan has studied in the University of Bonn (West Germany) and Jagiellonian

University in Krak´ow (Poland).