PhD — Àlex Reyes

Programme

PhD in Applied Mathematics — UPC

Advisor

Dr. Jaume Franch Bullich

Start

2026

Status

Preprint in preparation

Main paper

Paper preprint

Fractal modular dynamics and the Collatz conjecture

For every positive integer n and m applications of the Collatz function, the m-th iterate T^m(n) can be expressed as

T^m(n) = \frac{3^s \cdot n + R(C^m_k)}{2^m}

where s is the number of odd steps, C^m_k is the symbolic path of n (the sequence of parities along m iterations), and R(·) is a residual that encodes the arithmetic contribution of the path. This formulation lets us treat Collatz as a dynamical system on residue classes modulo 2^m and reveals a self-similar structure in the orbit tree.

Target: arXiv math.NT 2026

Research lines

Line 1

Symbolic encoding of orbits

Characterisation of the paths C^m_k and their relation to Collatz cycles in Z/2^mZ.

Line 2

Modular self-similarity

Study of the fractal pattern that appears in the residual function R(·) as the iteration depth m grows.

Line 3

Bounds on the length of flight

Estimates on the number of steps needed to reach 1 in terms of the symbolic path.

Visualisations

Coming soon

Interactive visualisations of the Collatz orbit tree and of the modular structure of the residual R(·). In preparation.

Conferences