67 research outputs found
An Inhomogeneous Transference Principle and Diophantine Approximation
In a landmark paper, D.Y. Kleinbock and G.A. Margulis established the
fundamental Baker-Sprindzuk conjecture on homogeneous Diophantine approximation
on manifolds. Subsequently, there has been dramatic progress in this area of
research. However, the techniques developed to date do not seem to be
applicable to inhomogeneous approximation. Consequently, the theory of
inhomogeneous Diophantine approximation on manifolds remains essentially
non-existent.
In this paper we develop an approach that enables us to transfer homogeneous
statements to inhomogeneous ones. This is rather surprising as the
inhomogeneous theory contains the homogeneous theory and so is more general. As
a consequence, we establish the inhomogeneous analogue of the Baker-Sprindzuk
conjecture. Furthermore, we prove a complete inhomogeneous version of the
profound theorem of Kleinbock, Lindenstrauss & Weiss on the extremality of
friendly measures. The results obtained in this paper constitute the first step
towards developing a coherent inhomogeneous theory for manifolds in line with
the homogeneous theory.Comment: 37 pages: a final section on further developments has been adde
Explicit bounds for rational points near planar curves and metric Diophantine approximation
The primary goal of this paper is to complete the theory of metric
Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007),
p.367-426] for non-degenerate planar curves. With this goal in mind, here
for the first time we obtain fully explicit bounds for the number of rational
points near planar curves. Further, introducing a perturbational approach we
bring the smoothness condition imposed on the curves down to (lowest
possible). This way we broaden the notion of non-degeneracy in a natural
direction and introduce a new topologically complete class of planar curves to
the theory of Diophantine approximation. In summary, our findings improve and
complete the main theorems of [Ann. of Math.(2) 166 (2007), p.367-426] and
extend the celebrated theorem of Kleinbock and Margulis appeared in [Ann. of
Math.(2), 148 (1998), p.339-360] in dimension 2 beyond the notion of
non-degeneracy.Comment: 24 page
A note on zero-one laws in metrical Diophantine approximation
In this paper we discuss a general problem on metrical Diophantine
approximation associated with a system of linear forms. The main result is a
zero-one law that extends one-dimensional results of Cassels and Gallagher. The
paper contains a discussion on possible generalisations including a selection
of various open problems.Comment: 12 pages, Dedicated to Wolfgang Schmidt on the occasion of his 75th
birthda
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