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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: I. Borosh and A. S. Fraenkel
Journal: Trans. Amer. Math. Soc. 211 (1975), 23-38
MSC: Primary 10K15
MathSciNet review: 0376591
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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 $ \vert p_i^\ast\vert \leq c\vert{p_i}{\vert^{{\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 $ \vert{\alpha _i} - {p_i}/{p_0}\vert < 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).


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DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0376591-6
PII: S 0002-9947(1975)0376591-6
Article copyright: © Copyright 1975 American Mathematical Society