Refined Slippery Bounds for Positive-Word Rotation Numbers

Definitions and mathematical statement

A positive word in the letters $a,b$ is a finite word containing no inverse letters. Define $h_a(w)$ and $h_b(w)$ to be the respective numbers of occurrences of $a$ and $b$ in $w$. Under the usual interpretation of the supplied alternating-block notation, \[ w=a^{\alpha_1}b^{\beta_1}\cdots a^{\alpha_m}b^{\beta_m},\qquad m\geq 1,\quad \alpha_i,\beta_i\in{\mathbb{Z}}_{>0}, \] these counts are \[ h_a(w)=\sum_{i=1}^{m}\alpha_i,\qquad h_b(w)=\sum_{i=1}^{m}\beta_i. \] The positivity of every block exponent and the intervening alternating blocks are conventional interpretations of the abbreviated source notation, requiring confirmation from the original context.

The supplied question asks whether, for every such positive word and every admissible pair of parameters $r,s$ for which $R(w,r,s)=p/q$, one has \[ R(w,r,s)-h_a(w)r-h_b(w)s\leq\frac{m}{q}. \] The record identifies this as a question about positive-word rotation numbers but does not define $R$, specify the domains of $r,s$, or state the denominator convention for $p/q$. Thus the quantifier over admissible parameters remains unresolved. Interpreting $q$ as the positive denominator in lowest terms would require $p\in{\mathbb{Z}}$, $q\in{\mathbb{Z}}_{>0}$, and $\gcd(|p|,q)=1$; that convention is not explicit in the supplied statement. A real-valued normalization of $R$ is also needed for the displayed subtraction and inequality. These missing qualifications prevent a fully specified conjecture from being recovered from this record alone.

Short English statement

The question asks whether the rotation-number quantity associated with a positive word exceeds the weighted contribution of its letters by at most $m/q$, where $m$ counts the displayed alternating block pairs and $q$ comes from $R(w,r,s)=p/q$. The definition and normalization of $R$, parameter domains, block conventions, and denominator convention require confirmation. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $30002129$, source problem code catalogue entry, and title "Refined Slippery Bounds for Positive-Word Rotation Numbers." The record gives proposed year 2012 and names no proposer.
• Supplied report citation: Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2013), Oberwolfach Reports, pp. 2139--2203, DOI 10.4171/owr/2012/35. https://doi.org/10.4171/owr/2012/35. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Calegari--Walker, Ziggurats and rotation numbers. https://arxiv.org/abs/1110.0080.
• Supplied literature reference: Mann, rigidity and flexibility of circle actions. https://e.math.cornell.edu/people/mann/papers/rigflexA.pdf.
• Supplied literature reference: Oberwolfach Report 35. https://doi.org/10.4171/owr/2012/35.
• Supplied dated literature curation, checked 2026-08-21: the status is recorded as open, with the assessment that the refined slippery inequality is established only in restricted settings and remains unresolved in the stated generality. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; the time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification claim; no source consultation or independent verification is asserted here.