In mathematics, a Brjuno number is an irrational number

Irrational number

In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

 α such that

where p_n/q_n are the convergents of the continued fraction

Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

expansion of α.
They were introduced by , who showed that germs of holomorphic functions with linear part e^2πiα are linearizable if α is a Brjuno number. showed that this condition is also necessary.

Brjuno function

The real Brjuno function B(x) is defined for irrational x and satisfies for all irrational x between 0 and 1.