Mathematicians just found a "speed limit" for how chaotic certain numbers can get, solving a mystery that’s been bugging them for ages.
April 10, 2026
Original Paper
A Liouville-Type Inequality for Values of Mahler M-Functions
arXiv · 2604.08208
The Takeaway
For years, researchers feared that certain complex numbers were so irrational they might defy standard classification. This new proof reveals these numbers are actually barred from the most extreme forms of mathematical randomness, establishing a fundamental boundary in the logic of our number system.
From the abstract
We establish a Liouville-type inequality for the values, at a common nonzero algebraic point, of arbitrary Mahler Mq-functions. As an application, we prove that no such value is a Liouville number, or even a U -number. This solves a long-standing problem in the field.