Open conjectures from Bergelson–Moreira–Richter's Hardy-field multiple ergodic averages: a joint ergodicity criterion for commuting transformations, a nilsequence structure theorem for weighted multicorrelations, and weakening Property (P) to Property (WP)

The open problems considered here are those stated in Section 6 of Bergelson–Moreira–Richter's paper on multiple ergodic averages along Hardy-field functions. The original single-transformation convergence conjecture (v1 Conjecture 6.1) was resolved affirmatively by Frantzikinakis (2023), and Questions 6.5 and 6.6 on intersectivity of polynomial shadows and integer-coefficient derivative spans were refuted by Ouyang–Rodríguez-Martín–Xu–Zhang (2026); none of these is part of the open core. The open core, still posed as conjectures in the authors' February 2026 corrected arXiv version, consists of three linked yes/no questions sharing one objective — completing the extension of polynomial multiple ergodic structure theory to general Hardy-field times: (i) a Berend–Bergelson-type joint ergodicity criterion for k commuting transformations along finite real linear combinations of monomials $t^c$, $c>0$ (Conjecture 6.1, possibly valid for all Hardy-field functions of polynomial growth per Remark 6.2); (ii) a nilsequence (Host–Kra-type) decomposition for weighted multicorrelation sequences along Hardy-field functions (Conjecture 6.3); and (iii) whether Theorem B, the paper's weighted convergence and recurrence theorem, remains true when the weight-compatibility Property (P) is weakened to Property (WP) (Conjecture 6.4). Partial results exist — the single-function case of (i) is proved by Donoso–Koutsogiannis–Sun, and the polynomial-iterate analogues of (i) and (ii) by Frantzikinakis–Kuca — but all three conjectures remain open as of mid-2026.

Background

A Hardy field $\mathcal H$ is a family of real-valued functions on a half-line $(t_0,\infty)$ that is closed under differentiation and under the field operations and whose every nonzero member is eventually nonzero; its elements have eventually monotone derivatives and enjoy strong equidistribution properties modulo 1. Examples include $t^c$ for $c>0$, $\log t$, and sums such as $t^{3/2}+\log t$; the paper works with members of polynomial growth, meaning functions eventually bounded by a power of $t$. The paper, by Vitaly Bergelson, Joel Moreira and Florian K. Richter (published in Advances in Mathematics 443 (2024); corrected arXiv v3, February 2026), studies multiple ergodic averages in which the usual polynomial iteration times are replaced by rounded Hardy-field values $[f(n)]$, where $[x]$ denotes the nearest integer to $x$. For a probability space $(X,\mathcal B,\mu)$ with an invertible measure-preserving transformation $T$ and functions $h_i\in L^\infty(X)$, the averages in question are $\frac1N\sum_{n=1}^N T^{[f_1(n)]}h_1\cdots T^{[f_k(n)]}h_k$. A system is totally ergodic if $T^m$ is ergodic for every $m\ge 1$; a collection of integer sequences $a_1(n),\dots,a_k(n)$ is jointly ergodic for a system if the corresponding multiple averages converge in $L^2$ to the product of the integrals $\int h_1\,d\mu\cdots\int h_k\,d\mu$. For classical polynomial iterates, the Host–Kra–Ziegler structure theory shows that the relevant multicorrelation sequences $\alpha(n)=\int h_0\cdot T^{[f_1(n)]}h_1\cdots T^{[f_k(n)]}h_k\,d\mu$ decompose as a nilsequence plus a small error: a nilmanifold is a quotient $Y=G/\Gamma$ of a nilpotent Lie group $G$ by a discrete cocompact subgroup $\Gamma$, a nilsequence is a sequence of the form $F(g^n y)$ with $F\in C(Y)$, $g\in G$, $y\in Y$, and this decomposition is the engine behind convergence and combinatorial recurrence results. Bergelson–Moreira–Richter extend this circle of ideas to Hardy-field times in two directions. Their Theorem B proves convergence and multiple recurrence for weighted averages $\frac1{W(N)}\sum_{n=1}^N w(n)T^{[f_1(n)]}h_1\cdots T^{[f_k(n)]}h_k$, where the weight is given by a Hardy-field function $W$ with increments $w(n)=W(n+1)-W(n)$, under a compatibility condition between the functions and the weight: Property (P) demands that for every $f$ in the derivative-span $\nabla\text{-}\mathrm{span}(f_1,\dots,f_k)$ (the real linear span of the functions together with all of their derivatives) and every real polynomial $p$, either $|f(t)-p(t)|\ll 1$ or $|f(t)-p(t)|\succ\log(W(t))$. Writing $\mathrm{span}^*(f_1,\dots,f_k)$ for the set of nonzero real linear combinations, the weaker Property (WP) asks the same alternative only for $f\in\mathrm{span}^*(f_1,\dots,f_k)$. Their combinatorial Theorem A gives multiple recurrence for sets of positive upper density under a polynomial-shadow intersectivity hypothesis. The state of the art around these results: Tsinas (Transactions of the AMS, 2023) proved joint ergodicity of Hardy-field sequences when all iterates are non-integer powers staying logarithmically away from rational polynomials (this appears as Theorem 1.12 in the published paper); Frantzikinakis (Advances in Mathematics, 2023) resolved the original 2020-version single-transformation conjecture; Koutsogiannis–Sun (2023) generalized total joint ergodicity; Donoso–Koutsogiannis–Sun (Israel Journal of Mathematics, 2025) proved the single-function joint ergodicity case for commuting transformations and posed the multi-function case as their Conjecture 1; Frantzikinakis–Kuca (Inventiones mathematicae, 2025) established the analogous joint ergodicity criterion and structure theorem for polynomial iterates of commuting transformations; and Tsinas (Ergodic Theory and Dynamical Systems, 2024) proved nilmanifold equidistribution along smooth functions of polynomial growth, a key input class for the Hardy-field nilstructure program. Ouyang–Rodríguez-Martín–Xu–Zhang (2026) constructed Bohr obstructions showing the real derivative-span recurrence criterion is essentially sharp, refuting two of the paper's questions. The paper's earlier companion (Advances in Mathematics, 2020) treats single and multiple recurrence along non-polynomial sequences with unweighted averages, and Reilly (2026) confirmed one of that companion's weighted-averages conjectures.

Problem Statement

The problem is the open core of the open-questions section of Bergelson–Moreira–Richter's Hardy-field multiple ergodic averages paper, in the numbering of its published/corrected version. The three components share one scientific objective — completing the extension of the polynomial multiple-ergodic structure theory (joint ergodicity criteria, nilsequence decomposition of multicorrelations, weighted convergence) to general Hardy-field iteration times — and each is a definite yes/no question. A proof or disproof of any one of (1)–(3) resolves that component; the shared objective is complete only once all three are settled. Throughout, $[x]$ denotes the nearest integer, $\mathrm{span}^*(f_1,\dots,f_k)$ denotes the set of nonzero real linear combinations of the $f_i$, and $\nabla\text{-}\mathrm{span}(f_1,\dots,f_k)=\mathrm{span}\{f_i^{(m)}:1\le i\le k,\ m\ge 0\}$ denotes the real linear span of the functions $f_i$ together with all of their derivatives.

(1) Conjecture 6.1. Let $f_1,\dots,f_k$ be functions of the form $f_i(t)=a_1t^{c_1}+\cdots+a_dt^{c_d}$ with real coefficients $a_1,\dots,a_d$ and real exponents $c_1,\dots,c_d>0$, and let $(X,\mathcal B,\mu,T_1,\dots,T_k)$ be a measure-preserving system with $k$ commuting invertible measure-preserving transformations. Suppose that (i) for every $h\in L^\infty(X)$ and all $i\ne j$,

$\frac1N\sum_{n=1}^N T_i^{[f_i(n)]}T_j^{-[f_j(n)]}h\longrightarrow\int_X h\,d\mu\quad\text{in }L^2(X),$

and (ii) for every $H\in L^\infty(X^k)$,

$\frac1N\sum_{n=1}^N \big(T_1^{[f_1(n)]}\times\cdots\times T_k^{[f_k(n)]}\big)H\longrightarrow\int_{X^k}H\,d\mu_k\quad\text{in }L^2(X^k).$

Is it then true that for all $h_1,\dots,h_k\in L^\infty(X)$,

$\frac1N\sum_{n=1}^N T_1^{[f_1(n)]}h_1\cdots T_k^{[f_k(n)]}h_k\longrightarrow\int h_1\,d\mu\cdots\int h_k\,d\mu\quad\text{in }L^2(X)?$

Remark 6.2 of the paper records the authors' suggestion that this conjecture may well hold for arbitrary functions $f_1,\dots,f_k$ from a Hardy field, not only for monomial sums.

(2) Conjecture 6.3. Let $f_1,\dots,f_k$ be functions of polynomial growth from a Hardy field $\mathcal H$, let $W$ be a compatible weight — in the sense of the paper, $W\in\mathcal H$ with $1\prec W(t)\ll t$ such that $f_1,\dots,f_k$ satisfy Property (P) with respect to $W$ — with increments $w(n)=W(n+1)-W(n)$, let $(X,\mathcal B,\mu,T)$ be an invertible measure-preserving system, let $h_0,\dots,h_k\in L^\infty(X)$, and let

$\alpha(n)=\int_X h_0\cdot T^{[f_1(n)]}h_1\cdots T^{[f_k(n)]}h_k\,d\mu$

be the multicorrelation sequence. Does there exist a nilmanifold $Y=G/\Gamma$, a continuous function $F\in C(Y)$, a point $y\in Y$, and elements $a_1,\dots,a_k\in G$ such that

$\alpha(n)=F\big(a_1^{[f_1(n)]}\cdots a_k^{[f_k(n)]}y\big)+\nu(n),\qquad n\in\mathbb N,$

with the error satisfying

$\lim_{N\to\infty}\frac1{W(N)}\sum_{n=1}^N w(n)\,|\nu(n)|=0?$

(3) Conjecture 6.4. Does Theorem B — the paper's theorem giving, for functions $f_1,\dots,f_k$ of polynomial growth from a Hardy field, a weight $W\in\mathcal H$ with $1\prec W(t)\ll t$ satisfying Property (P), and an invertible measure-preserving system, the convergence, recurrence, and combinatorial conclusions for the weighted averages

$\frac1{W(N)}\sum_{n=1}^N w(n)\,T^{[f_1(n)]}h_1\cdots T^{[f_k(n)]}h_k$

— remain valid when the weight-compatibility Property (P) (for all $f\in\nabla\text{-}\mathrm{span}(f_1,\dots,f_k)$ and $p\in\mathbb R[t]$, either $|f(t)-p(t)|\ll 1$ or $|f(t)-p(t)|\succ\log(W(t))$) is replaced by the weaker Property (WP) (the same alternative required only for all $f\in\mathrm{span}^*(f_1,\dots,f_k)$)?

The two further questions from the source section — on intersectivity of the integer polynomial shadows $\mathrm{poly}(f_1,\dots,f_k)\cap\mathbb Z[t]$ and on integer-coefficient derivative spans — have been answered negatively in the literature; they are not part of the open core.

References

1. Vitaly Bergelson, Joel Moreira, Florian K. Richter, "Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications", Advances in Mathematics 443 (2024), Article 109597. DOI: 10.1016/j.aim.2024.109597; arXiv:2006.03558 (corrected v3, February 2026). https://doi.org/10.1016/j.aim.2024.109597
2. Nikos Frantzikinakis, "Joint ergodicity of sequences", Advances in Mathematics 417 (2023), Article 108918. DOI: 10.1016/j.aim.2023.108918; arXiv:2102.09967. https://doi.org/10.1016/j.aim.2023.108918
3. Konstantinos Tsinas, "Joint ergodicity of Hardy field sequences", Transactions of the American Mathematical Society 376 (2023), 3191–3263. DOI: 10.1090/tran/8752; arXiv:2109.07941. https://doi.org/10.1090/tran/8752
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8. Kangbo Ouyang, Saúl Rodríguez-Martín, Leiye Xu, Shuhao Zhang, "Bohr obstructions to recurrence along Hardy-field sequences", arXiv:2605.17529 (2026, v2). https://arxiv.org/abs/2605.17529
9. Michael Reilly, "Uniform weighted averages and a conjecture of Bergelson, Moreira, and Richter", Ergodic Theory and Dynamical Systems (2026), 1–23. DOI: 10.1017/etds.2026.10325; arXiv:2602.20606. https://doi.org/10.1017/etds.2026.10325
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