Comments on: A problem on diophantine approximation http://cameroncounts.wordpress.com/2012/09/10/a-problem-on-diophantine-approximation/ always busy counting, doubting every figured guess . . . Wed, 19 Jun 2013 18:15:17 +0000 hourly 1 http://wordpress.com/ By: Peter Cameron http://cameroncounts.wordpress.com/2012/09/10/a-problem-on-diophantine-approximation/#comment-6866 Tue, 11 Sep 2012 20:11:01 +0000 http://cameroncounts.wordpress.com/?p=2559#comment-6866 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

]]> By: Peter Cameron http://cameroncounts.wordpress.com/2012/09/10/a-problem-on-diophantine-approximation/#comment-6804 Mon, 10 Sep 2012 16:14:11 +0000 http://cameroncounts.wordpress.com/?p=2559#comment-6804 Aha – I found it in Chapter 23 of Hardy and Wright. Thanks for the pointer to Kronecker.

]]> By: Peter Cameron http://cameroncounts.wordpress.com/2012/09/10/a-problem-on-diophantine-approximation/#comment-6799 Mon, 10 Sep 2012 15:22:46 +0000 http://cameroncounts.wordpress.com/?p=2559#comment-6799 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?

]]> By: Stanislav http://cameroncounts.wordpress.com/2012/09/10/a-problem-on-diophantine-approximation/#comment-6798 Mon, 10 Sep 2012 15:02:36 +0000 http://cameroncounts.wordpress.com/?p=2559#comment-6798 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 $R^n/Z^n$.

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