726 research outputs found
On Solutions to Some Polynomial Congruences in Small Boxes
We use bounds of mixed character sum to study the distribution of solutions
to certain polynomial systems of congruences modulo a prime . In particular,
we obtain nontrivial results about the number of solution in boxes with the
side length below , which seems to be the limit of more general
methods based on the bounds of exponential sums along varieties
On Point Sets in Vector Spaces over Finite Fields That Determine Only Acute Angle Triangles
For three points , and in the -dimensional
space \F_q^n over the finite field \F_q of elements we give a natural
interpretation of an acute angle triangle defined by this points. We obtain an
upper bound on the size of a set \cZ such that all triples of distinct points
\vec{u}, \vec{v}, \vec{w} \in \cZ define acute angle triangles. A similar
question in the real space \cR^n dates back to P. Erd{\H o}s and has been
studied by several authors
Tate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves
Let \E/\Q be a fixed elliptic curve over \Q which does not have complex
multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and
W. Duke have obtained an asymptotic formula for the number of primes
such that the reduction of \E modulo p has a trivial Tate-Shafarevich group.
Recent results of A. C. Cojocaru and C. David lead to a better error term. We
introduce a new argument in the scheme of the proof which gives further
improvement
Linear Congruences with Ratios
We use new bounds of double exponential sums with ratios of integers from
prescribed intervals to get an asymptotic formula for the number of solutions
to congruences with
variables from rather general sets
Approximation by Several Rationals
Following T. H. Chan, we consider the problem of approximation of a given
rational fraction a/q by sums of several rational fractions a_1/q_1, ...,
a_n/q_n with smaller denominators. We show that in the special cases of n=3 and
n=4 and certain admissible ranges for the denominators q_1,..., q_n, one can
improve a result of T. H. Chan by using a different approach
On Bilinear Exponential and Character Sums with Reciprocals of Polynomials
We give nontrivial bounds for the bilinear sums where
is a nontrivial additive character of the prime finite
field of elements, with integers , , a polynomial
and some complex weights ,
. In particular, for we obtain new bounds of bilinear
sums with Kloosterman fractions. We also obtain new bounds for similar sums
with multiplicative characters of
On Some Dynamical Systems in Finite Fields and Residue Rings
We use character sums to confirm several recent conjectures of V. I. Arnold
on the uniformity of distribution properties of a certain dynamical system in a
finite field. On the other hand, we show that some conjectures are wrong. We
also analyze several other conjectures of V. I. Arnold related to the orbit
length of similar dynamical systems in residue rings and outline possible ways
to prove them. We also show that some of them require further tuning
On the Restricted Divisor Function in Arithmetic Progressions
We obtain several asymptotic estimates for the sums of the restricted divisor
function \tau_{M,N}(k) = #\{1 \le m \le M, \ 1\le n \le N: mn = k\} over
short arithmetic progressions, which improve some results of J. Truelsen. Such
estimates are motivated by the links with the pair correlation problem for
fractional parts of the quadratic function , with a
real
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