FUNDAMENTALNAYA
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

Abstract

View as HTML
View as gif image

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
Diophantine exponent $w$_{0}(**x**)=1/n
for almost every point $$**x** on
a nondegenerate submanifold $M$ of $$**R**^{n}.
In this paper, the simultaneous inhomogeneous analogue of Sprindzuk's
conjecture is established.
More precisely, for any "inhomogeneous" vector $$** q ** Î **R**^{n}
we prove that the simultaneous inhomogeneous Diophantine exponent
$w$_{0}(**x**,** q **)
is $1/n$ for
almost every point $$**x**
on $M$.
The key result is an *inhomogeneous transference principle* which
enables us to deduce that the homogeneous exponent $w$_{0}(**x**)
is $1/n$ for
almost all $$**x** Î M if and only if, for
any $$** q ** Î
**R**^{n}, the inhomogeneous exponent
$w$_{0}(**x**,** q **)=1/n for almost
all $$**x** Î M.
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.

Location: http://mech.math.msu.su/~fpm/eng/k10/k105/k10501h.htm

Last modified: May 30, 2011