---
tag: 5WR304YRST
kind: concept
name: Brjuno number
msc: [37F50, 11J70]
sources:
  - url: https://en.wikipedia.org/wiki/Brjuno_number
same_as:
  wikidata: Q4971764
  wikipedia: Brjuno number
---

## Summary

A Brjuno number is an [[irrational number|0DZXAXEJQ9]] whose [[continued fraction|04P2RJ4WPS]] convergents have denominators that do not grow too fast.
The condition is that the series $\sum_n \log(q_{n+1}) / q_n$ converges.
Brjuno numbers are exactly the rotation numbers for which every holomorphic germ with that rotation at a fixed point can be linearized.
They are central to small divisor problems and to the existence of [[Siegel discs|KNDYA9T8ZK]].

## Explanations

### L1 human

Some irrational numbers are very close to fractions with small denominators, and others stay away from all such fractions.
A Brjuno number is one that is not approximated too well by fractions, measured in a precise, fairly generous way.
The golden mean is a Brjuno number, and in fact almost every real number is one.
The condition decides when a map that turns points near a fixed point by such an angle can be straightened into an exact rotation.