  Set systems with strongly restricted intersections, called
$\alpha$-intersecting families for a vector $\alpha$, were introduced recently
as a generalization of several well-studied intersecting families including the
classical oddtown and eventown. Given a binary vector $\alpha=(a_1, \ldots,
a_k)$, a collection $\mathcal F$ of subsets over an $n$ element set is an
$\alpha$-intersecting family modulo $2$ if for each $i=1,2,\ldots,k$, all
$i$-wise intersections of distinct members in $\mathcal F$ have sizes with the
same parity as $a_i$. Let $f_\alpha(n)$ denote the maximum size of such a
family. In this paper, we study the asymptotic behavior of $f_\alpha(n)$ when
$n$ goes to infinity. We show that if $t$ is the maximum integer such that
$a_t=1$ and $2t\leq k$, then $f_{\alpha(n)} \sim {(t! n)}^{\frac 1 t}$. More
importantly, we show that for any constant $c$, as the length $k$ goes larger,
$f_\alpha(n)$ is upper bounded by $O (n^c)$ for almost all $\alpha$.
Equivalently, no matter what $k$ is, there are only finitely many $\alpha$
satisfying $f_\alpha(n)=\Omega (n^c)$. This answers an open problem raised by
Johnston and O'Neill in 2023. All of our results can be generalized to modulo
$p$ setting for any prime $p$ smoothly.
