Abstract:
In the middle of 1990th, A.A. Karatsuba invented a new and original method of estimating of short Kloosterman sums, that is, the exponential sums of the folloving type:
$$
S(m,A)=\sum_{n\in A}\exp{\biggl(\frac{an^{*}+bn}{m}\biggr)},\quad nn^*\equiv 1 \pmod m.
$$
Here $m>2$, $A$ denotes some subset of reduced residual system $\mathbb{Z}_{m}^{*}$
modulo $m$, such that its cardinality $|A|$ does not exceed arbitrary small fixed power of $m$.
In the talk, we will discuss the last results concerning such sums and the further development of Karatsuba's method.