Optimizing Explicit Unit-Distance Lower-Bound Certificates
The 2026 disproof of Erdős's unit-distance conjecture and Sawin's quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes integer parameters whose choice is not fully optimized. This report treats Sawin's parameter selection as a nonlinear integer optimization problem and develops an open-source Python optimization and verification pipeline for certificates involving prime sets $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. After re...
| Authors | Michael T. M. Emmerich |
|---|---|
| Primary category | math.OC |
| Published | 2 June 2026 |
| Last updated at source | 9 June 2026 |
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