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A short while ago, Dvir proved the Kakeya conjecture over finite fields. Does this have any implications for restriction theorems over finite fields? I am aware only of implications going in the opposite direction (Mockenhaupt-Tao). Is there anything new known about restriction theorems over finite fields?

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up vote 12 down vote accepted

Not directly. The key connection between restriction and Kakeya in Euclidean settings is that thanks to Taylor expansion, a surface in Euclidean space looks locally flat, and so the Fourier transform of measures on that surface are a superposition of Fourier transforms of very flat measures, which by the uncertainty principle tend to propagate along tubes. The arrangement of these tubes is then governed by the Kakeya conjecture.

In finite fields, there is no notion of Taylor expansion, unless one forces it into existence by working on the tangent bundle of a surface, rather than a surface itself. So there is a connection between restriction and Kakeya on the former object, but not the latter. (This is discussed at the end of my paper with Mockenhaupt.)

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I take that restriction on any other "natural" surface (e.g. the paraboloid) still does not follow from finite-field Kakeya? – H A Helfgott Feb 22 '10 at 11:03
Not as far as I am aware, no. – Terry Tao Feb 23 '10 at 2:17

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