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There is a theorem (I can't remember its name) saying that for any irrational number $x$ and any $0<a<b<1$, there exists a positive integer $n$ such that $\{nx\}\in (a,b)$, where $\{\cdot\}$ denotes the fractional part.

Is it true that for any irrational numbers $x,y$ and any $0<a<b<1,0<c<d<1$, there exists a positive integer $n$ such that $\{nx\}\in (a,b)$ and $\{ny\}\in (c,d)$? Is there a theorem for this, or is it simple to prove?

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  • $\begingroup$ A result stronger than the one named in your opening paragraph goes by the name Weyl equidistribution, and implies that the proportion of multiplies lying in the interval is asymptotic to the measure of the interval. As for the question, think in terms of the torus, and refer to this MO answer: mathoverflow.net/questions/18174/… $\endgroup$ – Todd Trimble Nov 30 '15 at 3:43
  • $\begingroup$ I think my earlier comment was automatically deleted because it linked to the other question, and the system assumed that it should be deleted when the question was closed as a duplicate. Anyway, to clarify, the statement here is missing a condition that $\{1,x,y\}$ is independent over the rationals. A quick counterexample is $x=y, b\lt c$. $\endgroup$ – Douglas Zare Nov 30 '15 at 6:18