Additive shift is a widely used tool in the estimating of exponential sums and character sums.
It bases on the replacement of the summation variable $n$ by the expression of the type $n+x$ and the summation
over artificially introduced variable $x$. The transformation of the simple sum to multiple sum gives an additional
opportunities, which allow one on obtain the non-trivial bound for the initial sum. This shift was widely used by I.G. van der Corput, I.M. Vinogradov,
D.A. Burgess, A.A. Karatsuba and many other researchers. It became very useful tool also in dealing with character sums in finite fields and with multiple exponential sums.
E. Fouvry and P. Michel (1998) and then J. Bourgain (2005)
used successfully this shift to the estimation of Kloosterman sums.
E. Fouvry and P. Michel combine additive shift with deep-lying results from algebraic geometry.
On the contrary, the method of J. Bourgain is completely elementary. For example, it allows to the author to
give elementary proof of the estimate of Kloosterman sum prime modulo $q$ with primes in the case when its length $N$
exceeds $q^{\,1/2+\varepsilon}$.
In this paper, we give some new elementary applications of additive shift to weighted Kloosterman sums of the type
$$
\sum\limits_{n\le N}f(n)\exp{\biggl(\frac{2\pi ia}{q}\,(n+b)^{*}\biggr)},\quad (ab,q)=1.
$$
Here $q$ is prime and weight function $f(n)$ is equal to $\tau(n)$, that is, the number of divisors of $n$, or equal to
$r(n)$, which is the number of representations of $n$ by the sum of two squares of integers. The bounds for these sums are non-trivial for $N\ge q^{\,2/3+\varepsilon}$.
As a corollary of such estimates, we obtain some new results concerning the distribution of
the fractional parts of the following type
$$
\biggl\{\frac{a}{q}\,(uv+b)^{*}\biggr\},\quad \biggl\{\frac{a}{q}\,(u^{2}+v^{2}+b)^{*}\biggr\},
$$
where the integers $u$, $v$ run through the hyperbolic ($uv\le N$) and circle ($u^{2}+v^{2}\le N$)
domains, consequently.
