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Random Constructions for Sharp Estimates of Mizohata-Takeuchi Type
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Random Constructions for Sharp Estimates of Mizohata-Takeuchi Type
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A Mizohata-Takeuchi type estimate is a type of weighted Fourier restriction estimate. Using tools from high dimensional probability, we construct a large class of weights that satisfy sharp estimates of Mizohata-Takeuchi type. One can interpret our result as saying that with high probability, a generic weight satisfies a sharp inequality of Mizohata-Takeuchi type (up to an epsilon-loss).
Forward citations
Cited by 3 Pith papers
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Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates
For Borel E,F in the plane with dim_H E > 1, dim_H E + dim_H F > 2, and F having equal Hausdorff and packing dimension, some y in F has pinned distance set Delta_y(E) of positive Lebesgue measure.
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Rectangles, triangles and Schr\"{o}dinger waves
Constructs lattice point sets with many rectangles and few isosceles triangles to produce explicit counterexamples to the Mizohata-Takeuchi conjecture for the paraboloid via transference principles.
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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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