Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem. 

$$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$ 

subject to :

- $\left\lvert\left(\frac{\ln(j)}{2\pi}\right)t - z_j\right\rvert \leq \frac{r}{2\pi}  \quad \text{ for } 2 \leq j \leq n$.
      
- $z_j\geq 0 \quad \text{ for } 2 \leq j \leq n$. 

- $\sum_{j=2}^n z_j \geq 1$.

- $t \geq 0$.

    

Note that $(z_j)_{j=2}^n$ is a non-zero vector of natural numbers. Suppose, we write the optimal value $t^*$ as a function of $n$ and $r$ i.e. say 
$t^* = t^*(n,r)$.

1. How does $t^*(n,r)$ depend on $n$ and $r$?

2. Is it independent of $n$ and only dependent on $r$? 

3. Is it polynomial or exponential in $n$? 

4. Could we derive a good upper bound for $t^*(n,r)$?