For a given $S \subseteq \Bbb{N}$, the asymptotic density of $S$ is defined as
$$d_\text{asy}(S) := \lim_{n \to \infty} {\#(k \in S : k \le n) \over n}$$
If the limit exists. Wikipedia says this is also sometimes called "arithmetic density", but I want to use that phrase for a different definition of density - one which is supposed to be a measure of the proportion of arithmetic progressions contained in $S$
$$d(S) := \sup_{a \in \Bbb{N}} \left( {\#(0 \le b \lt a : a\Bbb{N} + b \subseteq S) \over a}\right)$$
Ie; for each $a$, it meaures the proportion of congruence classes modulo $a$ which are contained in $S$. I have shown a few properties of this density function:
$i)\ 0 \le d(S) \le 1$
$ii)$ if $S^c$ is finite, then $d(S) = 1$ (i.p. $d(\Bbb{N}) = 1$)
$iii)$ (trivial) if $S$ contains no infinite arithmetic progressions then $d(S) = 0$ (i.p. $d(\Bbb{N}^2) = 0$)
I want to know more about this notion of density - and especially its relation to asymptotic density (upper or lower in the case where $d_{asy}(S)$ does not exist)