Dickson's History of the Theory of Numbers, vol II p. 94 refers to some theorems of Kronecker on linear forms:

...find integers $w$ and $w^\prime$ such that $aw+a^\prime w^\prime$ takes a value as near as possible to $\xi$, where $a$, $a^\prime$ and $\xi$ are given real numbers. In general, consider a system of $p$ equations $$ > a_{i,1}w_1+\dots+a_{i,q}w_q=\xi_i\qquad(i=1,\dots,p), > $$ with real coefficients...

Dickson cites Kronecker's *Werke*, which I have, but my German is poor. Hardy and Wright has a chapter on 'Kronecker's Theorem', but not in this generality. I'm actually interested in the case $p=1$ and $\xi_1=1$; one equation with $q$ integer variables.

Can anyone suggest a modern reference in English?