Roth Recurrence in Groups of Intermediate Growth

Definitions and mathematical statement

For a finitely generated group $G$, choose a finite symmetric generating set $S$ and write \[ \beta_S(r)=\left|\{g\in G:|g|_S\le r\}\right|,\qquad r\in{\mathbb{N}},\ r\ge1, \] where $|g|_S$ is word length. Under the usual convention, $G$ has intermediate growth if \[ \lim_{r\to\infty}\frac{\log\beta_S(r)}{r}=0 \quad\text{and}\quad \nexists C>0,\ d\ge0\ \text{such that}\ \beta_S(r)\le Cr^d\ \text{for all }r\ge1. \] This property is independent of the finite generating set.

A measure-preserving action of $G$ on a measure space $(X,\mathcal A,\mu)$ consists of invertible measurable maps $T_g:X\to X$ satisfying $T_e=\operatorname{id}_X$, $T_{gh}=T_g\circ T_h$, and $\mu(T_g^{-1}A)=\mu(A)$ for every $g\in G$ and $A\in\mathcal A$. Write $g^nB=T_g^n(B)$ for $n\in{\mathbb{Z}}$.

On the existential reading of the unspecified elements $g_1,g_2$, the supplied question asks whether there exist a group $G$ of intermediate growth, a measure-preserving action of $G$ on $(X,\mathcal A,\mu)$, elements $g_1,g_2\in G$, and $B\in\mathcal A$ such that \[ \mu(B)>0 \quad\text{and}\quad \forall n\in{\mathbb{Z}}\setminus\{0\},\qquad \mu\left(T_{g_1}^{n}(B)\cap T_{g_2}^{n}(B)\right)=0. \] Unresolved qualifications. The supplied statement does not specify whether $\mu(X)=1$, whether $\mu(X)<\infty$, or whether $\mu(B)<\infty$. The displayed formulation preserves that omission; it cannot be identified with an intended probability-space recurrence problem without clarification. The source also gives no explicit quantifiers or additional hypotheses for $g_1,g_2$, such as commutativity or generation of $G$. Its displayed intersection contains exactly the two translated sets and no additional factor $B$. These details must be checked against the intended source formulation before mathematical acceptance.

Short English statement

The supplied question asks whether a group of intermediate growth can act by measure-preserving transformations with a positive-measure set whose two specified translates have zero-measure intersection at every common nonzero integer power. The measure-space assumptions and the intended hypotheses on the two group elements remain unspecified. The supplied label is open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30002207$, source problem code catalogue entry, title Roth Recurrence in Groups of Intermediate Growth. No individual proposer is supplied.
• Supplied source citation: Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory (2013), Oberwolfach Reports, pp. 2985--3059, DOI 10.4171/owr/2012/50. Original OWR report: https://doi.org/10.4171/owr/2012/50. The registry separately supplies proposed year 2012; the relationship between these year labels has not been independently checked.
• Supplied literature source, labeled "solvable exponential-growth counterexamples": https://doi.org/10.1017/S0143385703000427. No further bibliographic details are supplied.
• Literature assessment checked at 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem open and states that nilpotent groups satisfy the relevant Roth-type recurrence, that solvable groups of exponential growth admit counterexamples, and that the existence of such counterexamples for groups of intermediate growth remains unresolved. The supplied background explicitly requests independent verification of the time-sensitive status and cited claims; it does not state the precise hypotheses of those recurrence results or counterexamples.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. These are supplied editorial assertions, not independent mathematical verification in this draft.