1,014 research outputs found
Double Character Sums over Subgroups and Intervals
We estimate double sums with a multiplicative character
modulo where and is a subgroup of order
of the multiplicative group of the finite field of elements. A nontrivial
upper bound on can be derived from the Burgess bound if and from some standard elementary arguments if , where is arbitrary. We obtain a
nontrivial estimate in a wider range of parameters and . We also
estimate double sums and give an application to primitive
roots modulo with non-zero binary digits
Almost all primes have a multiple of small Hamming weight
Recent results of Bourgain and Shparlinski imply that for almost all primes
there is a multiple that can be written in binary as with or ,
respectively. We show that (corresponding to Hamming weight )
suffices.
We also prove there are infinitely many primes with a multiplicative
subgroup , for some
, of size , where the sum-product set
does not cover completely
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
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
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