Besicovitch one-dimensional rectifiability thresholds

How high must the lower one-density of a finite-length set be to force countable one-rectifiability? The Besicovitch threshold σ₁(H) asks this in a real Hilbert space H, including nonseparable H; the previous planar upper bound was 0.7.

Why it matters and what counts as progress

Why it matters: A smaller density threshold strengthens the geometric criterion forcing a finite-length set to lie on countably many rectifiable curves.

What counts as progress: Lower a proved threshold, construct a nonrectifiable example giving a lower bound, or extend a bound uniformly to all Hilbert spaces.

How the problem developed

1928 · Upper bound [1]
A. S. Besicovitch First bounded the planar density threshold for rectifiability by 1−10⁻²⁵⁷⁶ and identified the unresolved sharp-threshold question.
1938 · Upper bound [1]
A. S. Besicovitch Improved the planar upper bound to 3/4; a purely unrectifiable example gives the lower threshold bound 1/2.
2024 · Upper bound [1]
Camillo De Lellis, Federico Glaudo, Annalisa Massaccesi and Davide Vittone Improved the best known planar bound to 0.7 using linear programming, improving on the earlier Preiss–Tišer bound.
2 Sept 2026 · Reported claim [2]
Yongxi Lin, with AI-assisted proof development Reported a machine-checked threshold upper bound of 0.6934 for real Hilbert spaces, including the planar case.

AI contribution

A machine-checked bound of 0.6934 for Besicovitch's 1/2-problem

Yongxi Lin

For every real Hilbert space H, including nonseparable spaces, the one-dimensional Besicovitch rectifiability threshold σ₁(H) is at most 0.6934. In particular, a planar Borel set of finite H¹ measure is countably one-rectifiable when its lower one-density exceeds any fixed threshold greater than 0.6934 almost everywhere.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4 with Mathlib and exact rational Gram-certificate computation)

Human understandingFew