| $5/4=1.25$ | [[GIOW2018](#GIOW2018)] | Guth–Iosevich–Ou–Wang proved that if $\dim_H(E)>5/4$ then $\lvert\Delta(E)\rvert>0$. [GIOW2018-thm-5-4] |exact pinned row| $5/4=1.25$ | [[GIOW2018](#GIOW2018)] | Guth–Iosevich–Ou–Wang proved that if $\dim_H(E)>5/4$ then $\lvert\Delta(E)\rvert>0$. <a href="#GIOW2018-thm-5-4">[GIOW2018-thm-5-4]</a> |tracked since 2026-07-24 — no movement seen yet
Lower bound, last-listed row
| $1$ | [[Fal1986](#Fal1986)] | Falconer gave examples showing (in general dimension $d$) that one cannot expect $\lvert\Delta(E)\rvert>0$ below the threshold $d/2$; in $d=2$ this yields $s_\Delta(\mathbb{R}^2)\ge 1$. [[GIOW2018-lb-d-2]] |exact pinned row| $1$ | [[Fal1986](#Fal1986)] | Falconer gave examples showing (in general dimension $d$) that one cannot expect $\lvert\Delta(E)\rvert>0$ below the threshold $d/2$; in $d=2$ this yields $s_\Delta(\mathbb{R}^2)\ge 1$. [<a href="#GIOW2018-lb-d-2">[GIOW2018-lb-d-2]</a>] |tracked since 2026-07-24 — no movement seen yet
Cite this rowBounds Ledger — Falconer distance problem in $\mathbf{R}^2$ (34a)
upper bound: $5/4=1.25$ (tracked since 2026-07-24 — no movement seen yet)
lower bound: $1$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/34a.html