Heriot-Watt Mathematics Report Series
HWM98-69, 20 Jan 1999
Hausdorff dimension and a generalized form of simultaneous Diophantine approximation
B P Rynne and H Dickinson
Abstract
Suppose that $m$ and $n$ are positive integers,
$\btau = (\tau_1,\dots,\tau_m) \in \R_+^m$
is a vector of strictly positive numbers,
and $Q\subset\Z^n$ is an infinite set of integer vectors.
Let $X$ denote a general point in $\R^{mn}$, which we will write in the form
$X=(\x_{1},\dots, \x_{m})$, with $\x_{i} \in \R^n$, $i=1,\dots,m,$
and let $W_Q(m,n;\btau)$ be the set
$$
\{ X \in \R^{mn} :
\|\x_{i} . \q \| < |\q|^{-\tau_i} , \ 1 \le i \le m,\
\mbox{for infinitely many } \q \in Q \}.
$$
In this paper we obtain the Hausdorff dimension of this set.
We also consider a generalization of the set $W_Q(m,n;\btau)$,
where the error terms $|\q|^{-\tau_i}$ in the inequalities are replaced by $\psi_i(\q)$, for general functions $\psi_i$ satisfying a certain
condition `at infinity'.
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Full text: http://www.ma.hw.ac.uk/~bryan/ps_files/gnm.pdf
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