Under dim_H E >1, dim_H E + dim_H F >2 and F regular (equal Hausdorff and packing dimensions), there exists y in F such that the pinned distance set Δ_y(E) has positive Lebesgue measure.
Stein.Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 ofPrinceton Mathematical Series
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Compactness of bilinear singular integral operators holds under mild kernel regularity, with the critical exponent matching the best known condition from the classical bilinear T1 theorem, plus a new weak compactness property.
Introduces joint upper Banach densities for plane sets and proves a cross-set distance realization theorem plus maximal VC dimension for families of scaled curve translates with non-vanishing curvature.
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Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates
Under dim_H E >1, dim_H E + dim_H F >2 and F regular (equal Hausdorff and packing dimensions), there exists y in F such that the pinned distance set Δ_y(E) has positive Lebesgue measure.
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Compactness of bilinear singular integral with mild kernel regularity
Compactness of bilinear singular integral operators holds under mild kernel regularity, with the critical exponent matching the best known condition from the classical bilinear T1 theorem, plus a new weak compactness property.
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Joint upper Banach density, VC dimensions and Euclidean point configurations
Introduces joint upper Banach densities for plane sets and proves a cross-set distance realization theorem plus maximal VC dimension for families of scaled curve translates with non-vanishing curvature.