91 research outputs found
Direction problems in affine spaces
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
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
A Kakeya set in an affine plane of order is the point set
covered by a set of pairwise non-parallel lines. Large
Kakeya sets were studied by Dover and Mellinger; in [6] they showed that Kakeya
sets with size at least contain a large knot (a point of
lying on many lines of ). In this paper, we improve
on this result by showing that Kakeya set of size at least 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
We apply geometric incidence estimates in positive characteristic to prove
the optimal Fourier extension estimate for the paraboloid in the
four-dimensional vector space over a prime residue field. In three dimensions,
when is not a square, we prove an extension
estimate, improving the previously known exponent Comment: Final versio
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