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Abstract
Let
ξ ,
ζ
be quadratic real numbers in distinct quadratic fields. We establish
the existence of effectively computable, positive real numbers
τ and
c such that, for
every integer
q
with
q
>
c ,
we have
max { ∥ q ξ ∥ , ∥ q ζ ∥ }
> q − 1 + τ ,
where
∥ ⋅ ∥
denotes the distance to the nearest integer.
To the memory of Naum Ilich Feldman
(1918–1994)
Keywords
simultaneous approximation, Pell equation, linear form in
logarithms
Mathematical Subject Classification 2010
Primary: 11J13
Secondary: 11D09, 11J86
Milestones
Received: 29 July 2019
Revised: 9 March 2020
Accepted: 23 March 2020
Published: 5 November 2020