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
DOI:
https://doi.org/10.1090/S0002-9947-1975-0376591-6
MathSciNet review:
0376591
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Abstract | References | Similar Articles | Additional Information
Abstract: Let s be a positive integer, reals in [0, 1],
, and t the number of nonzero
. Let
be
disjoint sets of primes and S the set of all
-tuples of integers
satisfying
, where the
are integers satisfying
, and all prime factors of
are in
. Let
if
otherwise,
the set of all real s-tuples
satisfying
for an infinite number of
. The main result is that the Hausdorff dimension of
is
. Related results are obtained when also lower bounds are placed on the
. The case
was settled previously (Proc. London Math. Soc. 15 (1965), 458-470). The case
gives a well-known theorem of Jarník (Math. Z. 33 (1931), 505-543).
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- [6] D. Ridout, Rational approximations to algebraic numbers, Mathematika 4 (1957), 125–131. MR 0093508, https://doi.org/10.1112/S0025579300001182
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1975-0376591-6
Article copyright:
© Copyright 1975
American Mathematical Society