Intersective polynomials along the irreducibles in function fields

Let \(\mathbb {F}_q[t]\) ${\mathbb{F}}_q[t]$ be the polynomial ring over the finite field \(\mathbb {F}_q\) ${\mathbb{F}}_q$ of q elements. For a natural number \(N\ge 1,\) $N\ge 1,$ let \(\mathbb {G}_N\) ${\mathbb{G}}_N$ be the subset of \(\mathbb {F}_q[t]\) ${\mathbb{F}}_q[t]$ containing all polynomials of degree less than N. Let \(h\in \mathbb {F}_q[t][x]\) $h\in {\mathbb{F}}_q[t][x]$ be a polynomial of degree \(2\le k<p,\) $2\le k<p,$ the characteristic of \(\mathbb {F}_q.\) ${\mathbb{F}}_q.$ Suppose that for every \(d\in \mathbb {F}_q[t]\setminus \{0\},\) $d\in {\mathbb{F}}_q[t]\backslash \{0\},$ there exists \(m\in \mathbb {F}_q[t]\) $m\in {\mathbb{F}}_q[t]$ such that \(d\mid h(m)\) $d\mid h(m)$ and \((d,m)=1.\) $(d,m)=1.$ Let \(A\subseteq \mathbb {G}_N\) $A\subseteq {\mathbb{G}}_N$ with \(|A|=\delta q^N.\) $|A|=\delta q^N.$ Suppose further that \((A-A)\cap \left( h(\Omega )\setminus \{0\}\right) =\emptyset ,\) $(A-A)\cap \left(h,(,\mathrm{\Omega },),\backslash ,\{,0,\}\right)=\mathrm{\varnothing },$ where \(A-A\) $A-A$ is the difference set of A and \(\Omega \) $\mathrm{\Omega }$ denotes the set of all monic irreducible polynomials in \(\mathbb {F}_q[t]\) ${\mathbb{F}}_q[t]$ . It is proved that \(\delta \ll N^{-\mu }\) $\delta \ll N^{-\mu }$ for any \(0<\mu <1/(2k-2),\) $0<\mu <1/(2k-2),$ where the implied constant depends only on \(q,\ h\) $q,h$ and \(\mu .\) $\mu .$