FormalConjectures.Wikipedia.Falconer

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module

public import FormalConjecturesUtil

Falconer's distance set conjecture

If $E \subseteq \mathbb{R}^d$ is compact with $\dim_H E > \frac{d}{2}$, then the distance set $${ |x - y| \mid x, y \in E }$$ has positive Lebesgue measure.

References

K. Falconer, On the Hausdorff dimensions of distance sets

Wikipedia, Falconer's conjecture

open MeasureTheory Setopen scoped ENNReal EuclideanGeometry

Falconer's distance set conjecture, $d = 2$ case.

@[category research open, AMS 28 42]
lemma falconer_conjecture_two (E : Set <| ℝ²) (hc : IsCompact E) (hd : 2 < 2 * dimH E ) :
    0 < volume (image2 dist E E) := sorry

Falconer's distance set conjecture.

@[category research open, AMS 28 42]
lemma falconer_conjecture (d : ℕ) (E : Set <| ℝ^d) (hc : IsCompact E) (hd : d < 2 * dimH E ) :
    0 < volume (image2 dist E E) := sorry