Uniform Recurrence Rates for Minimal Interval Exchanges

Definitions and mathematical statement

Let $J=[0,1)$. An interval-exchange transformation on $J$ is a bijection $T:J\to J$ for which there is a finite partition \[ 0=a_0<a_1<\cdots<a_m=1 \] such that $T$ restricts to a translation on each half-open interval $[a_{i-1},a_i)$. Call $T$ minimal if $\{T^n(x):n\geq 0\}$ is dense in $J$ for every $x\in J$. Here $T^n$ denotes the $n$-fold iterate. For $x,y\in J$, define \[ \phi_1(x,y)=\liminf_{\substack{n\to\infty\\ n\in{\mathbb{N}},\ n\geq1}} n\lvert T^n(x)-y\rvert\in[0,+\infty]. \] The distance is ordinary absolute-value distance in ${\mathbb{R}}$.

Question. Does there exist a minimal interval-exchange transformation $T:J\to J$ such that \[ \forall x\in J\;\forall y\in J,\qquad \phi_1(x,y)=0? \] Equivalently, does such a transformation exist for which \[ \forall x,y\in J\;\forall\varepsilon>0\;\forall N\in{\mathbb{N}}\;\exists n\in{\mathbb{N}}, \qquad n\geq\max\{1,N\},\qquad n\lvert T^n(x)-y\rvert<\varepsilon? \] The universal quantifiers include partition endpoints belonging to $J$; the endpoint $1$ is excluded. No uniform choice of $n$ over $x,y$ is required.

The supplied wording contrasts this property with an almost-everywhere property, conventionally understood using product Lebesgue measure on $J\times J$. It does not supply the precise hypotheses of the referenced almost-everywhere results. The supplied dated assessment labels the question open and reports that the property is ruled out for interval exchanges on at most three intervals. These are supplied literature qualifications, not independently verified assertions here.

Short English statement

Does some minimal interval-exchange transformation of $[0,1)$ satisfy the following property: for every starting point $x$, every target $y$, and every $\varepsilon>0$, there are arbitrarily large positive integers $n$ with $\lvert T^n(x)-y\rvert<\varepsilon/n$? The supplied source asks for every pair, including interval endpoints, rather than merely almost every pair.

Sources

• Local UnsolvedMath registry, permanent ID $30001211$, title: Uniform Recurrence Rates for Minimal Interval Exchanges; source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The supplied record gives proposed year 2009 and names no individual proposer.
• Supplied report citation: Mini-Workshop: The Pisot Conjecture --- From Substitution Dynamical Systems to Rauzy Fractals and Meyer Sets (2009), Oberwolfach Reports, pp. 725--766, DOI 10.4171/owr/2009/13. https://doi.org/10.4171/owr/2009/13.
• Supplied literature source labeled "Boshernitzan--Chaika report text": https://ems.press/content/serial-article-files/46214?nt=1. This label is retained as supplied attribution; it does not identify an individual proposer of the question.
• Supplied literature source labeled "original OWR report": https://doi.org/10.4171/owr/2009/13.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open, describes the report's results as almost-everywhere statements, and reports exclusion of the desired property for at most three intervals. This is supplied dated curation, not an independently verified assessment. The record explicitly calls for independent verification of time-sensitive status and cited claims.