Integral points on convex curves

TL;DR

This paper provides bounds on the maximum number of integer lattice points that can lie on a convex curve in the plane, given constraints on length, curvature, and slope.

Contribution

It introduces new estimates for the number of integral points on convex curves based on geometric constraints, advancing understanding in Diophantine geometry.

Findings

Derived upper bounds for integral points on convex arcs.

Established relationships between curvature, length, and lattice points.

Provided new methods for estimating lattice points on convex curves.

Abstract

We estimate the maximal number of integral points which can be on a convex arc in the plane with given length, minimal radius of curvature and initial slope.