Let $x$ an irrationnal number between $0$ and $1$ and $N$ a positive integer. I try to find the value of : $\mathbb{min} \left(\{\mathbb{ceil}(mx), m \in \{1,...,N\}\} \right)$
Is there some situations where this quantity can be easily calculated ?
We can find a sequence to help us. Indeed, if we note $ A = \mathbb{min} \left(\{\mathbb{ceil}(mx), m \in \{1,...,N\}\} \right)$ then $A$ would be the minimum of the sequence : $u_0 = x$ ; $u_{n+1} = \mathbb{ceil} \left(u_n + x \left(\mathbb{floor} \left( \frac{1 - u_n}{x} \right) + 1 \right) \right) $ among $u_0, u_1, ... , u_N$