Counterexamples for Rational Points Near Curves

Source abstract

Let $δ\in [0,1/2]$ and $Q\geq 1$. Given a compact $C^{\infty}$-curve $\mathcal{C}$ in $\mathbb{R}^n$, denote by $N_{\mathcal{C}}(δ, Q)$ the number of pairs $(\mathbf{a}, q)\in \mathbb{Z}^n\times [Q/2,Q]$ such that the rational point $\mathbf{a}/q$ is $δ/q$-close to $\mathcal{C}$. We investigate the range of $δ$ in terms of $Q$ which is necessary for the heuristic $N_{\mathcal{C}}(δ, Q)\asympδ^{n-1}Q^2$ to be correct for the moment curve. For a nondegenerate curve $\mathcal{C}\subset \mathbb{R}^n$ and for sufficiently large $Q$, Hickman and the second author previously established that $N_{\mathcal{C}}(δ, Q)\lesssim_ν δ^{n-1}Q^2$ for all $δ\in[Q^{α(n+1)+ν},1/2)$ and any $ν>0$. Here $α(n)=-4n^{-2}+O(n^{-3})$ was explicitly computed. We show that this range is surprisingly close to being sharp in the asymptotic sense as $n\to\infty$, and can at most be extended to $δ\in[Q^{α(n)},1/2)$. In particular, the sharp exponent can only be about $O(n^{-3})$ better, even though it was widely believed before that an improvement of $O(n^{-1})$ should be possible. The complementary upper bound in this remaining range has been recently established by Gan--Guo--Oh.