A dimension-independent box bound from dual total correlation
Year: 2026
Type: preprint
Abstract: Let $S$ be a finite nonempty set in a product of $d$ coordinate spaces, and let $\mathrm{U}_S$ be its uniform law. We prove that some nonempty Cartesian box $B$ satisfies $ |S\triangle B|\leq 10\mathrm{DTC}(\mathrm{U}_S)|S|, $ where dual total correlation is computed using natural logarithms. The constant does not depend on the dimension or on the coordinate alphabets. The proof follows actual coordinate slices: their average disagreement is bounded linearly by mutual information, while an exact entropy identity retains the deficit available within each slice. Selecting one slice with both costs controlled makes the induction close with the same constant. The resulting Loomis--Whitney stability estimate has coefficient $20(d-1)$ under the usual normalized near-equality assumption. A two-point example shows that linear dependence on $d$ is necessary in that normalization. The constants are not claimed sharp. This deposit contains the preprint and its LaTeX source package, including one explanatory vector figure generated within the source. The manuscript gives the full proof and an appendix comparing the estimate with an earlier argument. AI assistance is disclosed in the manuscript. Version 1.0 is prepared for journal submission; it is not a peer-reviewed or accepted publication.
Author: Vaseghi, Sam
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221515006
Cites:
Cited by:
Related to: