Multiple ergodic averages — Problem 7

Definitions and mathematical statement

For an integer sequence $a(n)$, call two-fold recurrence of powers the following property: for every probability space $(X,\mathcal B,\mu)$, invertible measure-preserving map $R$, nonzero integers $r,s$, and measurable set $A$ of positive measure, there is $n\ge1$ with $a(n)\ne0$ and \[ \mu(A\cap R^{r a(n)}A \cap R^{s a(n)}A)>0. \] The set $R^kA$ denotes the image under the corresponding invertible iterate. Recurrence for commuting pairs instead requires that for every commuting pair $T,S$ and every positive-measure $A$, some such $n$ satisfy $\mu(A\cap T^{a(n)}A\cap S^{a(n)}A)>0$.

Does a sequence exist which has two-fold recurrence of powers but fails two-fold recurrence for commuting pairs? A positive answer must supply an integer sequence satisfying the universal first property and also one probability-preserving commuting system and positive-measure set $A$ for which \[ \mu(A\cap T^{a(n)}A\cap S^{a(n)}A)=0 \quad\text{whenever }a(n)\ne0. \] A negative answer would prove the implication for every integer sequence. Repeated terms and negative values are allowed; the condition $a(n)\ne0$ prevents a zero term from giving a vacuous recurrence. Neither property assumes ergodicity, and the powers hypothesis includes every nonzero integer pair, not only $1,2$.

Short English statement

Can recurrence along every pair of powers of one transformation fail for a general commuting pair?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800007 (AMR-047-0007), 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 7. Original source indexed by AMR-047-0007. https://arxiv.org/abs/1103.3808.