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Abstract
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We prove over fields of power series the analogues of several Diophantine
approximation results obtained over the field of real numbers. In particular we
establish the power series analogue of Kronecker’s theorem for matrices,
together with a quantitative form of it, which can also be seen as a transference
inequality between uniform approximation and inhomogeneous approximation.
Special attention is devoted to the one-dimensional case. Namely, we
give a necessary and sufficient condition on an irrational power series
which ensures that,
for some positive
,
the set
has full Hausdorff dimension.
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Keywords
Diophantine approximation, power series field, exponent of
homogeneous approximation, exponent of inhomogeneous
approximation, Hausdorff dimension
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Mathematical Subject Classification 2010
Primary: 11K55
Secondary: 11J04, 28A80
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Milestones
Received: 10 January 2019
Revised: 28 March 2019
Accepted: 1 April 2019
Published: 27 November 2019
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