For every uncountable Borel set in Euclidean space, there exists a compact Kakeya set of zero Lebesgue measure whose unit segments have centers in that set.
A Szemer´ edi type theorem for sets of positive density inR k.Israel J
3 Pith papers cite this work, alongside 112 external citations. Polarity classification is still indexing.
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Measurable sets in [0,R]² avoiding upward right triangles of area 1/2 satisfy |A| = O_c(R²/(log R)^c) for c<1/4 with Ω(R log R) example; for fixed-area triangles the bound sharpens to c<1/2 using a hyperbolic trilinear smoothing inequality and scale induction.
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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Some constructions of restricted Kakeya sets
For every uncountable Borel set in Euclidean space, there exists a compact Kakeya set of zero Lebesgue measure whose unit segments have centers in that set.
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On hyperbolic corners and unit-area triangles in planar sets of large measure
Measurable sets in [0,R]² avoiding upward right triangles of area 1/2 satisfy |A| = O_c(R²/(log R)^c) for c<1/4 with Ω(R log R) example; for fixed-area triangles the bound sharpens to c<1/2 using a hyperbolic trilinear smoothing inequality and scale induction.