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.