39 research outputs found
Character Sums and Congruences with n!
We estimate character sums with n!, on average, and individually. These
bounds are used to derive new results about various congruences modulo a prime
p and obtain new information about the spacings between quadratic nonresidues
modulo p. In particular, we show that there exists a positive integer \max \{n_i |
i=1,... 7\}=O(p^{11/12+\epsilon}), and we find the asymptotic formula for the
number of such representations. Finally, we show that products of 4 factorials
represent ``almost all''residue classes modulo p, and that products of 3
factorials n_1!n_2!n_3! with \max\{n_1, n_2, n_3\}=O(p^{5/6+\epsilon})$ are
uniformly distributed modulo p.Comment: 20 pages. Trans. Amer. Math. Soc. (to appear
On some congruences and exponential sums
Let be a fixed small constant, be the finite
field of elements for prime . We consider additive and multiplicative
problems in that involve intervals and arbitrary sets.
Representative examples of our results are as follows. Let be an
arbitrary subset of . If
and or if and then all, but elements of can be represented in the form with and , where depends only on . Furthermore,
let be an arbitrary interval of length and be a fixed
positive integer. If then the number of solutions of the
congruence
satisfies where depends only on
and
Character sums and products of factorials modulo
In this paper, we apply character sum estimates to study products of factorials modulo p
Exponential Sums and Congruences with Factorials
We estimate the number of solutions of certain diagonal congruences involving
factorials. We use these results to bound exponential sums with products of two
factorials and also derive asymptotic formulas for the number of
solutions of various congruences with factorials. For example, we prove that
the products of two factorials with are
uniformly distributed modulo , and that any residue class modulo is
representable in the form with .Comment: 21 page
On Congruences with Products of Variables from Short Intervals and Applications
We obtain upper bounds on the number of solutions to congruences of the type
modulo a prime with variables from some short intervals. We give some
applications of our results and in particular improve several recent estimates
of J. Cilleruelo and M. Z. Garaev on exponential congruences and on
cardinalities of products of short intervals, some double character sum
estimates of J. B. Friedlander and H. Iwaniec and some results of M.-C. Chang
and A. A. Karatsuba on character sums twisted with the divisor function
Multiplicative Congruences with Variables from Short Intervals
Recently, several bounds have been obtained on the number of solutions to
congruences of the type modulo a prime with variables
from some short intervals. Here, for almost all and all and also for a
fixed and almost all , we derive stronger bounds. We also use similar
ideas to show that for almost all primes, one can always find an element of a
large order in any rather short interval
On the Hidden Shifted Power Problem
We consider the problem of recovering a hidden element of a finite field
\F_q of elements from queries to an oracle that for a given x\in \F_q
returns for a given divisor . We use some techniques from
additive combinatorics and analytic number theory that lead to more efficient
algorithms than the naive interpolation algorithm, for example, they use
substantially fewer queries to the oracle.Comment: Moubariz Garaev (who has now become a co-author) has introduced some
new ideas that have led to stronger results. Several imprecision of the
previous version have been corrected to