  We consider the high-dimensional linear regression model and assume that a
fraction of the measurements are altered by an adversary with complete
knowledge of the data and the underlying distribution. We are interested in a
scenario where dense additive noise is heavy-tailed while the measurement
vectors follow a sub-Gaussian distribution. Within this framework, we establish
minimax lower bounds for the performance of an arbitrary estimator that depend
on the the fraction of corrupted observations as well as the tail behavior of
the additive noise. Moreover, we design a modification of the so-called
Square-Root Slope estimator with several desirable features: (a) it is provably
robust to adversarial contamination, and satisfies performance guarantees in
the form of sub-Gaussian deviation inequalities that match the lower error
bounds, up to logarithmic factors; (b) it is fully adaptive with respect to the
unknown sparsity level and the variance of the additive noise, and (c) it is
computationally tractable as a solution of a convex optimization problem. To
analyze performance of the proposed estimator, we prove several properties of
matrices with sub-Gaussian rows that may be of independent interest.
