Determining Improving Regions from Assouad Spectra

Definitions and mathematical statement

The supplied record refers to a set $E$, an improving region $T_E$, and the Assouad spectra of $E$ and its subsets. It does not specify the admissible sets $E$, their ambient metric space, the operator defining $T_E$, or the exponent and endpoint conventions for that region. Consequently, these objects cannot be given complete mathematical definitions from the supplied evidence alone.

For notation only, write \[ \mathcal S(F)=\left(\dim_{\mathrm A}^{\theta}F\right)_{0<\theta<1} \] for the Assouad spectrum referred to in the source, whenever it is defined. The source does not supply its covering convention or specify which subsets $F\subseteq E$ are to be used.

The source question is: for every admissible set $E$, determine whether, and in what precise manner, its improving region $T_E$ can be recovered from the Assouad spectrum of $E$ together with the Assouad spectra of subsets of $E$.

This remains a source-level question rather than a fully specified conjecture. In particular, the allowed subset data, their indexing, the meaning of recovery, and whether recovery includes the boundary of $T_E$ require clarification. No operator, ambient dimension, or admissible class of sets is inferred from the cited paper's title.

Short English statement

The question asks whether the Assouad spectra of a set and its subsets determine its improving region. A precise formulation requires the missing definition of that region, the allowed sets and subsets, and the endpoint conventions. The supplied curation labels the problem partially solved; that status has not been independently verified.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30005162$, title Determining Improving Regions from Assouad Spectra, exact source problem code catalogue entry. The record assigns proposed year 2022 and names no individual proposer.
• Supplied report citation: Real Analysis, Harmonic Analysis and Applications (2023), Oberwolfach Reports, pp. 1743--1803, DOI 10.4171/owr/2022/31. The literature-source label is Oberwolfach Report 31/2022. https://doi.org/10.4171/owr/2022/31. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Roos--Seeger, Spherical maximal functions and fractal dimensions. https://arxiv.org/abs/2004.00984.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that Roos--Seeger characterize the relevant improving region and solve broad classes, while exact general determination of $T_E$, especially boundary cases, remains incomplete. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that only clear formula delimiters, notation, grammar, and typography were repaired while mathematical content and scope were preserved. This supplied editorial label does not establish mathematical correctness or completeness of the formulation.