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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?

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2 Answers 2

up vote 1 down vote accepted

Take a look at Cassels - "An intorduction to diophantine approximation", Theorem VI in Ch1, where the theorem that Gerry mentioned is proved. I'm guessing that it appears also in Siegel's book about the geometry of numbers, although I don't have it with me at the moment.

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See also Ch. III, Section 5 of the Cassels book (Inhomogeneus approximation; Simultaneous approximation (Kronecker's theorem)). –  duje Mar 1 '12 at 9:58
@duje: This is what I was looking for, thanks! –  Stopple Mar 1 '12 at 23:26

I'm not sure if this is what you want, but Theorem 1C on page 27 of Wolfgang Schmidt, Diophantine Approximation, says

Suppose $\alpha_1,\dots,\alpha_n$ are real numbers and that $Q\gt1$ is an integer. Then there exist integers $q_1,\dots,q_n,p$ with $$1\le\max(|q_1|,\dots,|q_n|)\lt Q^{1/n}{\rm\ and\ }|\alpha_1q_1+\dots+\alpha_nq_n-p|\le{1\over Q}$$

No proof or attribution is given, but I think it goes back to Dirichlet and can be proved by pigeonhole as the case $n=1$.

EDIT: Thanks to the comment, I have a better understanding of the question. I'll make the trivial observation that to get the linear form close to $1$, you need some condition on the $\alpha_i$. For example, if the $\alpha_i$ are all even integers, then $|\alpha_1q_1+\dots+\alpha_nq_n-1|\ge1$. Similarly if the $\alpha_i$ are all integer multiples of $\pi$. So in contrast to the case for Theorem 1C, you'll need some hypothesis on the $\alpha_i$.

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Given $\alpha_1,\ldots,\alpha_n$ real I'm looking for integers $q_1,\ldots, q_n$ such that $|\alpha_1q_1+\ldots\alpha_nq_n-1|$ is 'small', where I'm not sure what can be achieved. Thanks for the reference to Schmidt; I'll look there tomorrow. –  Stopple Mar 1 '12 at 1:54
Theorem 1C is a special case of Theorem 1E from the same book by Schmidt. Its proof in given on page 29 and it is atributed to Dirichlet (1842). On page 32, it is shown that the statement of Theorems 1C and 1E is valid for all Q > 1 (not necessary integers). –  duje Mar 1 '12 at 9:40

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