Discrete imbedding theorems and Lebesgue constants

The order of growth of the Lebesgue constant for a “hyperbolic cross” is found:
$$L_R=\int_{T^2}\Bigl|\sum_{0<|\nu_1\nu_2|\le R_2}e^{2\pi i\nu x}\Bigr|\,dx\asymp R^{1.2},\quad R\to\infty.$$
Estimates are obtained by applying a discrete imbedding theorem. It is proved that among all convex domains in $E^2$, the square gives rise to a Lebesgue constant with the slowest growth $\ln^2R$.