Minkowski (box-counting) dimension of generalized Cantor set

I’m trying to solve this problem.

For $0<alpha, beta<1,$ let $K_{alpha, beta}$ be the Cantor set obtained as an intersection of the following nested compact sets. $K_{alpha, beta}^{0}=[0,1] .$ The set $K_{alpha, beta}^{1}$ is obtained by leaving the first interval of length $alpha$ and the last interval of length $beta,$ and removing the interval in between. To get $K_{alpha, beta}^{n},$ for each interval $I$ in $K_{alpha, beta}^{n-1},$ leave the first interval of length $alpha|I|$ and the last interval of length $beta|I|$ and remove the subinterval in between. Compute the Minkowski dimension of $K_{alpha, beta}$

---

My attempt.

If assume that $alpha le beta $ then

I believe that after $n$ generations the length of the longest interval $I$ in $K^n_{a,b} $ is $beta^{n}$ and the length of shortest interval $alpha^{n}$ and there are a total of $2^n$ intervals with lengths given by $alpha^{n-k} beta^k$ for $k in {0, …, n }$

But I don’t see how to use this information to get the dimension that is $ lim_{varepsilonto 0} frac{log{N(K, varepsilon)}}{log{varepsilon^{-1}}} $. I dont know how to find $N(K, varepsilon)$ the minimal amount of balls with diam = $varepsilon$ needed to cover $K_{alpha, beta}$.
I believe I should find an upper and lower bound for this so that the bounds agree but how to do this?