91 research outputs found

    Direction problems in affine spaces

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    This paper is a survey paper on old and recent results on direction problems in finite dimensional affine spaces over a finite field.Comment: Academy Contact Forum "Galois geometries and applications", October 5, 2012, Brussels, Belgiu

    Minimal symmetric differences of lines in projective planes

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    Let q be an odd prime power and let f(r) be the minimum size of the symmetric difference of r lines in the Desarguesian projective plane PG(2,q). We prove some results about the function f(r), in particular showing that there exists a constant C>0 such that f(r)=O(q) for Cq^{3/2}<r<q^2 - Cq^{3/2}.Comment: 16 pages + 2 pages of tables. This is a slightly revised version of the previous one (Thm 6 has been improved, and a few points explained

    A note on large Kakeya sets

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    A Kakeya set K\mathcal{K} in an affine plane of order qq is the point set covered by a set L\mathcal{L} of q+1q+1 pairwise non-parallel lines. Large Kakeya sets were studied by Dover and Mellinger; in [6] they showed that Kakeya sets with size at least q2−3q+9q^2-3q+9 contain a large knot (a point of K\mathcal{K} lying on many lines of L\mathcal{L}). In this paper, we improve on this result by showing that Kakeya set of size at least ≈q2−qq+32q\approx q^2-q\sqrt{q}+\frac{3}{2}q contain a large knot. Furthermore, we obtain a sharp result for planes of square order containing a Baer subplane.Comment: To appear in Advances in Geometr

    On the restriction problem for discrete paraboloid in lower dimension

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    We apply geometric incidence estimates in positive characteristic to prove the optimal L2→L3L^2 \to L^3 Fourier extension estimate for the paraboloid in the four-dimensional vector space over a prime residue field. In three dimensions, when −1-1 is not a square, we prove an L2→L329L^2 \to L^{\frac{32}{9} } extension estimate, improving the previously known exponent 6819.\frac{68}{19}.Comment: Final versio

    On the stability of sets of even type

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