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

Abstract

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