  We study high-dimensional linear models and the $\ell_1$-penalized least
squares estimator, also known as the Lasso estimator. In literature, oracle
inequalities have been derived under restricted eigenvalue or compatibility
conditions. In this paper, we complement this with entropy conditions which
allow one to improve the dual norm bound, and demonstrate how this leads to new
oracle inequalities. The new oracle inequalities show that a smaller choice for
the tuning parameter and a trade-off between $\ell_1$-norms and small
compatibility constants are possible. This implies, in particular for
correlated design, improved bounds for the prediction error of the Lasso
estimator as compared to the methods based on restricted eigenvalue or
compatibility conditions only.
