单分形曲线附近的有理点与强振荡原理
Rational Points near Monofractal Curves and the Strong Oscillation Principle
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中文总结 AI 辅助
本文证明了一大类确定性分形曲线（包括Takagi和Weierstrass函数）附近分母有界有理点数量的弱振荡原理，首次给出精确计数。
中文摘要 AI 辅助
在作者先前关于布朗运动附近有理点分布的研究中，他们猜想存在一个“振荡原理”，用以控制分母有界的有理点在单分形曲线图像附近数量的渐近行为。本文证明，对于一大类确定性分形曲线，该猜想的较弱形式成立。这些曲线包括经典的Takagi函数和Weierstrass处处不可微函数，实际上还包括在具有给定正则性指数的Hölder连续函数中占优（即“大量”）的一类函数。这构成了首次对确定性分形曲线建立所考虑有理点的精确计数。
英文摘要
In their previous work devoted to the distribution of rational points near Brownian motion, the authors conjectured the existence of an \emph{oscillation principle} governing the asymptotic behavior of the number of rational points with bounded denomi\-nators near the graph of a monofractal curve. In this note, a weaker form of this conjecture is shown to hold for a broad class of deterministic fractal curves. These include the classical Takagi and Weierstrass nowhere differentiable functions, and indeed a prevalent (i.e.~"large") class of functions among those which are Hölder continuous with a given exponent of regularity. This constitutes the first instance of deterministic fractal curves for which a precise count of the rational points under consideration is established.
