Consider rational, quadratic, or real numbers whose continued fraction representations satisfy periodic constraints. A typical instance is numbers whose continued fraction quotients are alternatively odd and even. Such sets have a somewhat fractal nature, and the Hausdorff dimension of the set of reals as well as the density of the set of rationals satisfying such constraints can be determined. Other consequences include a precise analysis of the height of constrained continued fractions. The methods rely on a transfer operator that generates the constraints and whose dominant spectral properties prove essential.
e2=exp(1)2= /1,2,1,1,4,1,1,6,1,.../= 

, 
U(x)= 

 [ 

] 
G  _{s}[f](z):= 

( 

)^{s} f( 

). 
G  _{ M, s}[f](z):= 

( 

)^{s} f( 

), 
G 

:= G 

° G 

° ... ° G 

. 
w(s)=(I  G 

) ^{1} [1] (0) 
l ( M  , s 

) = 1. 
 Q_{N}( M)  ~ c 

N 

; 
E[X_{N}( M)]~ 

log N, 
L= 

, 
L 

= 


l( M,s) 
½ ½ ½ ½ 

. 
This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.