357,385 research outputs found
On a generalization of distance sets
A subset in the -dimensional Euclidean space is called a -distance
set if there are exactly distinct distances between two distinct points in
and a subset is called a locally -distance set if for any point
in , there are at most distinct distances between and other points
in .
Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the
cardinalities of -distance sets on a sphere in 1977. In the same way, we are
able to give the same bound for locally -distance sets on a sphere. In the
first part of this paper, we prove that if is a locally -distance set
attaining the Fisher type upper bound, then determining a weight function ,
is a tight weighted spherical -design. This result implies that
locally -distance sets attaining the Fisher type upper bound are
-distance sets. In the second part, we give a new absolute bound for the
cardinalities of -distance sets on a sphere. This upper bound is useful for
-distance sets for which the linear programming bound is not applicable. In
the third part, we discuss about locally two-distance sets in Euclidean spaces.
We give an upper bound for the cardinalities of locally two-distance sets in
Euclidean spaces. Moreover, we prove that the existence of a spherical
two-distance set in -space which attains the Fisher type upper bound is
equivalent to the existence of a locally two-distance set but not a
two-distance set in -space with more than points. We also
classify optimal (largest possible) locally two-distance sets for dimensions
less than eight. In addition, we determine the maximum cardinalities of locally
two-distance sets on a sphere for dimensions less than forty.Comment: 17 pages, 1 figur
Bisector energy and few distinct distances
We introduce the bisector energy of an -point set in ,
defined as the number of quadruples from such that and
determine the same perpendicular bisector as and . If no line or circle
contains points of , then we prove that the bisector energy is
. We also prove the
lower bound , which matches our upper bound when is
large. We use our upper bound on the bisector energy to obtain two rather
different results:
(i) If determines distinct distances, then for any
, either there exists a line or circle that contains
points of , or there exist
distinct lines that contain points of . This result
provides new information on a conjecture of Erd\H{o}s regarding the structure
of point sets with few distinct distances.
(ii) If no line or circle contains points of , then the number of
distinct perpendicular bisectors determined by is
. This appears to
be the first higher-dimensional example in a framework for studying the
expansion properties of polynomials and rational functions over ,
initiated by Elekes and R\'onyai.Comment: 18 pages, 2 figure
Partial-Matching RMS Distance Under Translation: Combinatorics and Algorithms
We consider the problem of minimizing the RMS distance (sum of squared distances between pairs of points) under translation between two point sets A and B, in the plane, with (Formula presented.), in the partial-matching setup, in which each point in B is matched to a distinct point in A. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision (Formula presented.) of the plane and derive improved bounds on its complexity. Specifically, we show that this complexity is (Formula presented.), so it is only quadratic in |A|. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time. © 2017 Springer Science+Business Media New Yor
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
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