  We are aiming at sharp and explicit-in-dimension estimations of the
cardinality of $s$-dimensional hyperbolic crosses where $s$ may be large, and
applications in high-dimensional approximations of functions having mixed
smoothness. In particular, we provide new tight and explicit-in-dimension upper
and lower bounds for the cardinality of hyperbolic crosses. We apply them to
obtain explicit upper and lower bounds for Kolmogorov $N$-widths and
$\varepsilon$-dimensions of a modified Korobov class parametrized by positive
$a$ of $s$-variate periodic functions having mixed smoothness $r$, as a
function of three variables $N,s,a$ and $\varepsilon, s,a$, respectively, when
$N,s$ may be large, $\varepsilon$ may be small and $a$ may range from 0 to
infinity. Based on these results we describe a complete classification of
tractability for the problem of $\varepsilon$-dimensions of the modified
Korobov class. In particular, we prove the introduced exponential tractability
of this problem for $a>1$. All of these methods and results are also extended
to high-dimensional approximations of non-periodic functions by Jacobi
polynomials with powers in hyperbolic crosses.
