47,153 research outputs found
A new sum-product estimate in prime fields
In this paper we obtain a new sum-product estimate in prime fields. In
particular, we show that if satisfies then Our argument
builds on and improves some recent results of Shakan and Shkredov which use the
eigenvalue method to reduce to estimating a fourth moment energy and the
additive energy of some subset . Our main novelty
comes from reducing the estimation of to a point-plane incidence bound
of Rudnev rather than a point line incidence bound of Stevens and de Zeeuw as
done by Shakan and Shkredov.Comment: 16 page
New sum-product type estimates over finite fields
Let be a field with positive odd characteristic . We prove a variety
of new sum-product type estimates over . They are derived from the theorem
that the number of incidences between points and planes in the
projective three-space , with , is where denotes the maximum number of collinear planes.
The main result is a significant improvement of the state-of-the-art
sum-product inequality over fields with positive characteristic, namely that
\begin{equation}\label{mres} |A\pm A|+|A\cdot A| =\Omega
\left(|A|^{1+\frac{1}{5}}\right), \end{equation} for any such that
Comment: This is a revised version: Theorem 1 was incorrect as stated. We give
its correct statement; this does not seriously affect the main arguments
throughout the paper. Also added is a seres of remarks, placing the result in
the context of the current state of the ar
Direction problems in affine spaces
This paper is a survey paper on old and recent results on direction problems
in finite dimensional affine spaces over a finite field.Comment: Academy Contact Forum "Galois geometries and applications", October
5, 2012, Brussels, Belgiu
Four-variable expanders over the prime fields
Let be a prime field of order , and be a set in
with very small size in terms of . In this note, we show that
the number of distinct cubic distances determined by points in
satisfies which improves a result due to
Yazici, Murphy, Rudnev, and Shkredov. In addition, we investigate some new
families of expanders in four and five variables.
We also give an explicit exponent of a problem of Bukh and Tsimerman, namely,
we prove that
where is a quadratic polynomial in that is not
of the form for some univariate polynomial .Comment: Accepted in PAMS, 201
An Improved Point-Line Incidence Bound Over Arbitrary Fields
We prove a new upper bound for the number of incidences between points and
lines in a plane over an arbitrary field , a problem first
considered by Bourgain, Katz and Tao. Specifically, we show that points and
lines in , with , determine at most
incidences (where, if has positive
characteristic , we assume ). This improves on the
previous best known bound, due to Jones. To obtain our bound, we first prove an
optimal point-line incidence bound on Cartesian products, using a reduction to
a point-plane incidence bound of Rudnev. We then cover most of the point set
with Cartesian products, and we bound the incidences on each product
separately, using the bound just mentioned. We give several applications, to
sum-product-type problems, an expander problem of Bourgain, the distinct
distance problem and Beck's theorem.Comment: 18 pages. To appear in the Bulletin of the London Mathematical
Societ
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