21 research outputs found
Sets with few distinct distances do not have heavy lines
Let be a set of points in the plane that determines at most
distinct distances. We show that no line can contain more than points of . We also show a similar result for rectangular
distances, equivalent to distances in the Minkowski plane, where the distance
between a pair of points is the area of the axis-parallel rectangle that they
span
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
Improved Elekes-Szab\'o type estimates using proximity
We prove a new Elekes-Szab\'o type estimate on the size of the intersection
of a Cartesian product with an algebraic surface
over the reals. In particular, if are sets of real numbers and
is a trivariate polynomial, then either has a special form that encodes
additive group structure (for example ), or has cardinality . This is an improvement
over the previously bound . We also prove an asymmetric version of
our main result, which yields an Elekes-Ronyai type expanding polynomial
estimate with exponent . This has applications to questions in
combinatorial geometry related to the Erd\H{o}s distinct distances problem.
Like previous approaches to the problem, we rephrase the question as a
estimate, which can be analyzed by counting additive quadruples. The latter
problem can be recast as an incidence problem involving points and curves in
the plane. The new idea in our proof is that we use the order structure of the
reals to restrict attention to a smaller collection of proximate additive
quadruples.Comment: 7 pages, 0 figure
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