Paraboloid Lifting and Relative Branching for Pinned Distance Sets in ℝ³

The paper studies pinned distance sets for Ahlfors-regular subsets of ℝ³ and proves three unconditional results and one conditional result. For a set E ⊂ ℝ³ and a pin x ∈ E, the pinned distance set is Δₓ(E) = {‖x − y‖ : y ∈ E}, and the pinned squared distance set is Dₓ(E) = {‖x − y‖² : y ∈ E}. The general setting is an Ahlfors t-regular measure μ supported on E, meaning that there is a constant C_A ≥ 1 such that for all x ∈ E and 0 < r ≤ diam(E), C_A⁻¹ r^t ≤ μ(B(x,r)) ≤ C_A r^t. In this situation dim_H(E) = t and μ is doubling. The paper is concerned with the range t > 3/2, which is the conjectured Falconer threshold in ℝ³. The pinned Falconer problem asks whether some pin x ∈ E gives Δₓ(E) positive Lebesgue measure, or at least positive Hausdorff dimension. The paper works with Hausdorff dimension and with Ahlfors-regular measures throughout. The background result motivating the paper is that for every Borel set E ⊂ ℝ³ with dim_H(E) > 3/2 one has sup_{x∈E} dim_H(Δₓ(E)) ≥ 2/3 ≈ 0.6667. The paper aims to improve this benchmark in several geometric regimes. The first regime is the diffuse case, where E is hyperplane diffuse. The second is the product case E = A × F with dim_H A > 1. The third is the thin planar case, where E lies in a tube H(γ) around a plane H and the projection π_H(E) has dimension at most 1. The fourth is the thick planar case, where E ⊂ H(γ) and dim_H π_H(E) > 1; here the improvement is conditional on a Relative Branching Hypothesis. The paper explicitly does not claim an unconditional proof of q* = 0.7813… in the thick planar case. The notation used throughout is as follows. B(x,r) is the closed ball centered at x with radius r > 0. The tubular neighborhood of a plane H ≅ ℝ² is H(γ) = {(p,h) ∈ ℝ² × ℝ : |h| ≤ γ}. The orthogonal projection onto H is π_H : ℝ³ → H. The Hausdorff dimension is dim_H. The minimum number of δ-balls needed to cover E is N_δ(E). The Grassmannian of 1-dimensional linear subspaces of ℝᵏ is G(k,1). The Euclidean norm is ‖x‖. Constants C, c, C₁, C₂, … are positive and may change from line to line, and all logarithms are base 2 unless otherwise specified. The Hausdorff dimension is defined in the usual way: for E ⊂ ℝ³, s ≥ 0 and δ > 0, H^s_δ(E) = inf{∑ᵢ (diam Bᵢ)^s : E ⊂ ∪ᵢ Bᵢ, diam Bᵢ ≤ δ}, H^s(E) = lim_{δ→0} H^s_δ(E), and dim_H(E) = inf{s ≥ 0 : H^s(E) = 0}. Hyperplane diffuseness is defined as follows. A closed set J ⊂ ℝ³ is hyperplane diffuse if there exists γ > 0 such that for every x ∈ J, every 0 < r ≤ 1, and every affine plane H ⊂ ℝ³, one has J ∩ B(x,r) \ H(γr) ≠ ∅. A locally finite Borel measure μ on ℝ³ is (C,α)-absolutely decaying if there exist C ≥ 1 and α > 0 such that for every x ∈ supp μ, every 0 < r ≤ 1, every ε > 0, and every plane H, μ(B(x,r) ∩ H(εr)) ≤ C ε^α μ(B(x,r)). A standard fact is that if μ is doubling with support E = supp μ, then μ is absolutely decaying for some (C,α) if and only if E is hyperplane diffuse. In particular, an Ahlfors t-regular measure supported on a hyperplane-diffuse set is absolutely decaying. The paper uses the spread-out measure machinery. A scale r is a β-singular scale for (Θ,θ) if there exists an affine hyperplane H ⊂ ℝᵏ with Θ ∩ B(θ,r) ⊂ H(βr). For Θ ⊂ ℝᵏ, θ ∈ Θ, β ∈ (0,1], and δ ≤ Δ, the singular scale count is sₖ(θ | Θ, β, δ → Δ) = #{r ∈ 2^(−ℕ) ∩ [δ,Δ] : r is β-singular for (Θ,θ)}, and sₖ(θ | Θ, β, δ) = sₖ(θ | Θ, β, δ → 1). A finite nonzero Borel measure ν on ℝᵏ is (δ,β,C)ₖ-spread-out if for every s ∈ ℕ ∪ {0} and every Borel E ⊂ supp ν with sₖ(x | E, β, δ → 1) ≥ s for every x ∈ E, one has ν(E) ≤ C · 2^(−βs). A Borel measure ν on G(d+1,1) is spread-out if, after restriction to any big-cell chart and push-forward to ℝᵈ, it is (δ,β,C)_d-spread-out for some β > 0 and C ≥ 1, uniformly in δ ∈ 2^(−ℕ). The branching function is central to the paper. For δ = 2^(−n) and a finite set P ⊂ ℝ, define N_δ(P) = #{I ∈ 𝒟_δ : I ∩ P ≠ ∅}, where 𝒟_δ are the dyadic intervals of length δ. The branching function is b_P(n) = log₂ N_{2^(−n)}(P). For a measure μ supported on E, b_μ(n) is the branching function of the support at scale 2^(−n), restricted to a set of μ-measure at least 1/2; under Ahlfors regularity this agrees with the full branching function up to absolute constants. For x = (p,h) ∈ E and δ = 2^(−n), the paper writes bₓ(n) = b_{Dₓ(E)}(n) for the branching function of the pinned squared distance set Dₓ(E), and b_p⁰(n) = b_{D_p⁰(A)}(n) for the branching function of the planar squared distance set D_p⁰(A) = {‖p − a‖² : a ∈ A}, where A = π_H(E). Local stickiness is defined at a scale. Fix c ∈ (0,1) and δ = 2^(−n). The configuration E_δ is locally c-sticky at scale δ relative to x = (p,h) ∈ E if for every planar interval J ⊂ [c₀,C₀] of length δ with |E_δ(J)| ≥ c_A δ^(1−t), where c_A = c_A(t,C_A) > 0 and E_δ(J) = {(a,u) ∈ E_δ : ‖p − a‖² ∈ J}, one has diam({(h − u)² : (a,u) ∈ E_δ, ‖p − a‖² ∈ J}) ≤ cδ. The paper notes that the threshold δ^(1+α) appearing in some formulations is replaced here by the per-interval threshold cδ for stickiness, while the threshold δ^(1−α) appears in the non-sticky density, which measures the fraction of scales at which the vertical spread W_J exceeds 2^(−k(1−α)). These two thresholds serve different purposes. The non-sticky density is defined as follows. Let α ∈ (0,1). Let S_α(x,n) ⊂ {1,2,…,n} be the set of scales k such that δₖ = 2^(−k) is not locally c-sticky at scale δₖ relative to x. Define β_α(x,n) = |S_α(x,n)| / n. For the relative branching statement, the paper defines the stronger density β_k(α) = #{J ∈ 𝒥_k : W_J ≥ 2^(−k(1−α))} / |𝒥_k|, where 𝒥_k is the set of occupied planar intervals at scale 2^(−k), and W_J = diam({(h − u)² : (a,u) ∈ E_{2^(−k)}, ‖p − a‖² ∈ J}). The paper cites several tools. The first is a non-concentration condition: if μ is a measure on ℝᵏ satisfying μ(B(θ,r) ∩ H(εr)) ≤ (1 − ε) μ(B(θ,r)) for all θ ∈ supp μ, all 0 < r ≤ r₀, and all hyperplanes H, then for every a ∈ supp μ and every ρ ∈ 2^(−ℕ) ∩ (0,r₀], the restriction μ|_{B(a,ρ)} is (δ,β,C)-spread-out for every δ ∈ 2^(−ℕ), with β = β(ε) > 0 and C = C(ε,ρ) ≥ 1 both independent of δ. The second tool is that if ν is a spread-out measure on ℝᵈ, then for every Borel E ⊂ ℝᵈ, dim_H(Δ^y(E)) = min(dim_H(E),1) for ν-a.e. y ∈ ℝᵈ. The third tool is the Iosevich–Yavicoli planar bound: for every Borel E′ ⊂ ℝ² with dim_H(E′) > 1, there exists an exceptional set Z ⊂ ℝ² with dim_H(Z) ≤ 1 such that for every p ∈ E′ \ Z, dim_H(Δ_p(E′)) ≥ q* = 0.7813… The fourth tool is a relative form of a branching proposition: under a probability measure μ on ℝᵏ, a δ-separated set P_δ with μ(P_δ) ≥ c₀, a short arc Θ ⊂ G(k+1,1) of 1-planes, projections π₀,π₁ corresponding to the endpoints, and the assumption that for each scale k a β_k-fraction of the π₀-dyadic slablets have transverse width at least 2^(−k(1−α)), the regular branching function g of the refined incidence set satisfies g(n) ≥ g₀(n) + c ∑_{k=1}^n β_k − O(εn) − O(1), where g₀ is the regular branching function of the reference projection π₀ and c = c(α,k,c₀) > 0. The paper states this relative branching proposition as a hypothesis: the proof of the thick planar theorem is conditional on its validity, while the diffuse, product and thin planar theorems, the large-scale transfer lemma, and the regular-branching-to-Frostman proposition are unconditional given the standard tools. The first main result, the diffuse case, is unconditional. It states that if E ⊂ ℝ³ is Ahlfors t-regular with t > 3/2 and E is hyperplane diffuse for some γ > 0, then for μ-a.e. x ∈ E one has dim_H(Δₓ(E)) = 1, and in particular sup_{x∈E} dim_H(Δₓ(E)) = 1. The proof has three parts. First, a quantitative non-concentration lemma: if E is Ahlfors t-regular with t > 0 and x is a point of E that is γ-diffuse at all scales r ≤ r₀, then for every 0 < r ≤ r₀ and every plane H, μ(B(x,r) ∩ H(γr/2)) ≤ (1 − c γ^t) μ(B(x,r)), where c = c(t,C_A) > 0. The proof uses the existence of y ∈ E ∩ B(x,r) with dist(y,H) ≥ γr, so that B(y,γr/4) ⊂ B(x,r) \ H(γr/2). By Ahlfors regularity, μ(B(y,γr/4)) ≥ c₁(γr/4)^t and μ(B(x,r)) ≤ C₁ r^t, hence μ(B(x,r) \ H(γr/2)) ≥ c γ^t μ(B(x,r)) with c = c₁/(4^t C₁), and subtracting gives the claim. Second, under the same hypotheses, for μ-a.e. a ∈ E and every ρ ∈ (0,r₀], the restriction μ|_{B(a,ρ)} is (δ,β,C)-spread-out for every δ ∈ 2^(−ℕ), with β = β(ε₀) > 0 and C = C(ε₀,ρ) ≥ 1 both independent of δ, where ε₀ = c γ^t. This follows by combining the quantitative non-concentration lemma with the spread-out restriction tool. Third, the diffuse theorem follows: since E is hyperplane diffuse, there exists γ > 0 such that every x ∈ E is γ-diffuse at all scales r ≤ 1. By the spread-out restriction statement, for μ-a.e. a ∈ E and every ρ ∈ (0,1] the restriction ν = μ|_{B(a,ρ)} is spread-out with constants independent of δ. One lifts ν to a measure on G(4,1) via the standard identification y ↦ −2y of ℝ³ with a big-cell chart of G(4,1), and the push-forward remains spread-out. Applying the spread-out pinned-distance theorem with d = 3 and the paraboloid identification of pinned distances gives dim_H(Δ_y(E)) = min(dim_H(E),1) = 1 for ν-a.e. y, and choosing y = x ∈ E completes the proof. The paper develops a paraboloid lifting and a tubular identity. Define Φ : ℝ³ → ℝ⁴ by Φ(y) = (y,‖y‖²). For each x ∈ ℝ³ define ξₓ = (−2x,1) ∈ ℝ⁴ and θₓ = span(ξₓ) ∈ G(4,1). With the unnormalized convention Π_θ(q) = ⟨ξ,q⟩ for a generator ξ of θ, one has Π_{θₓ}(Φ(y)) = ‖x − y‖² − ‖x‖². The shift −‖x‖² is independent of y and does not affect dimension, so Π_{θₓ}(Φ(y)) is identified with ‖x − y‖² up to translation. The planar model is defined by Φ₀(a) = (a,0,‖a‖²) ∈ ℝ⁴, η_p = span((−2p,0,1)) ∈ G(4,1), and Π_{η_p}(Φ₀(a)) = ‖p − a‖² − ‖p‖². The tubular identity states that if x = (p,h) ∈ E and y = (a,u) ∈ E with p,a ∈ H and |h|,|u| ≤ γ, then ‖x − y‖² = ‖p − a‖² + (h − u)² and |Π_{θₓ}(Φ(y)) − Π_{η_p}(Φ₀(a))| = |u² − 2hu| ≤ 3γ². If h = 0, the bound improves to |u²| ≤ γ². The proof writes x − y = (p − a) + (h − u)e₃,