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

Daishi Kiyohara

TL;DR

The paper addresses counting lattice points near or on planar curves by bridging harmonic analysis and additive combinatorics. It develops an $\ell^2$ decoupling theory for non-degenerate $C^{n,\alpha}$ curves in $\mathbb{R}^n$, generalizing the moment-curve results to arbitrary nondegenerate curves and finite sets with doubling constant $K$ and minimal separation $s$. Using a curve-lifting framework à la Bombieri–Pila, it transfer counts from planar curves to higher-dimensional lifted curves and apply determinant-method-style bounds, yielding explicit lattice-point estimates in neighborhoods with exponents determined by monomial degrees. A key contribution is showing that non-degeneracy implies a finiteness condition for hyperplane intersections, enabling the determinant method, and hence extending lattice-point bounds to finite sets beyond lattices. Overall, the work connects $\ell^2$ decoupling with additive combinatorics to obtain sharp incidence-type bounds for curves in higher dimensions and their planar projections.

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.

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

TL;DR

The paper addresses counting lattice points near or on planar curves by bridging harmonic analysis and additive combinatorics. It develops an decoupling theory for non-degenerate curves in , generalizing the moment-curve results to arbitrary nondegenerate curves and finite sets with doubling constant and minimal separation . Using a curve-lifting framework à la Bombieri–Pila, it transfer counts from planar curves to higher-dimensional lifted curves and apply determinant-method-style bounds, yielding explicit lattice-point estimates in neighborhoods with exponents determined by monomial degrees. A key contribution is showing that non-degeneracy implies a finiteness condition for hyperplane intersections, enabling the determinant method, and hence extending lattice-point bounds to finite sets beyond lattices. Overall, the work connects decoupling with additive combinatorics to obtain sharp incidence-type bounds for curves in higher dimensions and their planar projections.

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 decoupling inequality for non-degenerate curves in . 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.
Paper Structure (12 sections, 12 theorems, 43 equations)

This paper contains 12 sections, 12 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Main theorems
  3. $\ell^2$ decoupling for non-degenerate curves and application to lattice points
  4. $\ell^2$ decoupling for a non-degenerate curve
  5. Proof of the main theorem
  6. Special case: generalized arithmetic progressions
  7. Curve-lifting method
  8. Lift of a planar curve
  9. Bombieri-Pila's estimate of lattice points on a planar curve
  10. Lattice points on a planar curve
  11. Proof of Theorem \ref{['thm:lat']}
  12. Non-degeneracy implies the finiteness condition about intersection with hyperplanes

Key Result

Theorem 1.1

Let $\Gamma$ be a $C^{n,\alpha}$ non-degenerate curve inside $\mathbb{R}^n$ for positive $\alpha$ with $W(\Gamma)(t)>c_0>0$ for all $t\in [0,1]$. Let $A$ be a finite subset of $\mathbb{R}^n$ with doubling constant $K$ and minimal separation $s$. Then we have where $\mathcal{N}_{\delta}(\Gamma)$ denotes the $\delta$-neighborhood of $\Gamma$. Here the implicit constant depends on $\epsilon$, $\|\ga

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 18 more