A Deterministic Fractal Set Derived from the Sequence of Prime Numbers

TL;DR

This paper constructs a deterministic fractal set based on prime numbers modulo 16, revealing a universal fractal structure linked to number theory with specific dimensional properties.

Contribution

It introduces a new prime-based fractal set with proven properties and generalizes the construction to arbitrary bases, bridging number theory and fractal geometry.

Findings

The fractal set has Hausdorff and box-counting dimensions of 1/4.

The set is non-empty, compact, nowhere dense, with measure zero.

Dimension is independent of specific residue sequences, depending only on branching and contraction ratios.

Abstract

We introduce a novel deterministic fractal set PF in the unit interval whose construction is driven by the sequence of prime numbers modulo 16. At each step of the recursive construction, two subintervals are retained based on the residues of consecutive primes, yielding a Cantor-like set with a uniform contraction ratio of 1/16 and a branching number of 2. We prove that PF is a non-empty, compact, nowhere dense set of Lebesgue measure zero. Its Hausdorff dimension and box-counting dimension are both equal to 1 4 . The dimension is universal in the sense that it does not depend on the specific choice of the residue sequence, but only on the branching number and the contraction ratio. A generalization to arbitrary bases and branching numbers is also provided. This construction establishes a rigorous link between number theory and fractal geometry, offering a deterministic fractal whose structure is entirely encoded by the distribution of primes.