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PreprintCC BY 4.0arXiv

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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Cite

BibTeXRIS
BibTeX
@misc{adiceam2026rational,
  title = {{Rational Points near Monofractal Curves and the Strong Oscillation Principle}},
  author = {Faustin Adiceam and Volodymyr Pavlenkov and Evgeniy Zorin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16377},
  eprint = {2609.16377},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16377},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Rational Points near Monofractal Curves and the Strong Oscillation Principle
AU  - Faustin Adiceam
AU  - Volodymyr Pavlenkov
AU  - Evgeniy Zorin
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.16377
UR  - https://arxiv.org/abs/2609.16377
AB  - 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.
ER  -