Collatz on the Dyadic Rationals in [0.5, 1) with Fractals: How Bit Strings Change Their Length Under 3x + 1

Abstract

The reduced Collatz map (Syracuse function) can be stated as “for any odd positive integer x, calculate $3x+1$ and then divide by 2 until the result is odd.” We calculate the change in bit string length caused by this map. The result arises from a novel reformulation of the Collatz process “for any fraction in [0.5, 1) with a binary representation of finite length, append $01010101\dots .$ If the resulting number is smaller than 2/3 then multiply by 3/2 otherwise multiply by 3/4.” The $<2/3$ domain is where strings may shrink versus $>2/3$ for may grow. If the Collatz map has non-trivial periodic orbits then they will arise from a fractal that has been added to a map that lacks periodic orbits. If there are an infinite number of such orbits then as their length increases, they will make roughly 0.71 visits to the may-shrink domain for each visit to the may-grow domain.

Keywords:

• Collatz conjecture
• 3x + 1 problem
• Syracuse function
• bit strings
• binary strings

2010 AMS SUBJECT CLASSIFICATION:

• 11B37 Recurrences
• 11-04 Explicit machine computation and programs

Acknowledgments

The author thanks Dmitri Kamenetsky and Richard Taylor for their comments on drafts of this article. He also thanks the anonymous reviewers for their work in providing constructive feedback, and for their extensive work to repair the statement and proof of Lemma 5.