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The Falconer distance conjecture

A distance set with full one-dimensional Hausdorff dimension need not have positive length.

Progress records 1

Understand the problem ↓

1 record · How we rate progress

Understanding the problem

Read the background

A distance set with full one-dimensional Hausdorff dimension need not have positive length. The conjectured target is positive Lebesgue measure at the strict half-dimension threshold.

Compactness and the strict dimension inequality are retained. No endpoint statement at dimension d/2 is made.

Why it matters

If correct, attains the conjectured strict half-dimension threshold in every stated dimension.

What counts as progress

A sourced resolution or substantial advance on the precise target, with hypotheses, prior context, and verification evidence stated separately.

What to watch for

  • Assess the precise statement and assumptions before comparing results.
  • Compactness and the strict dimension inequality are retained. No endpoint statement at dimension d/2 is made.
  • Track independent assessments, dependencies, corrections, and withdrawals separately from importance.

Background review: 2026-10-11. Selected sources:

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