Lattice points on a curve via \(\ell^2\) decoupling (Q6591605)

\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).