# Find two integers such that |m+ αn + β| < ε

An assignment I am working on (Problem 4-4 in Lee's Introduction to Smooth Manifolds) reduces to the following problem. Given $$\alpha, \beta, \varepsilon\in\mathbb{R}$$, with $$\alpha$$ irrational and $$\varepsilon > 0$$, show there exist two integers, $$m, n$$ such that $$|m - \alpha \cdot n + \beta| < \epsilon.$$

If $$\beta\in \mathbb{Z}$$, then we could simply use Dirichlet's approximation theorem, but I've not been able to prove it in general.

One thing I've tried is approximating $$\beta$$ as a rational number $$\tilde \beta = p/q \in \mathbb{Q}$$, which leads to $$|m - \alpha n + p/q | < \epsilon$$ which can be manipulated into $$|qm + p - \alpha q n | < \epsilon$$. We can then let $$\tilde m = qm + p$$ and $$\tilde n = q n$$, and apply DAT to show $$\tilde m$$ and $$\tilde n$$ exist such that $$|\tilde m + \alpha \tilde n| < q\epsilon$$. One problem with this approach, though, is that when we solve for $$m = \frac{\tilde m - p}{q}$$ we aren't guaranteed to get an integer.

• Search for Kronecker's density theorem. You only need the 1-dimensional case. The proof is pretty much the same as with Dirichlet's. You can find integers $m,n$ such that the fractional part of $m-n\alpha$ is in $(0,\epsilon)$. But then $2m-2n\alpha,3m-3n\alpha,\ldots$ make a sequence with increments less than $\epsilon$. Surely you can approximate anything in $(0,1)$ well enough with one of those. Commented Oct 25, 2019 at 5:48
• Wow, that's oddly specific to what I need. Thanks! Commented Oct 25, 2019 at 5:56
• Actually, it looks like that does not apply. Using the notation from the Wikipedia article (en.wikipedia.org/wiki/Kronecker%27s_theorem), the condition only holds if $\beta r \in \mathbb{Z}$ for some $r \in \mathbb{Z}$, which is impossible if $\beta$ is irrational. Commented Oct 25, 2019 at 6:06
• That WP-page looks very weird to me. The version of Kronecker's density theorm I know of says that if the set $1,\alpha_1,\alpha_2,\ldots,\alpha_n$ of real numbers is linearly independent over $\Bbb{Z}$ (in particular all the $\alpha_i$ are irrational), then the fractional parts of the vectors $m(\alpha_1,\ldots,\alpha_n)$ are dense in $[0,1]^n$. Commented Oct 25, 2019 at 6:18
• On Wolfram MathWorld, the theorem is stated in exactly the form I need. Thanks! mathworld.wolfram.com/KroneckersApproximationTheorem.html Commented Oct 25, 2019 at 6:34

For any $$\alpha, \beta, \epsilon \in \mathbb{R}$$, with $$\alpha$$ irrational and $$\epsilon > 0$$, then there exists integers $$m$$ and $$n$$ with $$n>0$$, such that $$|m - \alpha n + \beta| < \epsilon.$$