Hausdorff dimension of the multiplicative golden mean shift

We compute the Hausdorff dimension of the "multiplicative golden mean shift" defined as the set of all reals in $[0,1]$ whose binary expansion $(x_k)$ satisfies $x_k x_{2k}=0$ for all $k\ge 1$, and show that it is smaller than the Minkowski dimension.Comment: 5 pages, to appear in Comptes Rendus Mathematique; minor errors correcte

On Veech's proof of Sarnak's theorem on the M\"{o}bius flow

We present Veech's proof of Sarnak's theorem on the M\"{o}bius flow which say that there is a unique admissible measure on the M\"{o}bius flow. As a consequence, we obtain that Sarnak's conjecture is equivalent to Chowla conjecture with the help of Tao's logarithmic Theorem which assert that the logarithmic Sarnak conjecture is equivalent to logaritmic Chowla conjecture, furthermore, if the even logarithmic Sarnak's conjecture is true then there is a subsequence with logarithmic density one along which Chowla conjecture holds, that is, the M\"{o}bius function is quasi-generic.Comment: 11 pages. Some misprints are corrected, and some details about the proof of the main result is added. We further add :"W. Veech in his letter indicated to me that there is only four persons in the world who has a copy of his notes including me.