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A Deterministic Fractal Set Derived from the Sequence of Prime Numbers

Zhengqiang Li

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.

A Deterministic Fractal Set Derived from the Sequence of Prime Numbers

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.
Paper Structure (9 sections, 10 theorems, 23 equations)

This paper contains 9 sections, 10 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of the Fractal Prime Set $P_F$
  4. Computation of the Fractal Dimension
  5. Box-counting dimension
  6. Hausdorff dimension
  7. Connection with Prime Distribution and Statistical Self-similarity
  8. Generalization
  9. Conclusion

Key Result

Lemma 6

For every $n\ge1$,

Theorems & Definitions (18)

  • Definition 1: Generalized Moran set
  • Definition 2: Hausdorff dimension
  • Definition 3: Box-counting dimension
  • Definition 4: Prime residue sequence
  • Definition 5: Recursive construction
  • Lemma 6: Basic properties of the construction
  • Theorem 7: Topological nature of $P_F$
  • Theorem 8: Hexadecimal representation
  • Theorem 9
  • Definition 10: Natural mass distribution
  • ...and 8 more