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Restriction and Kakeya phenomena for finite fields. (English) Zbl 1072.42007

A Besicovitch set (or Kakeya set) is a set which contains a unit line segment in each direction. The Kakeya conjecture is: does a Besicovitch set in \(\mathbb R^n\) must have Hausdorff dimension \(n\)? A typical restriction theorem is the Tomas-Stein inequality \[ \| \mathcal F^{-1}(fd\sigma)\| _{L^2(n+1)/(n-1)}(\mathbb R^n) \leq C_n\| f\| _{L^2(S^{n-1})}, \] where \(f\) is an \(L^2\)-function on the sphere \(S^{n-1}\) and \(d\sigma\) is a surface measure on the sphere. These restriction and Kakeya phenomena in the Euclidean space have been intensively studied. The authors investigate these restriction and Kakeya phenomena in the vector spaces \(F^n\) over a finite field \(F\) of characteristic greater than \(2\). They mention that the finite field case serves as a good model for the Euclidean case in that many of the technical difficulties ( such as small angle issues, small separation issues, etc.) are eliminated, although certain tools in the Euclidean space are not available (e.g., Taylor series approximation, combinatorial arguments based on the ordering of \(\mathbb R\)).

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
11T24 Other character sums and Gauss sums
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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