A note on Farey sequences and Hausdorff dimension

TL;DR

This paper introduces a novel fractal parameter-based classification of Farey sequences, linking them to Hausdorff dimension, and establishes algebraic equations for each class, revealing new structural insights.

Contribution

It proposes a new framework connecting Farey sequences with fractal geometry through a Hausdorff dimension parameter, providing algebraic characterizations for each class.

Findings

Farey sequences can be classified into equivalence classes labeled by a fractal parameter h.

Each class h satisfies properties similar to Farey series.

For each h, an algebraic equation characterizes the class.

Abstract

We prove that the Farey sequences can be express into equivalence classes labeled by a fractal parameter which looks like a Hausdorff dimension $h$ defined within the interval 1 < h < 2. The classes $h$ satisfy the same properties of the Farey series and for each value of $h$ there exists an algebraic equation.