New gaps on the Lagrange and Markov spectra

Let L and M denote the Lagrange and Markov spectra, respectively. It is known that L subset of M and that M \ L not equal empty set. In this work, we exhibit new gaps of L and M using two methods. First, we derive such gaps by describing a new portion of M \ L near to 3.938: this region (together with three other candidates) was found by investigating the pictures of L recently produced by V. Delecroix and the last two authors with the aid of an algorithm explained in one of the appendices to this paper. As a by-product, we also get the largest known elements of M \ L and we improve upon a lower bound on the Hausdorff dimension of M \ L obtained by the last two authors together with M. Pollicott and P. Vytnova (heuristically, we get a new lower bound of 0.593 . 593 on the dimension of M \ L ). Secondly, we use a renormalisation idea and a thickness criterion (reminiscent from the third author's PhD thesis) to detect infinitely many maximal gaps of M accumulating to Freiman's gap preceding the so-called Hall's ray [4.52782956616. . 52782956616 . .. , infinity) C L .