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Let $\tau,\sigma\in(0,\infty)$ with $\frac{\tau}{\sigma}\notin\mathbb Q$. A common version of Kronecker's Approximation Theorem is the following:

For each $x\in \mathbb R$ and $\epsilon>0$, there are $m,n\in\mathbb N$ such that $|x+n\tau-m\sigma|<\epsilon$.

This answer to a recent question of mine provides the following variant, which gives an $\epsilon$-dependent bound for $n$:

For each $\epsilon>0$, there is some $N(\epsilon)\in\Bbb N$ such that for all $x\in \mathbb [0,\infty)$ there are $m\in\mathbb N$ and $n\in\{0,\ldots,N(\epsilon)\}$ such that $|x+n\tau-m\sigma|<\epsilon$.

The linked answer contains a (simple and correct) proof, so I have absolutely no questions about that.

My question is the following: Does anyone know of a citable reference (book) which contains this statement?

Thanks a lot in advance.

Mars Plastic
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This result can be found in Modular Functions and Dirichlet Series in Number Theory by Tom M. Apostol in section 7.4. If I recall correctly, the stated result in the text is slightly more general than the result you put forth, but nevertheless I think your statement of the theorem follows immediately from it. In section 7.5, there is another result generalizing the theorem to multiple dimensions.

  • Sections 7.4 and 7.5 give common versions of Kronecker's Approximation Theorem, but none of these include the bound on $n$ that I am specifically looking for. I know how to derive "my" version from the common ones. I am merely looking for a reference that already states "my" version explicitly. – Mars Plastic Jun 11 '19 at 17:11