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
SN - 1239-629X
VL - 37
SP - 557
EP - 562
JO - Annales Academiae Scientiarum Fennicae Mathematica
JF - Annales Academiae Scientiarum Fennicae Mathematica
IS - 1
ER -