Lattice point visibility along powers of quadratic polynomials

Source abstract

We study the growth of the number of invisible lattice points along powers of quadratic polynomials. Let $f(x)=Ax^2+Bx+C\in\mathbb{Z}[x]$ have a positive leading coefficient and nonzero discriminant, and let $F(x)=f(x)^m$ with $m\geq 2$. For $m\geq 3$ we prove that the number of invisible lattice points in $[1,N]^2$ has order $N\log N$, and when $m=2$ the number of invisible lattice points satisfies $N\log N \ll_F\#\mathrm{Invisible}_F(N)\ll_F N(\log N)^4$. These estimates refine a previous result of the authors.