Let $\alpha$ be an irrational with $0<\alpha<1$. Consider the function given by \begin{align*} f: &\mathbb{N}\longrightarrow \mathbb{N}\\ &x\longmapsto [ \alpha\cdot x]\end{align*} where $[r]$ is the largest integer that is less than or equal to $r$ for $r\in\mathbb{R}$.

Let $N=\left[\frac{1}{\alpha}\right]$ be the integer part of $\frac{1}{\alpha}$, and consider the following numbers:

$A_1(n)=\#\{k\leq n : f(k)<f(k+1)\}$

$A_2(n)=\#\{k\leq n : f(k)=f(k+N)\}$

The questions are the following:

1) Is it true that $\displaystyle{\lim_{n\to\infty} \dfrac{A_1(n)}{n}=\alpha}$?

2) What is the value of $\displaystyle{\lim_{n\to\infty} \dfrac{A_2(n)}{n}}$, if it exists?

The problem above came out when I was trying to find examples of classes of finite ordered structures that satisfies strong conditions on their definable sets (for my Ph.D. Thesis). I translated the problem in simple terms, and I would really appreciate any help.

(If it is too easy, a hint will be good. If the problem is related with something in number theory, combinatorics, ergodic theory or anything else, I would also appreciate if you tell me how)