This is an announcement for the paper “Embeddings and Lebesgue-type inequalities for the greedy algorithm in Banach spaces” by P.M. Berná<https://arxiv.org/find/math/1/au:+Berna_P/0/1/0/all/0/1>, O. Blasco<https://arxiv.org/find/math/1/au:+Blasco_O/0/1/0/all/0/1>, G. Garrigós<https://arxiv.org/find/math/1/au:+Garrigos_G/0/1/0/all/0/1>, E. Hernández. T. Oikhberg<https://arxiv.org/find/math/1/au:+Oikhberg_E/0/1/0/all/0/1>. Abstract: We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary complete seminormalized biorthogonal systems in Banach spaces. The bounds are given only in terms of the upper democracy functions of the basis and its dual. We also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces. Finally, the asymptotic optimality of these inequalities is illustrated in various examples of non necessarily quasi-greedy bases. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.07513