63 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
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
Metric considerations concerning the mixed Littlewood Conjecture
The main goal of this note is to develop a metrical theory of Diophantine
approximation within the framework of the de Mathan-Teulie Conjecture, also
known as the `Mixed Littlewood Conjecture'. Let p be a prime. A consequence of
our main result is that, for almost every real number \alpha,
\liminf_{n\rar\infty}n(\log n)^2|n|_p\|n\alpha\|=0.Comment: 17 pages, corrected various oversights
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