19 research outputs found
Point-curve incidences in the complex plane
We prove an incidence theorem for points and curves in the complex plane.
Given a set of points in and a set of curves with
degrees of freedom, Pach and Sharir proved that the number of point-curve
incidences is . We
establish the slightly weaker bound
on the number of incidences between points and (complex) algebraic
curves in with degrees of freedom. We combine tools from
algebraic geometry and differential geometry to prove a key technical lemma
that controls the number of complex curves that can be contained inside a real
hypersurface. This lemma may be of independent interest to other researchers
proving incidence theorems over .Comment: The proof was significantly simplified, and now relies on the
Picard-Lindelof theorem, rather than on foliation
Improved estimates on the number of unit perimeter triangles
We obtain new upper and lower bounds on the number of unit perimeter
triangles spanned by points in the plane. We also establish improved bounds in
the special case where the point set is a section of the integer grid.Comment: 8 pages, 1 figur
VC-Dimension of Hyperplanes over Finite Fields
Let be the -dimensional vector space over the finite
field with elements. For a subset and a fixed
nonzero , let , where
is the indicator function of the set . Two of the
authors, with Maxwell Sun, showed in the case that if and is sufficiently large, then the VC-dimension of
is 3. In this paper, we generalize the result to arbitrary
dimension and improve the exponent in the case .Comment: 9 pages, 1 figur
Higher Distance Energies and Expanders with Structure
We adapt the idea of higher moment energies, originally used in Additive
Combinatorics, so that it would apply to problems in Discrete Geometry. This
new approach leads to a variety of new results, such as
(i) Improved bounds for the problem of distinct distances with local
properties.
(ii) Improved bounds for problems involving expanding polynomials in
(Elekes-Ronyai type bounds) when one or two of the sets have
structure.
Higher moment energies seem to be related to additional problems in Discrete
Geometry, to lead to new elegant theory, and to raise new questions