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I am trying, essentially, to prove that for incommensurable numbers $a, b \in \mathbb{R}$, and some $\epsilon > 0$, there exist integers $n, m \in \mathbb{Z}$ such that $|na + mb| < \epsilon$.

There are several other problems equivalent or very closely related to this. For example, a ray with irrational slopes under the quotient map $\mathbb{R}^n \longrightarrow \mathbb{R}^n / \mathbb{Z}^n$ is dense, or the fact that $(-n, n) \subseteq Im(\sin a_1x + \dots + \sin a_nx)$ where $a_1 \dots a_n$ all incommensurable. I have seen similar questions posted in the following StackOverflow entries, but they are always solved without justification. I am trying to actually prove that these facts are true (they will mostly imply each other).

Minimum of sum of incommensurable functions

Period of the sum/product of two functions

Range of a sum of sine waves

It seems like the result is well known, so I assume I'm missing something obvious.

andynopandy
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Suppose $b$ is irrational. I will first prove the result when $a=1$. For each positive integer $m$ there exists a unique integer $n_m$ in the interval $[-mb,-mb+1)$. Let $x_m=n_m+mb$ so $x_m \in [0,1)$ for each $m$. Choose a positive integer $k$ such that $\frac 1 k <\epsilon$. Consider the number $x_1,x_2,...,x_{k+1}$ in $[0,1)$. If any two of these differ by $\frac 1 k$ or more then the difference between the largest and the smallest of them would exceed $1$ which is a contradiction. In other words there exist $i,j$ such that $0<|x_i-x_j| <\frac 1 k <\epsilon$. Thus $0<|(n_i+ib)-(n_j+jb)| <\epsilon$. [The fact that $b$ is irrational is required to say that $(n_i+ib)-(n_j+jb) \neq 0]$. This proves the result when $a=1$. A simple modification gives the result for any $a$.

Kavi Rama Murthy
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