Rate at which $\kappa(n)$ approaches $1$

Upper bound, last-listed row

| $1$ | [AJM2025] | Stated in the discussion of Problem 11. [AJM2025-prob11] (A supporting mechanism is the lower bound $\kappa(n)\ge 1+c\frac{\log n}{n}$ for $n\not\equiv 0 \pmod 4$.) [AJM2025-thm6] |

exact pinned row

| $1$ | <a href="#AJM2025">[AJM2025]</a> | Stated in the discussion of Problem 11. <a href="#AJM2025-prob11">[AJM2025-prob11]</a> (A supporting mechanism is the lower bound $\kappa(n)\ge 1+c\frac{\log n}{n}$ for $n\not\equiv 0 \pmod 4$.) <a href="#AJM2025-thm6">[AJM2025-thm6]</a> |

tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $17/92 \approx 0.18478$ | [AJM2025] | The authors state that (unconditionally) their method permits $\kappa(n)\le 1+\frac{1}{n^\alpha}$ for all sufficiently large $n$ with $\alpha=17/92-\delta$ for any small $\delta>0$. [AJM2025-thm1-alpha] |

exact pinned row

| $17/92 \approx 0.18478$ | <a href="#AJM2025">[AJM2025]</a> | The authors state that (unconditionally) their method permits $\kappa(n)\le 1+\frac{1}{n^\alpha}$ for all sufficiently large $n$ with $\alpha=17/92-\delta$ for any small $\delta>0$. <a href="#AJM2025-thm1-alpha">[AJM2025-thm1-alpha]</a> |

tracked since 2026-07-24 — no movement seen yet