Sets of the form $\mathscr A+\mathscr B$ and finite continued fractions

Estimates are obtained for the number of proper irreducible fractions with denominator $p$ such that an initial and an end segment of their expansion in a continued fraction have bounded partial quotients. These results are connected with an estimate of incomplete Kloosterman sums over sets of the form $\mathscr A+\mathscr B\subset\mathbb Z_p$. Results on the distribution in $\mathbb Z_p$ of the elements of sets of the form $(\mathscr A+\mathscr B)^k$ and $k\cdot(\mathscr A+\mathscr B)^{-1}$ are obtained.
Bibliography: 21 titles.