Lattice points on a curve via \(\ell^2\) decoupling
type: Item
instance of: scholarly article
title: Lattice points on a curve via \(\ell^2\) decoupling
author: Daishi Kiyohara
published in: International Journal of Number Theory
publication date: 2024-08-22T00:00:00Z
review text: \textit{V. Jarník} [Math. Z. 24, 500--518 (1925; JFM 51.0153.01)] proved that a strictly convex closed curve of length \(l\ge 3\) has at most \(3(2\pi)^{-1/3}l^{2/3}+O(l^{1/3})\) lattice points. \textit{H. P. F. Swinnerton-Dyer} [J. Number Theory 6, 128--135 (1974; Zbl 0285.10020], motivated by the question of G. Andrews, looked at the number of lattice points of \(N\)-dilations of a strictly convex plane curve around the origin as \(N\rightarrow \infty\). In particular, he looked at the least upper bound for the exponent of \(N\) that would bound the number of points on the \(N\)-dilations of such curves. Among other results, he proved that for three times continuously differentiable strictly convex curve \(\Gamma\), \(|N\Gamma\cap\mathbb{Z}^2|=O(N^{3/5+\epsilon})\), where \(\epsilon>0\). The best-conjectured exponent here is \(1/2\), due to \textit{W. M. Schmidt} [Monatsh. Math. 99, 45--72 (1985; Zbl 0551.10026)], who improved and generalised Swinnerton-Dyer's results. \textit{E. Bombieri} and \textit{J. Pila} [Duke Math. J. 59, No. 2, 337--357 (1989; Zbl 0718.11048)] extended these results to plane curves defined by non-algebraic real analytic functions on closed bounded intervals.\N\NThe current paper extends Bombieri and Pila's lattice results to arbitrary finite sets using recent work on \( \ell^2\) decoupling. Specifically, Theorem 1.1 shows that given a finite subset \(A\subset \mathbb{R}^n\) and a non-degenerate \(C^{n,\alpha}\) curve \(\Gamma\) inside \(\mathbb{R}^n\) with a strictly positive Wronskian at \(t \in [0,1]\). Then the number of points of \(A\) in a neighbourhood of \(\Gamma\) is bounded above by \(|A|^{\frac{2}{n(n+1)}}\) times constants coming from additive combinatorics (such as the doubling constant and minimal separation of \(A\)). By applying curve-lifting methods, the theorem above allows us to bound the number of lattice points on a planar curve (this is shown in Theorem 1.2).
reviewed by: Pavlo Yatsyna
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 11P21
zbMATH DE Number: 7900420
zbMATH Keywords: lattice points
zbMATH Keywords: \(\ell^2\) decoupling
zbMATH Keywords: curve-lifting
zbMATH Keywords: additive combinatorics
MaRDI profile type: Publication
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DOI: 10.1142/S1793042124500994
description: scientific article; zbMATH DE number 7900420