TY - JOUR

T1 - On the Mattila-Sjölin theorem for distance sets

AU - Iosevich, Alex

AU - Mourgoglou, Mihalis

AU - Taylor, Krystal

PY - 2012/2

Y1 - 2012/2

N2 - We extend a result, due to Mattila and Sjölin, which says that if the Hausdorff dimension of a compact set E⊂ Rd, d ≥2, is greater than d+1/2, then the distance set Δ(E) = {|x-y|:x,y ∈ E} contains an interval. We prove this result for distance sets ΔB(E) = {||x-y||B: x,y ∈ E}, where ||·||B is the metric induced by the norm defined by a symmetric bounded convex body B with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also obtain some detailed estimates pertaining to the Radon-Nikodym derivative of the distance measure.

AB - We extend a result, due to Mattila and Sjölin, which says that if the Hausdorff dimension of a compact set E⊂ Rd, d ≥2, is greater than d+1/2, then the distance set Δ(E) = {|x-y|:x,y ∈ E} contains an interval. We prove this result for distance sets ΔB(E) = {||x-y||B: x,y ∈ E}, where ||·||B is the metric induced by the norm defined by a symmetric bounded convex body B with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also obtain some detailed estimates pertaining to the Radon-Nikodym derivative of the distance measure.

KW - Arithmetic of the lattice

KW - Bilinear operators

KW - Distribution of angles

KW - Erdos problems

KW - Falconer distance problem

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U2 - 10.5186/aasfm.2012.3732

DO - 10.5186/aasfm.2012.3732

M3 - Article

AN - SCOPUS:84905827397

VL - 37

SP - 557

EP - 562

JO - Annales Academiae Scientiarum Fennicae Mathematica

JF - Annales Academiae Scientiarum Fennicae Mathematica

SN - 1239-629X

IS - 1

ER -