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 .
An irrational number is called a Brjuno number when the infinite sum
converges to a finite number.
Here:
Consider the golden ratio :
Then the nth convergent can be found via the recurrence relation:
It is easy to see that for , as a result
and since it can be proven that for any irrational number, is a Brjuno number. Moreover, a similar method can be used to prove that any irrational number whose continued fraction expansion ends with a string of 1's is a Brjuno number.
By contrast, consider the constant with defined as
Then , so we have by the ratio test that diverges. is therefore not a Brjuno number.
The Brjuno numbers are important in the one-dimensional analytic small divisors problems. Bruno improved the diophantine condition in Siegel's Theorem by showing that germs of holomorphic functions with linear part are linearizable if is a Brjuno number. showed in 1987 that Brjuno's condition is sharp; more precisely, he proved that for quadratic polynomials, this condition is not only sufficient but also necessary for linearizability.
Intuitively, these numbers do not have many large "jumps" in the sequence of convergents, in which the denominator of the ()th convergent is exponentially larger than that of the th 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
and satisfies
for all irrational between 0 and 1.
Yoccoz's variant of the Brjuno sum defined as follows:
where:
This sum converges if and only if the Brjuno sum does, and in fact their difference is bounded by a universal constant.