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).
Formal definition An irrational number α {\displaystyle \alpha } is called a Brjuno number when the infinite sum
B ( α ) = ∑ n = 0 ∞ log q n + 1 q n {\displaystyle B(\alpha )=\sum _{n=0}^{\infty }{\frac {\log q_{n+1}}{q_{n}}}}
converges to a finite number. Here:
q n {\displaystyle q_{n}} is the denominator of the nth convergent p n q n {\displaystyle {\tfrac {p_{n}}{q_{n}}}} of the continued fraction expansion of α {\displaystyle \alpha } .
B {\displaystyle B} is a Brjuno function
Examples Consider the golden ratio 𝜙:
ϕ = 1 + 5 2 = 1 + 1 1 + 1 1 + 1 1 + 1 ⋱ . {\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}.}
Then the nth convergent p n q n {\displaystyle {\frac {p_{n}}{q_{n}}}} can be found via the recurrence relation:
{ p n = p n − 1 + p n − 2 with p 0 = 1 , p 1 = 2 , q n = q n − 1 + q n − 2 with q 0 = q 1 = 1. {\displaystyle {\begin{cases}p_{n}=p_{n-1}+p_{n-2}&{\text{ with }}p_{0}=1,p_{1}=2,\\q_{n}=q_{n-1}+q_{n-2}&{\text{ with }}q_{0}=q_{1}=1.\end{cases}}}
It is easy to see that q n + 1 < q n 2 {\displaystyle q_{n+1}<q_{n}^{2}} for n ≥ 2 {\displaystyle n\geq 2} , as a result
log q n + 1 q n < 2 log q n q n for n ≥ 2 {\displaystyle {\frac {\log {q_{n+1}}}{q_{n}}}<{\frac {2\log {q_{n}}}{q_{n}}}{\text{ for }}n\geq 2}
and since it can be proven that ∑ n = 0 ∞ log q n q n < ∞ {\displaystyle \sum _{n=0}^{\infty }{\frac {\log q_{n}}{q_{n}}}<\infty } for any irrational number, 𝜙 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 α = [ a 0 , a 1 , a 2 , … ] {\displaystyle \alpha =[a_{0},a_{1},a_{2},\ldots ]} with ( a n ) {\displaystyle (a_{n})} defined as
a n = { 10 if n = 0 , 1 , q n q n − 1 if n ≥ 2. {\displaystyle a_{n}={\begin{cases}10&{\text{ if }}n=0,1,\\q_{n}^{q_{n-1}}&{\text{ if }}n\geq 2.\end{cases}}}
Then q n + 1 > q n 2 q n q n − 1 {\displaystyle q_{n+1}>q_{n}^{\frac {2q_{n}}{q_{n-1}}}} , so we have by the ratio test that ∑ n = 0 ∞ log q n + 1 q n {\displaystyle \sum _{n=0}^{\infty }{\frac {\log q_{n+1}}{q_{n}}}} diverges. α {\displaystyle \alpha } is therefore not a Brjuno number.
Importance The Brjuno numbers are important in the one-dimensional analytic small divisors problems. Brjuno improved the diophantine condition in Siegel's Theorem by showing that germs of holomorphic functions with linear part e 2 π i α {\displaystyle e^{2\pi i\alpha }} are linearizable if α {\displaystyle \alpha } is a Brjuno number. Jean-Christophe Yoccoz (1995) 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.
Properties 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.
Brjuno function
Brjuno sum The Brjuno sum or Brjuno function B {\displaystyle B} is
B ( α ) = ∑ n = 0 ∞ log q n + 1 q n {\displaystyle B(\alpha )=\sum _{n=0}^{\infty }{\frac {\log q_{n+1}}{q_{n}}}}
where:
q n {\displaystyle q_{n}} is the denominator of the nth convergent p n q n {\displaystyle {\tfrac {p_{n}}{q_{n}}}} of the continued fraction expansion of α {\displaystyle \alpha } .
Real variant
The real Brjuno function B ( α ) {\displaystyle B(\alpha )} is defined for irrational numbers α {\displaystyle \alpha }
B : R ∖ Q → R ∪ { + ∞ } {\displaystyle B:\mathbb {R} \setminus \mathbb {Q} \to \mathbb {R} \cup \{+\infty \}}
and satisfies
B ( α ) = B ( α + 1 ) B ( α ) = − log α + α B ( 1 / α ) {\displaystyle {\begin{aligned}B(\alpha )&=B(\alpha +1)\\B(\alpha )&=-\log \alpha +\alpha B(1/\alpha )\end{aligned}}}
for all irrational α {\displaystyle \alpha } between 0 and 1.
Yoccoz's variant Yoccoz's variant of the Brjuno sum defined as follows:
Y ( α ) = ∑ n = 0 ∞ α 0 ⋯ α n − 1 log 1 α n , {\displaystyle Y(\alpha )=\sum _{n=0}^{\infty }\alpha _{0}\cdots \alpha _{n-1}\log {\frac {1}{\alpha _{n}}},}
where:
α {\displaystyle \alpha } is irrational real number: α ∈ R ∖ Q {\displaystyle \alpha \in \mathbb {R} \setminus \mathbb {Q} }
α 0 {\displaystyle \alpha _{0}} is the fractional part of α {\displaystyle \alpha }
α n + 1 {\displaystyle \alpha _{n+1}} is the fractional part of 1 α n {\displaystyle {\frac {1}{\alpha _{n}}}}
This sum converges if and only if the Brjuno sum does, and in fact their difference is bounded by a universal constant.
See also Irrationality measure Markov constant
References Brjuno, Alexander D. (1971), "Analytic form of differential equations. I, II", Trudy Moskovskogo Matematičeskogo Obščestva, 25: 119–262, ISSN 0134-8663, MR 0377192 Lee, Eileen F. (Spring 1999), "The structure and topology of the Brjuno numbers" (PDF), Proceedings of the 1999 Topology and Dynamics Conference (Salt Lake City, UT), Topology Proceedings, vol. 24, pp. 189–201, MR 1802686 Marmi, Stefano; Moussa, Pierre; Yoccoz, Jean-Christophe (2001), "Complex Brjuno functions", Journal of the American Mathematical Society, 14 (4): 783–841, doi:10.1090/S0894-0347-01-00371-X, ISSN 0894-0347, MR 1839917 Yoccoz, Jean-Christophe (1995), "Théorème de Siegel, nombres de Bruno et polynômes quadratiques", Petits diviseurs en dimension 1, Astérisque, vol. 231, pp. 3–88, MR 1367353
Notes