density over modulo M

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density over modulo M

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#1 h Apr 20, 2025, 4:19 AM

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Let $M$ be a positive integer and let $\alpha$ be an irrational number. Show that for every integer $0\leq a < M$ , there exists a positive integer $n$ such that $M \mid \lfloor{n \alpha}\rfloor-a$ .

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#2 h Apr 20, 2025, 9:16 AM

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Let $x_n= \lfloor{n \alpha}\rfloor$
Then $x_n$ is increasing , because $n \alpha$ is increasing and unbounded, and $x_{n+1}-x_n \leq 1$ so $x_n$ take all non-negative values

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#3 h Apr 21, 2025, 2:44 PM

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RagvaloD wrote:

Let $x_n= \lfloor{n \alpha}\rfloor$
Then $x_n$ is increasing , because $n \alpha$ is increasing and unbounded, and $x_{n+1}-x_n \leq 1$ so $x_n$ take all non-negative values

Sorry, it is supposed to be “ $\alpha$ is an arbitrary irrational number” xD. I was confused.

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#4 h Apr 22, 2025, 7:44 AM

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Solution

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