Let *n* be a positive integer, and *c* a vector in **R**^{n} whose components are linearly independent over **Q**. Is it true that any line with direction vector *c* passes arbitrarily close to a point with integer coordinates?

This is true for *n* = 2, by the simplest result about approximating irrationals by rationals. Maybe it is too much to ask for it to hold in general; if so, I’d like a specific example and a specific counterexample.

This problem is connected to a question about homogeneous Cayley objects; I will explain the connection in the series on this topic.

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## About Peter Cameron

I count all the things that need to be counted.

It’s true. Follows from the so-called Kronecker thm. Effectively, it says that if $x_1,..,x_n$ are l.i. over rationals, then the subsemigroup of $R^n/Z^n$ generated by $x=(x_1,…,x_n)$ mod $Z^n$ is dense in the n-dim. torus $R^n/Z^n$.

This can’t be quite right – if n=1 then x_1 has to be irrational for the theorem to hold.

Also, my rather casual search suggests that Kronecker only proved the 1-dimensional case.

Is there a reference to the general case?

Aha – I found it in Chapter 23 of Hardy and Wright. Thanks for the pointer to Kronecker.

Oliver Jenkinson gave me a reference from the dynamical systems viewpoint: see http://books.google.co.uk/books?id=9nL7ZX8Djp4C&pg=PA29&lpg=PA29&dq=%22anticipates+some+of+the+most+fruitful+methods%22&source=bl&ots=oSmhT8jIIH&sig=PSQHzCkWQl2FweMtkL7h7Qv7Q0I&hl=en#v=onepage&q=%22anticipates%20some%20of%20the%20most%20fruitful%20methods%22&f=false