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 $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side length below $p^{1/2}$, 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 $\vec{u}$,$\vec{v}$ and $\vec{w}$ in the $n$-dimensional space \F_q^n over the finite field \F_q of $q$ 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

Character sums with division polynomials

We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, ...$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least $q^{1/2 + \epsilon}$ for some fixed $\epsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences which has recently been brought up by K. Lauter and the second author

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 $p\le x$ 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

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 $$\sum_{u = 1}^{U} \sum_{v=1}^V \alpha_u \beta_v \mathbf{\,e}_p(u/f(v))$$ where $\mathbf{\,e}_p(z)$ is a nontrivial additive character of the prime finite field ${\mathbb F}_p$ of $p$ elements, with integers $U$, $V$, a polynomial $f\in {\mathbb F}_p[X]$ and some complex weights $\{\alpha_u\}$, $\{\beta_v\}$. In particular, for $f(X)=aX+b$ we obtain new bounds of bilinear sums with Kloosterman fractions. We also obtain new bounds for similar sums with multiplicative characters of ${\mathbb F}_p$

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 $$\sum_{j=1}^n a_j x_jy_j^{-1} \equiv a_0 \pmod p,$$ with variables from rather general sets