Kakeya sets of curves

Details Kakeya sets of curves Kakeya sets of curves

2005-02-04

arxiv.org/abs/math/0502102v1

GAFA Geometric and Functional Analysis 15 (2005) 1319-1362

Mathematics

Classical Analysis and ODEs

32 pages. To appear in GAFA

Scientific paper

10.1007/s00039-005-0540-9

We investigate analogues for curves of the Kakeya problem for straight lines. These arise from H"ormander-type oscillatory integrals in the same way as the straight line case comes from the restriction and Bochner-Riesz problems. Using some of the geometric and arithmetic techniques developed for the straight line case by Bourgain, Wolff, Katz and Tao, we are able to prove positive results for families of parabolas whose coefficients satisfy certain algebraic conditions.

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