12 research outputs found
Detecting all regular polygons in a point set
In this paper, we analyze the time complexity of finding regular polygons in
a set of n points. We combine two different approaches to find regular
polygons, depending on their number of edges. Our result depends on the
parameter alpha, which has been used to bound the maximum number of isosceles
triangles that can be formed by n points. This bound has been expressed as
O(n^{2+2alpha+epsilon}), and the current best value for alpha is ~0.068.
Our algorithm finds polygons with O(n^alpha) edges by sweeping a line through
the set of points, while larger polygons are found by random sampling. We can
find all regular polygons with high probability in O(n^{2+alpha+epsilon})
expected time for every positive epsilon. This compares well to the
O(n^{2+2alpha+epsilon}) deterministic algorithm of Brass.Comment: 11 pages, 4 figure
Incidences between points and generalized spheres over finite fields and related problems
Let be a finite field of elements where is a large odd
prime power and , where , , and for all . A -sphere is a set of the form , where . We prove bounds on the number of incidences between a point set
and a -sphere set , denoted by
, as the following.
We prove this estimate by studying the spectra of directed graphs. We also
give a version of this estimate over finite rings where is
an odd integer. As a consequence of the above bounds, we give an estimate for
the pinned distance problem. In Sections and , we prove a bound on the
number of incidences between a random point set and a random -sphere set in
. We also study the finite field analogues of some
combinatorial geometry problems, namely, the number of generalized isosceles
triangles, and the existence of a large subset without repeated generalized
distances.Comment: to appear in Forum Mat
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