Visible lattice points along curves

TL;DR

This paper introduces the concept of level of visibility to analyze lattice points along curves, providing asymptotic formulas for counts of points visible at various levels, advancing understanding of lattice point visibility.

Contribution

It proposes a new concept of level of visibility, enabling detailed analysis of visible and invisible lattice points along curves, with derived asymptotic formulas.

Findings

Asymptotic formulas for lattice points at different visibility levels

Enhanced analysis of visibility along curves

New framework for lattice point visibility studies

Abstract

This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously. By proposing the concept of level of visibility, we are able to analyze more carefully about both the "visible" points and the "invisible" points in the definition of previous research. We prove asymptotic formulas for the number of lattice points in different levels of visibility.