Multiple ergodic averages — Problem 28

Definitions and mathematical statement

For positive integers $n$, put $a(n)=\lfloor n\sin n\rfloor$, where the angle is measured in radians and the floor is the greatest integer at most its argument. The sequence has both positive and negative values. Let $T$ be an invertible measure-preserving map on an arbitrary probability space $(X,\mathcal B,\mu)$.

Prove both of the following assertions for every integer $\ell\ge1$. For all integers $k_1,\ldots,k_\ell$ and all functions $f_i\in L^\infty(\mu)$, the averages \[ \frac1N\sum_{n=1}^N\prod_{i=1}^\ell f_i\circ T^{k_i a(n)} \] converge in $L^2(\mu)$. For all nonzero integers $k_1,\ldots,k_\ell$ and every measurable $A$ with $\mu(A)>0$, some $n\ge1$ with $a(n)\ne0$ satisfies \[ \mu\left(A\cap T^{k_1a(n)}A\cap\cdots \cap T^{k_\ell a(n)}A\right)>0. \] These are, respectively, multiple convergence and multiple recurrence of powers. Negative exponents are interpreted using $T^{-1}$, and the nonzero condition in recurrence excludes a vacuous return. The system need not be ergodic and the sequence is fixed independently of it. No pointwise convergence or particular formula for the norm limit is requested. The oscillating integer sequence itself is the target; replacing it by $n$, $|a(n)|$, or a monotone rearrangement would be a different problem.

Short English statement

Show that the oscillating sequence floor(n sin n) supports every finite-order recurrence and mean-convergence problem for powers of one transformation.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800028 (AMR-047-0028), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Frantzikinakis - Some open problems on multiple ergodic averages (2016), Problem 28. Original source indexed by AMR-047-0028. https://arxiv.org/abs/1103.3808.