Do you think the following is true or not?

Let $x,y,\epsilon \in \mathbb{R^{+ \star}}$. Can I find $\tilde{\epsilon}$ such that $\tilde{\epsilon}<\epsilon$ and both $\frac{x}{\tilde{\epsilon}}$ and $\frac{y}{\tilde{\epsilon}}$ are integers?

If it's false, do you think it can be true with an additional assumption on $x,y$ (being rationnal or integer)?

I would also like to generalize to a finite familly of real numbers $(x_1,\dots,x_n)$ and $\epsilon$?

Thank you in advance!

  • $\begingroup$ What's $\mathbb{R^{+ \star}}$? $\endgroup$ – user99914 Nov 21 '15 at 10:36
  • $\begingroup$ Strictly positive real numbers. $\endgroup$ – Taylorien Nov 21 '15 at 10:45

HINT: if both $x/\tilde{\epsilon}, y/\tilde{\epsilon}$ are integers, then their quotient is a rational number. But their quotient is $x/y$...

  • $\begingroup$ Thank you! If $x,y$ are rational, do you think it is possible? $\endgroup$ – Taylorien Nov 21 '15 at 10:43
  • $\begingroup$ Of course. If $h,k$ are the denominators of $x,y$ respectively, then take $\tilde{\epsilon} = 1/(ahk)$ where $a \ge 1$ is a large enough number. $\endgroup$ – Crostul Nov 21 '15 at 10:45

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