In mathematics, a Brjuno number (sometimes spelled Bruno or Bryuno) is a special type of irrational number named for Russian mathematician Alexander Bruno, who introduced them in Brjuno (1971).
An irrational number is called a Brjuno number when the infinite sum
converges to a finite number.
Here:
The Brjuno numbers are important in the one–dimensional analytic small divisors problems. Bruno improved the diophantine condition in Siegel's Theorem, showed that germs of holomorphic functions with linear part are linearizable if is a Brjuno number. Jean-Christophe Yoccoz (1995) showed in 1987 that this condition is also necessary, and for quadratic polynomials is necessary and sufficient.
Intuitively, these numbers do not have many large "jumps" in the sequence of convergents, in which the denominator of the (n + 1)th convergent is exponentially larger than that of the nth convergent. Thus, in contrast to the Liouville numbers, they do not have unusually accurate diophantine approximations by rational numbers.
The Brjuno sum or Brjuno function is
where:
The real Brjuno function is defined for irrational numbers [1]
and satisfies
for all irrational between 0 and 1.
Yoccoz's variant of the Brjuno sum defined as follows:[2]
where:
This sum converges if and only if the Brjuno sum does, and in fact their difference is bounded by a universal constant.