I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2010, VOLUME 16, NUMBER 5, PAGES 3-17
Simultaneous inhomogeneous Diophantine approximation on manifolds
V. V. Beresnevich
S. L. Velani
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In 1998, Kleinbock and Margulis proved Sprindzuk's conjecture
pertaining to metrical Diophantine approximation (and indeed the
stronger Baker--Sprindzuk conjecture).
In essence, the conjecture stated that the simultaneous homogeneous
for almost every point on
a nondegenerate submanifold of .
In this paper, the simultaneous inhomogeneous analogue of Sprindzuk's
conjecture is established.
More precisely, for any "inhomogeneous" vector
we prove that the simultaneous inhomogeneous Diophantine exponent
almost every point
The key result is an inhomogeneous transference principle which
enables us to deduce that the homogeneous exponent
almost all if and only if, for
any , the inhomogeneous exponent
The inhomogeneous transference principle introduced in this paper is
an extremely simplified version of that recently discovered by us.
Nevertheless, it should be emphasised that the simplified version has
the great advantage of bringing to the forefront the main ideas while
omitting the abstract and technical notions that come with describing
the inhomogeneous transference principle in all its glory.
Last modified: May 30, 2011