  In this paper we present results on asymptotic characteristics of
multivariate function classes in the uniform norm. Our main interest is the
approximation of functions with mixed smoothness parameter not larger than
$1/2$. Our focus will be on the behavior of the best $m$-term trigonometric
approximation as well as the decay of Kolmogorov and entropy numbers in the
uniform norm. It turns out that these quantities share a few fundamental
abstract properties like their behavior under real interpolation, such that
they can be treated simultaneously. We start with proving estimates on finite
rank convolution operators with range in a step hyperbolic cross. These results
imply bounds for the corresponding function space embeddings by a well-known
decomposition technique. The decay of Kolmogorov numbers have direct
implications for the problem of sampling recovery in $L_2$ in situations where
recent results in the literature are not applicable since the corresponding
approximation numbers are not square summable.
