Positive Measure and Renormalization for a Disturbed Binary Achievement Set
Positive measure, infinite gaps, and exact scale recursion
Establishes positive measure and quantitative bounds for a disturbed binary achievement set, together with an exact renormalization identity. It isolates the remaining interior question rather than settling the complete topological classification.
- Identifier
- MF-PRISM-MATH-2026-13
- Series
- Mathematics
- Edition
- Version 1.1
- Length
- 11 pages
- Reserved DOI
- 10.5281/zenodo.22728336 (record reserved)
What is this paper trying to establish?
What can be proved about the measure and self-similarity of the disturbed binary achievement set before its interior is classified?
The central idea.
Establishes positive measure and quantitative bounds for a disturbed binary achievement set, together with an exact renormalization identity. It isolates the remaining interior question rather than settling the complete topological classification.
Partial progress: positive measure and renormalization do not settle the remaining Cantor-set-versus-Cantorval classification.
This manuscript develops a method and finite evidence, but it does not resolve the full open problem. The unresolved boundary is part of the published result.
What changes if the method holds?
The results narrow the remaining classification problem and supply exact structure that future work can use without pretending the Cantor-set-versus-Cantorval question is finished.
Where scrutiny should concentrate.
The paper identifies the following points as the highest-value targets for independent review.
- 01Measure bounds for nested outer approximations and possible overlap overcounting.
- 02Exact pair tails, gaps, and the renormalization family.
- 03The unresolved interior question and the boundary between a positive-measure Cantor set and a Cantorval.
An open, inspectable research artifact.
The public record links the manuscript to its release, source package, review materials, and persistent identifier. Status travels with the paper; a reserved DOI is not presented as a published Zenodo record.
- Document
- Preprint
- Review status
- Open for independent review
- Published
- 2026-09-12
- DOI status
- Reserved · 10.5281/zenodo.22728336
- Canonical record
- MF-PRISM-MATH-2026-13