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.

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About Peter Cameron

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.

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