Lattice point visibility along powers of quadratic polynomials

Abstract

We study the growth of the number of invisible lattice points along powers of quadratic polynomials. Let $f(x)=A{x}^2+Bx+C\in \mathbb{Z}[x]$ have a positive leading coefficient and nonzero discriminant, and let $F(x)=f(x{)}^m$ with $m\ge 2$. For $m\ge 3$ 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.

16 pages. Comments welcome!