Avoiding secants of given size in finite projective planes

Let $q$ be a prime power and $k$ be a natural number. What are the possible cardinalities of point sets ${S}$ in a projective plane of order $q$, which do not intersect any line at exactly $k$ points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this paper we show a series of results which point out the existence of all or almost all possible values $m\in [0, q^2+q+1]$ for $|S|=m$, provided that $k$ is not close to the extremal values $0$ or $q+1$. Moreover, using polynomial techniques we show the existence of a point set $S$ with the following property: for every prescribed list of numbers $t_1, \ldots t_{q^2+q+1}$, $|S\cap \ell_i|\neq t_i$ holds for the $i$th line $\ell_i$, $\forall i \in \{1, 2, \ldots, q^2+q+1\}$.