On this forum, I've found following statement on densities of regular languages:
The density of a regular language is of the form $\Theta \left( n^k \lambda ^ n \right)$ for some integer $k \geq 0$ and real $\lambda \geq 0$.
In the OP, the density of a language is defined as follows:
$p_L(n) = | L \cap \Sigma^n |$
But the language consisting of words of even length over arbitrary alphabet is clearly regular, and for it we have: $$p_L(n) = [2 \mid n] |\Sigma|^n$$ So, the statement definitely doesn't hold here
Would the statement hold, if we redefine density as follows?
$p_L(n) = | L \cap \Sigma^{\leq n} |$
And generally, what could be meant by the statement and are there any references on similar statements?
