109,169 research outputs found
Bounds on sets with few distances
We derive a new estimate of the size of finite sets of points in metric
spaces with few distances. The following applications are considered:
(1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform
intersecting families of subsets;
(2) we refine the bound of Delsarte-Goethals-Seidel on the maximum size of
spherical sets with few distances;
(3) we prove a new bound on codes with few distances in the Hamming space,
improving an earlier result of Delsarte.
We also find the size of maximal binary codes and maximal constant-weight
codes of small length with 2 and 3 distances.Comment: 11 page
Bounds on three- and higher-distance sets
A finite set X in a metric space M is called an s-distance set if the set of
distances between any two distinct points of X has size s. The main problem for
s-distance sets is to determine the maximum cardinality of s-distance sets for
fixed s and M. In this paper, we improve the known upper bound for s-distance
sets in n-sphere for s=3,4. In particular, we determine the maximum
cardinalities of three-distance sets for n=7 and 21. We also give the maximum
cardinalities of s-distance sets in the Hamming space and the Johnson space for
several s and dimensions.Comment: 12 page
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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