On non-linear Kloosterman sum

Exponential sums of a special type — so-called Kloosterman sums — play key role in the series of number-theoretic problems concerning the distribution of inverse residues in the residual rings of given modulo $q$. At the same time, in many cases, the estimates of such sums are based on A. Weil's bound of so-called complete Kloosterman sum of prime modulo. This bound allows one to estimate Kloosterman sums of length $N\ge q^{0.5+\varepsilon}$ for any fixed $\varepsilon>0$ with power-saving factor. Weil's bound was proved originally by methods of algebraic geometry. Later, S. A. Stepanov gave an elementary proof of this bound, but this proof was also complete enough. The aim of this paper is to give an elementary proof of Kloosterman sum of length $N\ge q^{0.5+\varepsilon}$, which also leads to power-saving factor. This proof is based on the trick of “additive shift” of the variable of summation which is widely used in different problems of number theory.
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