  A current strand of research in high-dimensional statistics deals with
robustifying the available methodology with respect to deviations from the
pervasive light-tail assumptions. In this paper we consider a linear mean
regression model with random design and potentially heteroscedastic,
heavy-tailed errors, and investigate support estimation in this framework. We
use a strictly convex, smooth variant of the Huber loss function with tuning
parameter depending on the parameters of the problem, as well as the adaptive
LASSO penalty for computational efficiency. For the resulting estimator we show
sign-consistency and optimal rates of convergence in the $\ell_\infty$ norm as
in the homoscedastic, light-tailed setting. In our analysis, we have to deal
with the issue that the support of the target parameter in the linear mean
regression model and its robustified version may differ substantially even for
small values of the tuning parameter of the Huber loss function. Simulations
illustrate the favorable numerical performance of the proposed methodology.
