Bergelson–Moreira–Richter conjecture

For every function $f$ in the broad class of non-polynomial functions considered by Bergelson, Moreira, and Richter, every $ell\textbackslash in cmathbb{N}$ , every invertible measure-preserving system $(X,cmathscr{B},cmu,T)$ , and every $A\textbackslash in cmathscr{B}$ with $cmu(A)>0$ , the return-time set $\{n\textbackslash in cmathbb{N}: cmu(A\textbackslash \cap T^{-[f(n)]}A\textbackslash \cap T^{-2[f(n)]}A\textbackslash \cap \cdots\textbackslash \cap T^{-ell[f(n)]}A)>0\}$ is thick, meaning that it contains arbitrarily long intervals of natural numbers. The supplied sources do not specify the full admissibility hypotheses defining the broad class; they explicitly identify the subclass of smooth functions satisfying $x^{d-1}prec f(x)prec x^d$ for some $d\textbackslash in cmathbb{N}$ .

References

Primary source

arXiv

Additional references

Weighted averages and applications to sets of multiple recurrence — arXiv — Vitaly Bergelson, Michael Reilly

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims to settle the conjecture by proving a much broader recurrence theorem, but the claim has not been independently verified.

The conjecture concerns recurrence and thickness of return-time sets along broad classes of non-polynomial sequences. Bergelson, Moreira, and Richter developed the foundational results in work submitted in 2017 and revised in 2020.

Known results

Bergelson, Moreira, and Richter, 2017/2020: for a broad class of non-polynomial functions, relevant single- and multiple-return sets are thick, containing arbitrarily long intervals.
Their result answers the corresponding multiple-recurrence problem for $f(n)=n^\alpha$ when $\alpha\geq 1$ is noninteger; integer exponents admit counterexamples.
Bergelson, Moreira, and Richter, 2020: related Hardy-field work established convergence, recurrence, and combinatorial consequences.

September 2026 claimed resolution

On September 15, 2026, the preprint Weighted averages and applications to sets of multiple recurrence by Vitaly Bergelson and Michael Reilly was reported as proving a weighted recurrence theorem that includes the Bergelson–Moreira–Richter conjecture and extends thickness results to a broad class of smooth non-polynomial sequences. The exact admissibility hypotheses are not given in the retrieved summary, so the resolution remains unverified.

Current status (as of September 2026): the conjecture has a claimed proof in a new preprint, but independent verification is absent and the exact hypotheses and proof status remain open.

Sources

No solutions have been posted yet.