Falconer Distance Conjecture

Prove Falconer’s distance conjecture: For any compact set E ⊂ R^d with d ≥ 2 and Hausdorff dimension dim_H(E) > d/2, demonstrate that the Lebesgue measure of the distance set A(E) = {|x − y| : x, y ∈ E} is positive.

Background

The paper reviews the Falconer distance conjecture as a central motivating open problem in geometric measure theory, relating the Hausdorff dimension of a set E in Euclidean space to the Lebesgue measure of its distance set. The authors survey progress on exponent thresholds achieved in different dimensions and settings, situating their work on generalized distances and configuration realizations within this broader context.

This conjecture has inspired numerous developments in harmonic analysis and fractal geometry, with partial results proving positivity of the distance set’s measure under dimension thresholds strictly larger than d/2. The authors’ techniques on generalized Radon transforms and realizability of trees of configurations contribute to the landscape of methods addressing such configuration problems, though the full conjecture remains open.

References

The celebrated Falconer distance conjecture (see e.g. [7], [19]) states that if the Hausdorff dimension of a compact set E C Rd, d ≥ 2, is greater than , then the Lebesgue measure of the distance set A(E) = {|x -y| : x, y € E} is positive.

— Realizing trees of configurations in thin sets  (2401.11597 - Greenleaf et al., 2024) in Section 1, Introduction

The threshold $d/2$ would be sharp. The problem remains open in every dimension $d \ge 2$ \footnote{Well after a full draft of this paper was completed, OpenAI posted a file claiming that a closely related problem has been solved. The claim, even if accepted by the mathematical community, would not recover our estimates in dimension $d\ge 3$.},

— Spread-out Furstenberg sets II: general configurations and fractal restricted projections  (2610.13682 - Cohen et al., 9 Oct 2026) in Section “Distance sets with spread-out pins”