What is Falconer’s Conjecture?

Alex Iosevich · Notices of the American Mathematical Society · 2019

The Statement of the ProblemMany problems in mathematics take the following form.Suppose that 𝑋, 𝑌 are sets and 𝑓 ∶ 𝑋 → 𝑌 is a function.Suppose that 𝑋 is sufficiently large and 𝑓 is suitably nontrivial.Then 𝑓(𝑋) takes up a substantial portion of 𝑌.A classical example of this phenomenon is Picard's Little Theorem, which says that any entire analytic function whose range omits two points must be a constant function.Let 𝑋 = 𝐸 × 𝐸, 𝑌 = ℝ, and 𝑓(𝑥, 𝑦) = |𝑥 -𝑦|, where 𝐸 is a compact subset of ℝ 𝑑 , 𝑑 ≥ 2, and |𝑥| = √ 𝑥 2 1 + 𝑥 2 2 + ⋯ + 𝑥 2 𝑑 .The Falconer distance problem asks how large does the Hausdorff dimension of 𝐸 needs to be to ensure that the Lebesgue measure of the distance set Δ(𝐸) = {|𝑥 -𝑦| ∶ 𝑥, 𝑦 ∈ 𝐸} is positive.In this context, it is sufficient to think of Hausdorff dimension of a compact set 𝐸, denoted by 𝑑𝑖𝑚 ℋ (𝐸), in the following way.There exists a Borel measure supported on 𝐸 such that for every 𝛼 < 𝑑𝑖𝑚 ℋ (𝐸), the 𝛼-energy integral(1)The background and the details pertaining to the Hausdorff dimension and energy integrals are beautifully described in Falconer's "Geometry of Fractal Sets" ([5]), and Mattila's "Fourier Analysis and Hausdorff Dimensions" ([12]).This problem can be viewed as a more delicate variant of the celebrated Steinhaus Theorem, which says that if 𝐸 ⊂

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