Let n≥ 2, let (X,B,mu) be a probability space, and let T_1,…,T_n be commuting measure-preserving transformations of X. For f_1,…,f_n∈ L^∞(X), define the multiple ergodic averages A_N(f_1,…,f_n):=frac1N∑_k=1^N∏_j=1^nf_j circ T_j^k. For r>2, prove that there is a constant C_n,r, depending only on n and r, such that |bigl(A_N(f_1,…,f_n)bigr)_N≥1|_V^r(N;L^2(X))≤ C_n,r∏_j=1^n|f_j|_L^∞(X), where |(a_N)|_V^r(N;L^2):=sup_N_0<…<N_J(|a_N_0|_L^2^r+∑_i=1^J|a_N_i-a_N_i-1|_L^2^r)^1/r. The associated Avigad–Rute question asks for such quantitative variation bounds uniformly for every number of commuting transformations and every admissible variation exponent.

Avigad–Rute open question on norm variation of multiple ergodic averages

Let $n\ge 2$ , let $(X,\mathcal{B},\mu)$ be a probability space, and let $T_1,\ldots,T_n$ be commuting measure-preserving transformations of $X$ . For $f_1,\ldots,f_n\in L^\infty(X)$ , define the multiple ergodic averages $A_N(f_1,\ldots,f_n):=\frac{1}{N}\sum_{k=1}^{N}\prod_{j=1}^{n}f_j\circ T_j^k$ . For $r>2$ , prove that there is a constant $C_{n,r}$ , depending only on $n$ and $r$ , such that $\left\|\bigl(A_N(f_1,\ldots,f_n)\bigr)_{N\ge1}\right\|_{V^r(\mathbb{N};L^2(X))}\le C_{n,r}\prod_{j=1}^{n}\|f_j\|_{L^\infty(X)}$ , where $\|(a_N)\|_{V^r(\mathbb{N};L^2)}:=\sup_{N_0<\cdots<N_J}\left(\|a_{N_0}\|_{L^2}^r+\sum_{i=1}^{J}\|a_{N_i}-a_{N_{i-1}}\|_{L^2}^r\right)^{1/r}$ . The associated Avigad–Rute question asks for such quantitative variation bounds uniformly for every number of commuting transformations and every admissible variation exponent.

References

Primary source

arXiv

Additional references

A blueprint for the formalization of norm-variation of multiple ergodic averages for commuting transformations — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new Lean formalization claims to settle the quantitative strengthening, but its mathematical status has not been independently verified.

The Avigad–Rute question asks for quantitative variation bounds for multiple ergodic averages, conjecturally for every $r \ge 2$ and any number of commuting transformations. It was recorded as an open problem by Avigad and Rute in 2015.

Known results

Tao, 2008: qualitative $L^2$ norm convergence for finitely many commuting transformations.
Do, Oberlin, and Palsson, 2014: variational estimates in a restricted two-transformation setting.
Zorin-Kranich, 2023: an $r$ -variation estimate for three commuting transformations when $r>4$ ; the cases $r \ge 2$ and more than three transformations remain open.

August 2026 Lean formalization

A blueprint reports a completed Lean $4$ formalization of the quantitative norm-variation result and its underlying estimates, claiming a machine-checked resolution of the tracked question. This claim is unverified, and the source does not establish independent peer review of the mathematical significance.

Current status (as of August 2026): A Lean $4$ formalization claims the general quantitative result, but the claim is unverified; absent that formalization, the strongest published partial result remains $r>4$ for three transformations, with the broader question open.

Sources

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