A generalization of Jarník’s theorem on Diophantine approximations to Ridout type numbers

Abstract:

Let s be a positive integer, $c > 1,{\mu _0}, \ldots ,{\mu _s}$ reals in [0, 1], $\sigma = \Sigma _{i = 0}^s\;{\mu _i}$, and t the number of nonzero ${\mu _i}$. Let ${\Pi _i}\;(i = 0, \ldots ,s)$ be $s + 1$ disjoint sets of primes and S the set of all $(s + 1)$-tuples of integers $({p_0}, \ldots ,{p_s})$ satisfying ${p_0} > 0,{p_i} = p_i^\ast {p’_i}$, where the $p_i^\ast$ are integers satisfying $|p_i^\ast | \leq c|{p_i}{|^{{\mu _i}}}$, and all prime factors of ${p’_i}$ are in ${\Pi _i},i = 0, \ldots ,s$. Let $\lambda > 0$ if $t = 0,\lambda > \sigma /\min (s,t)$ otherwise, ${E_\lambda }$ the set of all real s-tuples $({\alpha _1}, \ldots ,{\alpha _s})$ satisfying $|{\alpha _i} - {p_i}/{p_0}| < p_0^{ - \lambda }\;(i = 1, \ldots ,s)$ for an infinite number of $({p_0}, \ldots ,{p_s}) \in S$. The main result is that the Hausdorff dimension of ${E_\lambda }$ is $\sigma /\lambda$. Related results are obtained when also lower bounds are placed on the $p_i^\ast$. The case $s = 1$ was settled previously (Proc. London Math. Soc. 15 (1965), 458-470). The case ${\mu _i} = 1\;(i = 0, \ldots ,s)$ gives a well-known theorem of Jarník (Math. Z. 33 (1931), 505-543).