Euclidean View-Obstruction in Dimensions Eight to Ten and the Asymptotic Deficit
Year: 2026
Type: preprint
Abstract: The Euclidean view-obstruction problem asks for the critical radius of balls centered at the half-integer lattice that meet every positive ray from the origin. Writing Δ_n for its square, we determine Δ_8 = 8/17, Δ_9 = 1087/1980 and Δ_10 = 7/11 by computer-assisted proofs using Kahlon's seven-dimensional theorem. The upper bounds combine full-support subspace averaging, projected lattice coverings and exact finite certificates; dimension ten also requires a finite search through short integer relations. Explicit directions attain all three values, including a ten-dimensional direction previously listed by Gritsenko, Skoruppa and Zagier. In arbitrary dimension, we express their theta-block invariant in geometric normalization and derive the known bounds c log(n+1) ≤ n/12 - Δ_n ≤ C log^3(n+1), with repeated coordinates allowed throughout. Thus Δ_n ~ n/12, while the deficit tends to infinity more slowly than every positive power of n. The supplementary archive contains the executable verifiers, exact certificates and archived verification records.
Authors: Tan, Hu, Zhang, Ying
Institutions: Chinese Academy of Sciences, Soochow University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225452974