Does goodness for equidistribution alone imply joint ergodicity of integer sequences (Frantzikinakis's Problem 1)?

Frantzikinakis's 2023 criterion characterizes joint ergodicity of integer sequences by two conditions: goodness for Host-Kra seminorm estimates and goodness for equidistribution. The paper's Problem 1 asks whether the equidistribution condition alone suffices: if the sequences $a_1,\dots,a_\ell\colon\mathbb{N}\to\mathbb{Z}$ are good for equidistribution, in that every nontrivial real linear combination of them satisfies the Weyl exponential-sum vanishing condition defined below, must the sequences be jointly ergodic for every ergodic measure-preserving system? The global problem remains open: it is settled positively only for a single sequence ($\ell=1$, by the spectral theorem) and for nilsystems, while the analogous fixed-system version is known to fail even for $\ell=1$.

Background

A measure-preserving system $(X,\mu,T)$ is a probability space together with an invertible, measure-preserving transformation $T$; the system is ergodic if every $T$-invariant integrable function is constant almost everywhere. For a tuple of integer sequences $a_1,\dots,a_\ell\colon\mathbb{N}\to\mathbb{Z}$, the associated multiple ergodic averages are

$$\mathbb{E}_{n\in[N]}\, T^{a_1(n)}f_1\cdots T^{a_\ell(n)}f_\ell,$$

where $\mathbb{E}_{n\in[N]}=\frac{1}{N}\sum_{n=1}^{N}$ denotes a Cesàro average and $T^k f=f\circ T^k$. The sequences are jointly ergodic for the system if for all bounded measurable $f_1,\dots,f_\ell\in L^\infty(\mu)$ these averages converge in $L^2(\mu)$ to the product of the integrals $\int f_1\,d\mu\cdots\int f_\ell\,d\mu$, and jointly ergodic without qualification if this holds for every ergodic system. Joint ergodicity has direct combinatorial content: by Furstenberg's correspondence principle, it implies that every set of integers of positive upper density contains patterns of the form $m, m+a_1(n),\dots,m+a_\ell(n)$ for some $m,n\in\mathbb{N}$.

Joint ergodicity imposes a purely number-theoretic necessary condition. Writing $e(t)=e^{2\pi i t}$ and testing the definition on eigenfunctions of the system (functions satisfying $Tf=e(t)f$), one finds that the sequences must be good for equidistribution: for every $(t_1,\dots,t_\ell)\in[0,1)^\ell$ not all zero,

$$\lim_{N\to\infty}\mathbb{E}_{n\in[N]}\,e\big(a_1(n)t_1+\cdots+a_\ell(n)t_\ell\big)=0,$$

which, by Weyl's equidistribution criterion, implies that for every frequency vector $(t_1,\dots,t_\ell)\in[0,1)^\ell$ no nonzero integer multiple of which is an integer vector, the combination $t_1a_1+\cdots+t_\ell a_\ell$ is equidistributed modulo 1.

The main structural theorem of the source paper (Theorem 1.1 of Frantzikinakis, 'Joint ergodicity of sequences') shows that this condition, together with one analytic condition, is also sufficient: the sequences are jointly ergodic if and only if they are (i) good for seminorm estimates and (ii) good for equidistribution. Condition (i) is phrased with the Host–Kra seminorms $\lVert\cdot\rVert_s$, a hierarchy of seminorms on $L^\infty(\mu)$ defined recursively from the correlations of a function under iterates of $T$; they arise from the Host–Kra factors, the finite-step characteristic factors of multiple ergodic averages, and $\lVert f\rVert_s$ vanishes exactly when $f$ is orthogonal to the corresponding finite-step Host–Kra factor. The family is good for seminorm estimates if there exists $s\in\mathbb{N}$ such that for every ergodic system, whenever $f_1,\dots,f_\ell\in L^\infty(\mu)$ satisfy $\lVert f_m\rVert_s=0$ for some $m$ and the functions $f_{m+1},\dots,f_\ell$ are eigenfunctions of $T$, the above averages tend to $0$ in $L^2(\mu)$. Thus joint ergodicity splits into a mixing part (seminorm control) and a spectral part (equidistribution), and the question is whether the spectral part alone forces the mixing part.

Two facts support a positive answer. First, for a single sequence ($\ell=1$) the answer is positive by the spectral theorem. Second, for nilsystems — systems given by translation on a compact nilmanifold $G/\Gamma$, a homogeneous space of a nilpotent Lie group by a discrete cocompact subgroup — the Host–Kra seminorms trivialize after finitely many steps, and equidistribution alone does imply joint ergodicity (Corollary 1.4 of the source paper). On the other hand, the fixed-system version of the question has a negative answer even for $\ell=1$: the sequence $\lfloor\log_2 n\rfloor$ is good for equidistribution in the system-relative sense for every weakly mixing system (relative to a fixed system the condition is checked only for the frequencies of its nonconstant eigenfunctions, and weakly mixing systems have none, so it holds vacuously on them), yet it is ergodic only on systems conjugate to a single point, as shown by Donoso, Koutsogiannis, Kuca, Sun, and Tsinas in 2025. Hence the quantification over all ergodic systems in the global question is essential, and the known counterexample machinery does not transfer to it.

Problem Statement

Problem (Frantzikinakis, Problem 1, Section 1.8 of 'Joint ergodicity of sequences'). Let $\ell\ge 2$ and let $a_1,\dots,a_\ell\colon\mathbb{N}\to\mathbb{Z}$ be integer sequences such that

$$\lim_{N\to\infty}\mathbb{E}_{n\in[N]}\,e\big(a_1(n)t_1+\cdots+a_\ell(n)t_\ell\big)=0$$

for every $(t_1,\dots,t_\ell)\in[0,1)^\ell$ with $(t_1,\dots,t_\ell)\neq(0,\dots,0)$; that is, the collection is good for equidistribution. Must it be jointly ergodic, meaning that

$$\lim_{N\to\infty}\left\lVert\mathbb{E}_{n\in[N]}\,T^{a_1(n)}f_1\cdots T^{a_\ell(n)}f_\ell-\int f_1\,d\mu\cdots\int f_\ell\,d\mu\right\rVert_{L^2(\mu)}=0$$

for every ergodic measure-preserving system $(X,\mu,T)$ and all $f_1,\dots,f_\ell\in L^\infty(\mu)$? Prove this, or exhibit a counterexample consisting of an integer $\ell\ge 2$, sequences $a_1,\dots,a_\ell\colon\mathbb{N}\to\mathbb{Z}$ good for equidistribution, an ergodic measure-preserving system $(X,\mu,T)$, and functions $f_1,\dots,f_\ell\in L^\infty(\mu)$ for which the displayed convergence fails. The case $\ell=1$ is settled positively by the spectral theorem, so the open content is the general case $\ell\ge 2$; by the source paper's Theorem 1.1, an affirmative answer is equivalent to the statement that goodness for equidistribution implies goodness for seminorm estimates. The quantification over all ergodic systems is essential: the analogous question for a single fixed system has a negative answer even for $\ell=1$.

References

1. Nikos Frantzikinakis, Joint ergodicity of sequences, Advances in Mathematics 417 (2023), article 108918. DOI: 10.1016/j.aim.2023.108918; arXiv:2102.09967, https://arxiv.org/abs/2102.09967
2. Borys Kuca, Joint ergodicity - 40 years on, survey preprint (March 2026). arXiv:2603.18974, https://arxiv.org/abs/2603.18974
3. Sebastián Donoso, Andreas Koutsogiannis, Borys Kuca, Wenbo Sun, Konstantinos Tsinas, Resolving the joint ergodicity problem for Hardy sequences, preprint (2025). arXiv:2506.20459, https://arxiv.org/abs/2506.20459
4. Sebastián Donoso, Andreas Koutsogiannis, Borys Kuca, Wenbo Sun, Konstantinos Tsinas, Seminorm estimates and joint ergodicity for pairwise independent Hardy sequences, preprint (2024). arXiv:2410.15130, https://arxiv.org/abs/2410.15130
5. Nikos Frantzikinakis, Borys Kuca, Joint ergodicity for commuting transformations and applications to polynomial sequences, Inventiones mathematicae 239 (2025), no. 2, 621–706. DOI: 10.1007/s00222-024-01313-w; arXiv:2207.12288, https://arxiv.org/abs/2207.12288
6. Konstantinos Tsinas, Joint ergodicity of Hardy field sequences, Transactions of the American Mathematical Society 376 (2023), no. 5, 3191–3263. DOI: 10.1090/tran/8752; arXiv:2109.07941, https://arxiv.org/abs/2109.07941