127,351 research outputs found
An improved sum-product estimate for general finite fields
This paper improves on a sum-product estimate obtained by Katz and Shen for
subsets of a finite field whose order is not prime
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
A sum-product theorem in function fields
Let be a finite subset of \ffield, the field of Laurent series in
over a finite field . We show that for any there
exists a constant dependent only on and such that
. In particular such a result is
obtained for the rational function field . Identical results
are also obtained for finite subsets of the -adic field for
any prime .Comment: Simplification of argument and note that methods also work for the
p-adic
On additive shifts of multiplicative almost-subgroups in finite fields
We prove that for sets with and a fixed holds
In particular,
and
The first estimate improves the bound by Roche-Newton and Jones.
In the general case of a field of order we obtain similar estimates
with the exponent under the condition that does not have
large intersection with any subfield coset, answering a question of
Shparlinski.
Finally, we prove the estimate
for Gauss sums over , where is a non-trivial additive
character and . The estimate gives an improvement over
the classical Weil bound when
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
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
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