12 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
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