Heriot-Watt Mathematics Report Series
HWM00-2, 25 Feb 2000
Simultaneous Diophantine approximation on manifolds and Hausdorff dimension
B P Rynne
Abstract
We show that given any $m$-dimensional $C^k$ manifold in $R^n$
there are manifolds $M_p$, $M_z$ arbitrarily $C^k$ close to $M$
such that for all sufficiently large $\tau$
$$
\dim S_{\tau}(M_z) = 0, \quad \dim S_{\tau}(M_p) > 0,
$$
where
$$
S_{\tau}(M) := \{ x} \in M : \|q {\br x} \| < q^{-\tau}\
\text{for infinitely many $q \in N$ \}.
$$
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Publication
B P Rynne
, Simultaneous Diophantine approximation on manifolds and Hausdorff dimension, J Number Theory (accepted), 98, 1-9, (2003).
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