Additive shift is a widely used tool for estimating exponential sums and character sums. According to it, the summation variable
n
is replaced by an expression of the type
n
+
x
with the subsequent ...summation over the artificially introduced variable
x
. The transformation of a simple sum into a multiple one gives additional opportunities for obtaining a nontrivial bound for the initial sum. This technique was widely used by I.G. van der Corput, I.M. Vinogradov, D.A. Burgess, A.A. Karatsuba, and many other researchers. It became a useful tool for dealing with character sums in finite fields and with multiple exponential sums. E. Fouvry and P. Michel (1998) and J. Bourgain (2005) successfully used this shift to estimate Kloosterman sums. Fouvry and Michel combined additive shift with profound results from algebraic geometry. On the contrary, the method of J. Bourgain is completely elementary. For example, it allowed the author to give an elementary proof of an estimate for the Kloosterman sum modulo a prime
q
with primes in the case when its length
N
exceeds
. In this paper, we give some new elementary applications of additive shift to weighted Kloosterman sums of the type
where
q
is a prime and the weight function
is equal to
, which is the number of divisors of
n
, or to
, which is the number of representations of
n
by a sum of two squares of integers. The bounds obtained for these sums are nontrivial for
. As a corollary of such bounds, we obtain some new results concerning the distribution of fractional parts of the type
where
u
and
are integers running through hyperbolic (
) and circular (
) domains, respectively.
This survey is an extended version of the mini-course read by the author in November 2015 during the Chinese–Russian workshop on exponential sums and sumsets. This workshop was organized by Professor ...Chaohua Jia (Institute of Mathematics, Academia Sinica) and Professor Ke Gong (Henan University) at the Academy of Mathematics and System Science, CAS (Beijing). The author is warmly grateful to them for support and hospitality. The survey consists of the Introduction, three sections, and the Conclusion. The basic definitions and results concerning complete Kloosterman sums are given in the Introduction. A method for estimating incomplete Kloosterman sums modulo a growing power of a fixed prime is described in Section 1. This method is based on an idea of A.G. Postnikov, according to which the estimation of such sums reduces to estimating exponential sums with a polynomial in the exponent by applying I.M. Vinogradov’s mean value theorem. A.A. Karatsuba’s method for estimating incomplete sums to an arbitrary modulus is described in Section 2. This method is based on a fairly accurate estimate for the number of solutions of a symmetric congruence involving inverse residues to a given modulus. This estimate plays the same role in the problems under consideration as Vinogradov’s mean value theorem in estimating corresponding exponential sums. The method of J. Bourgain and M.Z. Garaev is described in Section 3. This method is based on a profound sum-product estimate and on an improvement of Karatsuba’s bound for the number of solutions of a symmetric congruence. The Conclusion contains a number of recent results concerning estimates of short Kloosterman sums without proofs.
On Nonlinear Kloosterman Sums Korolev, M. A.
Doklady. Mathematics,
12/2022, Letnik:
106, Številka:
Suppl 2
Journal Article
Recenzirano
Exponential sums of a special type—the so-called Kloosterman sums—play a key role in numerous number-theoretic problems concerning the distribution of inverse residues in residue rings modulo a given
...q
. In many cases, estimates of such sums are based on A. Weil’s bound for complete Kloosterman sums over primes. This bound allows one to estimate Kloosterman sums of length
for any fixed
with a power-saving factor. Weil’s bound was originally proved using methods of algebraic geometry. Later, S.A. Stepanov gave an elementary proof of this bound, but this proof was also complicated enough. The aim of this paper is to give an elementary proof of an estimate for the Kloosterman sum of length
, which also leads to a power-saving factor. This proof is based on the trick of additive shift of the summation variable, which is widely used in various number theory problems.
The problem of the solvability of the congruence
in primes
,
,
, is addressed. Here
,
is the inverse of the residue
, i.e.,
,
, and
,
,
, and
are arbitrary integers with
. The analysis of this ...congruence is based on new estimates of the Kloosterman sums with primes. The main result of the study is an asymptotic formula for the number of solutions in the case when the modulus
is divisible by neither
nor
.
The article reflects main problems and opinions of social workers about possibilities of improving medical social care of population and providing social medical services. This group of specialists ...is forced to implement significant amount of work that is not regulated by their official duties. The lack of necessary competencies does not allow social workers to provide medical social assistance to consumers of medical social services to sufficient extent. To ensure interaction between medical and social workers largely falls on shoulders of the latter ones. They have to undertake a number of medical functions (examination, temporary stop of bleeding, blood pressure measurement, etc.). It conceives appropriate to transfer some of functions of junior and paramedical staff to social workers. The study results can be applied as methodological basis of increasing accessibility of medical social care and provided social and medical services to various groups of population.
A new estimate of the Kloosterman sum with primes modulo a prime number
q
is obtained, in which the number of summands can be of order
q
0.5
+
ε
. This estimate refines results obtained earlier by J. ...Bourgain (2005) and R. Baker (2012).
A new elementary proof of an estimate for incomplete Kloosterman sums modulo a prime
q
is obtained. Along with Bourgain’s 2005 estimate of the double Kloosterman sum of a special form, it leads to an ...elementary derivation of an estimate for Kloosterman sums with primes for the case in which the length of the sum is of order
q
0.5+
ε
, where
ε
is an arbitrarily small fixed number.
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