15,242 research outputs found
Distinct Distance Estimates and Low Degree Polynomial Partitioning
We give a shorter proof of a slightly weaker version of a theorem from Guth and Katz (Ann Math 181:155–190, 2015): we prove that if L is a set of L lines in R[superscript 3] with at most L[superscript 1/2] lines in any low degree algebraic surface, then the number of r-rich points of is L is ≲ L[superscript (3/2) + ε] r[superscript -2]. This result is one of the main ingredients in the proof of the distinct distance estimate in Guth and Katz (2015). With our slightly weaker theorem, we get a slightly weaker distinct distance estimate: any set of N points in R[superscript 2] c[subscript ε]N[superscript 1-ε] distinct distances
A restriction estimate using polynomial partitioning
If is a smooth compact surface in with strictly positive
second fundamental form, and is the corresponding extension operator,
then we prove that for all , . The proof uses polynomial partitioning arguments
from incidence geometry.Comment: 42 pages. Minor revisions. Accepted for publication in JAM
The flecnode polynomial: a central object in incidence geometry
We give a brief exposition of the proof of the Cayley-Salmon theorem and its
recent role in incidence geometry. Even when we don't use the properties of
ruled surfaces explicitly, the regime in which we have interesting results in
point-line incidence problems often coincides with the regime in which lines
are organized into ruled surfaces.Comment: 12 pages. An expository note submitted to ICM proceeding
Incidences between points and lines in three dimensions
We give a fairly elementary and simple proof that shows that the number of
incidences between points and lines in , so that no
plane contains more than lines, is (in the precise statement, the constant
of proportionality of the first and third terms depends, in a rather weak
manner, on the relation between and ).
This bound, originally obtained by Guth and Katz~\cite{GK2} as a major step
in their solution of Erd{\H o}s's distinct distances problem, is also a major
new result in incidence geometry, an area that has picked up considerable
momentum in the past six years. Its original proof uses fairly involved
machinery from algebraic and differential geometry, so it is highly desirable
to simplify the proof, in the interest of better understanding the geometric
structure of the problem, and providing new tools for tackling similar
problems. This has recently been undertaken by Guth~\cite{Gu14}. The present
paper presents a different and simpler derivation, with better bounds than
those in \cite{Gu14}, and without the restrictive assumptions made there. Our
result has a potential for applications to other incidence problems in higher
dimensions
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