AI 中文总结
本文研究布朗运动图像上的有理点计数问题，确立了除临界区域外所有$\delta$对应的计数函数几乎必然渐近展开，完善了相关丢番图逼近理论，为连接数论与多重分形分析搭建了新桥梁。
AI 中文摘要
给定一个实值函数 $f$，令 $\mathcal{N}_f(\delta, Q)$ 为函数 $f$ 图像的 $(\delta/Q)$-管状邻域内分母不超过 $Q\ge 1$ 的有理点的数量。有一个启发性猜想预测，只要 $\delta$ 足够大（在合适的意义下），这类点的数量会像该邻域的面积一样增长。为证明该启发性猜想，人们对正则曲线付出了大量努力。这一努力在 Vaughan 与 Velani（2006）以及 Huang（2015）的工作中达到了顶点，他们证明了当映射 $f$ 满足包括二阶连续可微在内的若干假设，且 $\delta\gg Q^{-1+\epsilon}$（其中 $\epsilon>0$）时，$\mathcal{N}_f(\delta, Q)$ 存在渐近展开。本文研究迄今尚未探索的、对曲线施加最小正则性条件的情形。更确切地说，本文研究的是映射 $f$ 是布朗运动图像的几乎必然实现的情形。主要结果确立了对所有 $\delta$ 值（临界区域除外），计数函数存在几乎必然渐近展开，从而远超当前正则曲线相关理论的范围。证明中的一个关键组成部分是面积启发性猜想的推导，这依赖于确定布朗运动图像管状邻域面积的几乎必然渐近行为。该结果有两个主要推论：其一，它完善了由 Sprindžuk（1979）开创的布朗运动图像上丢番图逼近理论中的计数方面；其二，它暗示了存在一种理论，能将曲线上附近有理点的分析与其局部 Hölder 正则性及精细尺度振荡统一起来。因此，它在数论与多重分形分析之间搭建了一座看似全新的桥梁。
英文摘要
Given a real-valued function $f$, let $\mathcal{N}_f(δ, Q)$ be the number of rational points with denominators at most $Q\ge 1$ in the $(δ/Q)$-tubular neighbourhood of the graph of the function $f$. A heuristic predicts that the number of such points grows like the area of the neighbourhood provided that $δ$ is big enough (in a suitable sense). Considerable efforts have been committed to prove this heuristic for regular curves. This culminated in the works by Vaughan \& Velani~(2006) and by Huang~(2015) establishing an asymptotic expansion for $\mathcal{N}_f(δ, Q)$ provided that $δ\gg Q^{-1+ε}$ for some $ε>0$ when the map $f$ is, among other assumptions, twice continuously differentiable. The present work deals with the thus-far unexplored regime where minimal regularity conditions are imposed on the curve. More precisely, it is concerned with the case where the map $f$ is an a.s. realisation of the graph of Brownian motion. The main result establishes the existence of an almost sure asymptotic expansion for the counting function for all values of $δ$, with the exception of a critical regime, thereby going well beyond the theory currently available for regular curves. A key ingredient in the proof is the derivation of the area heuristic, which relies on establishing the a.s. asymptotics of the area of the tubular neighbourhood of the graph of Brownian motion. This result has two main consequences: firstly, it completes the counting aspect of the theory of Diophantine approximation on the graph of Brownian motion initiated by Sprindžuk (1979). Secondly, it hints at the existence of a theory unifying the analysis of rational points near a curve on the one hand and, on the other, its local Hölder regularity and fine-scale oscillations. It thus builds a seemingly new bridge between Number Theory and Multifractal Analysis.
