New explicit-in-dimension estimates for the cardinality of high-dimensional hyperbolic crosses and approximation of functions having mixed smoothness

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 $\epsilon$-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 $\epsilon ,s,a$, respectively, when $N,s$ may be large, $\epsilon$ 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 $\epsilon$-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.