Lattice points on a curve via $\ell^2$ decoupling

TL;DR

This paper advances the understanding of lattice points on curves by extending classical estimates to more general sets using $\,\ell^2$ decoupling inequalities, with applications to planar curves and considerations of set separation.

Contribution

It introduces an $\,\ell^2$ decoupling inequality for non-degenerate curves in higher dimensions and extends lattice point estimates to arbitrary finite sets.

Findings

Established $\,\ell^2$ decoupling inequality for non-degenerate curves

Extended lattice point estimates to arbitrary finite sets with minimal separation

Reviewed and applied curve-lifting method for planar curves

Abstract

This paper extends Bombieri and Pila's estimate of lattice points on curves to arbitrary finite sets by incorporating considerations of minimal separation and the doubling constant. We derive the estimate by establishing the $\ell^2$ decoupling inequality for non-degenerate curves in $\mathbb{R}^n$. Additionally, we review the curve-lifting method introduced in Bombieri and Pila's work and establish the estimate of lattice points in the neighborhood of a planar curve.