PreprintarXiv licencearXiv
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)=Ax2+Bx+C∈ℤ[x] have a positive leading coefficient and nonzero discriminant, and let F(x)=f(x)m with m≥ 2. For m≥ 3 we prove that the number of invisible lattice points in [1,N]2 has order Nlog N, and when m=2 the number of invisible lattice points satisfies Nlog N ≪F#InvisibleF(N)≪F N(log N)4. These estimates refine a previous result of the authors.
Related
- Density one for lattice point visibility along polynomials with at least two distinct roots
- Powers in orbits and semigroups generated by polynomials
- Invariance of Hurwitz-Stability of Polynomials of Degree Five under Positive Hadamard Powers
- Fixed points and exact periods of Chebyshev polynomials modulo odd prime powers
- Iterated Monodromy Groups of Quadratic Polynomials, I
BibTeX
@misc{lobsenz2026lattice,
title = {{Lattice point visibility along powers of quadratic polynomials}},
author = {Abraham Lobsenz and Tristan Phillips},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05027},
eprint = {2609.05027},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05027},
}
RIS
TY - UNPB
TI - Lattice point visibility along powers of quadratic polynomials
AU - Abraham Lobsenz
AU - Tristan Phillips
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05027
UR - https://arxiv.org/abs/2609.05027
AB - We study the growth of the number of invisible lattice points along powers of quadratic polynomials. Let f(x)=Ax^2+Bx+C∈ℤ[x] have a positive leading coefficient and nonzero discriminant, and let F(x)=f(x)^m with m≥ 2. For m≥ 3 we prove that the number of invisible lattice points in [1,N]^2 has order Nlog N, and when m=2 the number of invisible lattice points satisfies Nlog N ≪_F#Invisible_F(N)≪_F N(log N)^4. These estimates refine a previous result of the authors.
ER -