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The language $$L=\bigcup_{n\in\mathbb N}\{0,1\}^{\lfloor n^\delta\rfloor}0^{n-\lfloor n^\delta\rfloor}$$ is computable in $\mathrm L\subseteq\mathrm{NSPACE}(n)=\mathrm{CSL}$, and it has $s_L(n)=2^{\lfloor n^\delta\rfloor}$. The language $$L=\bigcup_{n\in\mathbb N}\{0,1\}^{\lfloor(\log n)^c\rfloor}0^{n-\lfloor(\log n)^c\rfloor}$$ is computable in $\mathrm L\... 4 An algebraic characterisation of the restricted temporal logic (fragment using only next and eventually, but not until) is given in [1]. The expressive power of this fragment is the set of regular languages whose syntactic monoid are locally$\cal L\$-trivial. I am not sure of the result you are looking for, but this article probably contains enough material ...