Jamneshan and Pan proved that for every number of commuting actions dd, every solvability length ll, and every ε>0\varepsilon>0 there are constants δ>0\delta>0 and K≥1K\geq 1, depending only on (d,l,ε)(d,l,\varepsilon), such that for every solvable group of derived length ll (indeed every group in a fixed uniformly amenable class) the set of multiple-recurrence times of any set of measure at least ε\varepsilon is KK-syndetic. Their paper poses two linked open questions: (i) does this uniform-syndeticity conclusion persist over the entire class of discrete amenable groups, with δ\delta and KK depending only on dd and ε\varepsilon? and (ii) does the Bergelson–McCutcheon non-commutative Roth theorem — which for two commuting measure-preserving actions of an arbitrary countable group locates the return times in a central∗^* (hence syndetic) set — admit an analogue for three or more commuting actions, together with the uncountable version needed for ultraproduct arguments? Both questions remain open as of 2026.

Background

Multiple recurrence studies when a measure-preserving dynamical system returns to a set of positive measure in structured configurations. For a discrete group Γ\Gamma acting on a probability space (X,μ)(X,\mu) by measure-preserving transformations, consider dd pairwise commuting such actions T1,…,Td ⁣:Γ↷(X,μ)T_1,\ldots,T_d\colon \Gamma \curvearrowright (X,\mu). Writing Tγ[1,i]=T1γ∘T2γ∘⋯∘TiγT^{[1,i]}_\gamma = T^\gamma_1 \circ T^\gamma_2 \circ \cdots \circ T^\gamma_i for the composite action, the multiple-recurrence time set of a measurable E⊆XE\subseteq X is {γ∈Γ:μ(E∩Tγ−1[1,1]E∩⋯∩Tγ−1[1,d]E)>0}\{\gamma\in\Gamma : \mu(E \cap T^{[1,1]}_{\gamma^{-1}} E \cap \cdots \cap T^{[1,d]}_{\gamma^{-1}} E) > 0\}. A subset of Γ\Gamma is (left) syndetic if finitely many left translates of it cover Γ\Gamma, and KK-syndetic if KK translates suffice. The composite-action form is forced in the non-abelian setting: Bergelson and Leibman exhibited solvable groups of exponential growth where the naive formulation with the individual actions TiγT^\gamma_i fails to give recurrence. An amenable group is one admitting a finitely additive left-invariant probability measure (equivalently, Følner sets); solvable groups are amenable. Austin's amenable ergodic Szemerédi theorem established convergence of the non-conventional averages 1∣Φ∣∑γ∈Φ∏i=1dfi∘Tγ[1,i]\frac{1}{|\Phi|}\sum_{\gamma\in\Phi} \prod_{i=1}^{d} f_i \circ T^{[1,i]}_\gamma along Følner sets for dd commuting actions of any amenable group, which yields qualitative (non-uniform) recurrence. The qualitative statement gives constants that a priori depend on the acting group; the modern question is when they can be made uniform over a class of groups. A class G\mathcal{G} of discrete groups is uniformly amenable (Keller) if its members share a single Følner-type function F(n,ε)F(n,\varepsilon) controlling the size of Følner sets for all subsets of size at most nn and all error parameters ε\varepsilon; the solvable groups of a fixed derived length (the least nn with the nn-th derived subgroup trivial) form such a class, while by a result of Wysoczański the class of all amenable groups does not. Jamneshan and Pan proved (their Theorem 1.5, more generally Theorem 3.2): for all d,l≥1d,l\geq 1 and ε>0\varepsilon>0 there exist δ>0\delta>0 and K≥1K\geq 1, depending only on ε,d,l\varepsilon,d,l, such that for every solvable group Γ\Gamma of derived length ll, every probability space, every dd commuting measure-preserving Γ\Gamma-actions, and every EE with μ(E)≥ε\mu(E)\geq\varepsilon, the set {γ:μ(E∩Tγ−1[1,1]E∩⋯∩Tγ−1[1,d]E)≥δ}\{\gamma : \mu(E \cap T^{[1,1]}_{\gamma^{-1}} E \cap \cdots \cap T^{[1,d]}_{\gamma^{-1}} E) \geq \delta\} is KK-syndetic in Γ\Gamma. The proof passes to an ultraproduct of the given systems — which lands on an uncountable group acting on an inseparable (Loeb) probability space — and applies an uncountable, probability-algebra version of Austin's theorem that the authors develop (their Theorem 1.10, valid for all amenable groups). This reduction requires the class of groups to be closed under the relevant (ultra)product constructions, which is where uniform amenability enters: D. Benson observed that a countable direct product of nilpotent groups need not be amenable, undermining the strategy for larger classes. A complementary, topological-combinatorial route comes from central sets theory: A central set of Γ\Gamma is a member of a minimal idempotent ultrafilter on Γ\Gamma (an idempotent point for ultrafilter composition that lies in the smallest two-sided ideal of the Stone–Čech compactification βΓ\beta\Gamma), a set is central∗^* if it meets every central set, and central∗^* sets are syndetic. Bergelson and McCutcheon's non-commutative Roth theorem (their Theorem 1.3) shows that for any countable group acting by measure-preserving transformations, with two commuting actions, the Roth-type return set is central∗^* — amenability is not needed at all for d=2d=2.

Problem Statement

The problem, posed by Jamneshan and Pan at the end of the introductory discussion of "Uniform syndeticity in multiple recurrence" (2024/2025), is to extend class-uniform syndeticity in multiple recurrence beyond uniformly amenable classes of groups. It has two concrete, linked components. (i) For the entire class of discrete amenable groups: given d≥1d\geq 1 and ε>0\varepsilon>0, do there exist constants δ>0\delta>0 and K≥1K\geq 1, depending only on dd and ε\varepsilon, such that for every discrete amenable group Γ\Gamma, every probability space (X,μ)(X,\mu), every dd pairwise commuting measure-preserving Γ\Gamma-actions T1,…,TdT_1,\ldots,T_d, and every measurable E⊆XE\subseteq X with μ(E)≥ε\mu(E)\geq\varepsilon, the set {γ∈Γ:μ(E∩Tγ−1[1,1](E)∩Tγ−1[1,2](E)∩⋯∩Tγ−1[1,d](E))≥δ}\{\gamma\in\Gamma : \mu(E \cap T^{[1,1]}_{\gamma^{-1}}(E) \cap T^{[1,2]}_{\gamma^{-1}}(E) \cap \cdots \cap T^{[1,d]}_{\gamma^{-1}}(E)) \geq \delta\} is KK-syndetic in Γ\Gamma, where Tγ[1,i]=T1γ∘⋯∘TiγT^{[1,i]}_\gamma = T^\gamma_1 \circ \cdots \circ T^\gamma_i? (ii) For arbitrary countable groups: prove or disprove that the Bergelson–McCutcheon non-commutative Roth theorem extends from two to three or more commuting measure-preserving actions — i.e., that the corresponding multiple-recurrence return set is central∗^* (in particular syndetic) — and, in the positive case, establish the uncountable (probability-algebra) version of that statement, which by the source paper's argument would then yield class-uniform syndeticity for much larger classes of groups, conceivably all groups, at least for two actions. The common scientific objective is to determine the true scope of uniformity in multiple recurrence: whether the syndeticity of return times with constants independent of the acting group is a general feature of amenability, and whether central-set techniques can replace amenability altogether for more than two commuting actions.

References

  1. Asgar Jamneshan and Minghao Pan, "Uniform syndeticity in multiple recurrence", Ergodic Theory and Dynamical Systems 45(2):504–525 (2025; first published online 2024). DOI: 10.1017/etds.2024.40, https://doi.org/10.1017/etds.2024.40; arXiv:2208.02833, https://arxiv.org/abs/2208.02833
  2. V. Bergelson and R. McCutcheon, "Central sets and a non-commutative Roth theorem", American Journal of Mathematics 129(5):1251–1275 (2007). DOI: 10.1353/ajm.2007.0031, https://doi.org/10.1353/ajm.2007.0031
  3. Tim Austin, "Non-conventional ergodic averages for several commuting actions of an amenable group", Journal d'Analyse Mathématique 130:243–274 (2016). DOI: 10.1007/s11854-016-0036-6, https://doi.org/10.1007/s11854-016-0036-6
  4. V. Bergelson and A. Leibman, "Failure of the Roth theorem for solvable groups of exponential growth", Ergodic Theory and Dynamical Systems 24(1):45–53 (2004). DOI: 10.1017/s0143385703000427, https://doi.org/10.1017/s0143385703000427
  5. Janusz Wysoczański, "On Uniformly Amenable Groups", Proceedings of the American Mathematical Society 102(4):933–938 (1988). DOI: 10.2307/2047337, https://doi.org/10.2307/2047337
  6. Polona Durcik, Rachel Greenfeld, Annina Iseli, Asgar Jamneshan, and José Madrid, "An uncountable ergodic Roth theorem and applications", Discrete and Continuous Dynamical Systems 42(11):5509–5540 (2022). DOI: 10.3934/dcds.2022111, https://doi.org/10.3934/dcds.2022111; arXiv:2101.00685, https://arxiv.org/abs/2101.00685
  7. Ethan Ackelsberg and Asgar Jamneshan, "Equidistribution in 2-Nilpotent Polish Groups and triple restricted sumsets", arXiv:2504.07865 (v1 April 2025; v3 March 2026), https://arxiv.org/abs/2504.07865
  8. Ethan Ackelsberg, Vitaly Bergelson, and Andrew Best, "Multiple recurrence and large intersections for abelian group actions", Discrete Analysis 2021:18, 91 pp. DOI: 10.19086/da.28877, https://doi.org/10.19086/da.28877; arXiv:2101.02811, https://arxiv.org/abs/2101.02811
  9. V. Bergelson, D. Robertson, and P. Zorin-Kranich, "Triangles in Cartesian Squares of Quasirandom Groups", Combinatorics, Probability and Computing 26(2):161–182 (2017). DOI: 10.1017/s0963548316000250, https://doi.org/10.1017/s0963548316000250
  10. Dimitrios Charamaras and Andreas Mountakis, "Finding product sets in some classes of amenable groups", Forum of Mathematics Sigma 13:e10 (2025). DOI: 10.1017/fms.2024.155, https://doi.org/10.1017/fms.2024.155; arXiv:2402.07779, https://arxiv.org/abs/2402.07779
  11. Vitaly Bergelson and Saúl Rodríguez-Martín, "Product sets in sets of returns and positivity of symmetric ergodic averages", arXiv:2608.02873 (2026), https://arxiv.org/abs/2608.02873