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Let $\mu \ge 2$ be a real number and let $ℳ\left(\mu \right)$ denote the set of real numbers approximable at order at least $\mu$ by rational numbers. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of $ℳ\left(\mu \right)$ is equal to $2/\mu$. We investigate the size of the intersection of $ℳ\left(\mu \right)$ with Ahlfors regular compact subsets of the interval $[0,1]$. In particular, we propose a conjecture for the exact value of the dimension of $ℳ\left(\mu \right)$ intersected with the middle-third Cantor set and give several results...

We design algorithms of “optimal" for several natural problems about first-order queries on structures of bounded degree. For that purpose, we first introduce a framework to deal with logical or combinatorial problems whose instances may admit of several solutions . One associates to such a problem several specific tasks: compute a (for the uniform probability distribution) solution ; without repetition each solution in some specific linear order where ; compute the solution...