This is an announcement for the paper "Lebesgue type inequalities for quasi-greedy bases" by Gustavo Garrigos, Eugenio Hernandez, and Timur Oikhberg. Abstract: We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N- term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C logN. We show with two examples that this bound is attained for quasi-greedy democratic bases. Archive classification: math.FA Mathematics Subject Classification: 41A65, 41A46, 41A17 Report Number: 01 Remarks: 19 pages Submitted from: eugenio.hernandez@uam.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.0946 or http://arXiv.org/abs/1207.0946