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arXiv · 2609.16377

Rational Points near Monofractal Curves and the Strong Oscillation Principle

Faustin Adiceam·Volodymyr Pavlenkov·Evgeniy Zorin
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Abstract

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.

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Faustin Adiceam, Volodymyr Pavlenkov, Evgeniy Zorin. 2026-09-14. Rational Points near Monofractal Curves and the Strong Oscillation Principle. https://arxiv.org/abs/2609.16377

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