Counting tame $SL_3$- and $SL_4$- frieze patterns over finite fields

In this article we count tame $SL_3$- and $SL_4$-frieze patterns with width $w$ over a finite field $K$, as well as some tame $SL_k$-frieze patterns for higher $k$. Let $n = w + k + 1$. We consider the sets $C_k(n)$ of tuples of $n$ points in the projective space $\mathbb{P}^{k-1}(K)$, such that $k$ consecutive points are always independent (the first and last point in the tuple are considered to be consecutive). First assume $\gcd(k,n) = 1$. In this case, we prove that the problem of counting tame $SL_k$-frieze patterns can be reduced to counting $C_k(n)$. We also show that $\lvert C_k(n) \rvert$ is essentially already known as long as $k$ and $n$ are coprime, and we derive the number of tame $SL_k$-frieze-patterns in that case. In the case $\gcd(k,n) \neq 1$, we define certain subsets $C_k^*(n)$ and show that it is sufficient to count these sets. Afterwards, we count $C_k^*(n)$ in the cases $k = 3$ and $k = 4$ and thus the number of tame $SL_3$- and $SL_4$-frieze patterns for any width $w$.