## A problem on diophantine approximation

Let n be a positive integer, and c a vector in Rn 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.

I count all the things that need to be counted.
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### 4 Responses to A problem on diophantine approximation

1. Stanislav says:

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.