# quasiperiodic continued fractions

Is anything known about continued fractions in which the sequence of integers is quasiperiodic?

Quasiperiodic is meant here in the sense of 1D quasicrystals. For example, draw an irrationally-sloped straight line through a 2D lattice: whenever the line intersects a vertical line of the lattice write integer $A$, and whenever the line intersects a horizontal line of the lattice write (different) integer $B$.

For example, one sequence generated by slope $(1-\sqrt{5})/2$ with $A=1,\,B=2$ is

$1,\,2,\, 2,\, 1,\, 2,\, 2,\, 1,\, 2,\, 1,\, 2,\, 2,\, 1,\, 2,\, 1,\, 2,\, 2,\, 1,\, 2,\, 2,\, 1,\, \ldots$

in which case the expansion (integer part zero) evaluates to

$0.70326125114456817\ldots$

My understanding is that the continued fraction expansion of a 'generic' irrational number is expected to comprise numbers approximately following the Gauss-Kuzmin distribution

$p(k)=-\log_2\left(1-\left(1+k\right)^{-2}\right).$

In this sense, quadratic irrationals are non-generic, having periodic continued fraction expansions. I'm wondering if the quasiperiodic case might be close enough to periodic that it's covered by some results applying to quadratic irrationals but not to generic irrationals.

Any thoughts would be greatly appreciated!

• One approach could be: Consider the sequence $-f_{n-1}/f_n$ of rational approximants of the irrational slope $(1-\sqrt{5})/2$, i.e. $-2/3,-3/5, -5/8, \ldots$. Each element gives a periodic sequence of 1s and 2s, the period of which is $f_n$. The corresponding periodic continuous fraction equals a quadratic irrational. If one studies the series of these quadratic irrationals, there might be a chance to get insight about the quasiperiodic limit ($n \to \infty$). – Andreas Rüdinger Mar 8 '17 at 21:06
• What is the number of this sequence in OEIS? – Alexey Ustinov Mar 9 '17 at 12:49
• I'm not sure what question you are asking. It seems that you're defining a continued fraction by a predetermined pattern, and then asking a statistical question about that continued fraction that you could just answer by directly looking at the pattern you started with. – Greg Martin Mar 9 '17 at 17:32
• @ Alexey Ustinov: Not identical, but closely related (and quasiperiodic in the sense above and with slope golden ratio) is the series oeis.org/A003849 – Andreas Rüdinger Mar 10 '17 at 21:27