Bounding the number of lattice points near a convex curve by curvature

TL;DR

This paper establishes explicit bounds on the number of lattice points close to convex curves, relating these bounds to geometric properties like curvature and length, with optimal constants and general lattice settings.

Contribution

It provides the first explicit bounds on lattice points near convex curves involving curvature and affine invariants, with optimal constants and broader lattice applicability.

Findings

Bounds depend explicitly on curvature and length

Optimal constants achieved in several estimates

Results extend to general lattices beyond the integer lattice

Abstract

We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our estimates hold for lattices more general than the usual lattice of integral points in the plane.