In mathematics, and more particularly in the analytic theory of regular continued fractions, an infinite regular continued fraction x is said to be restricted, or composed of restricted partial quotients, if the sequence of denominators of its partial quotients is bounded; that is and there is some positive integer M such that all the (integral) partial denominators ai are less than or equal to M.[1][2] A regular periodic continued fraction consists of a finite initial block of partial denominators followed by a repeating block; if then ζ is a quadratic irrational number, and its representation as a regular continued fraction is periodic.
Historically, mathematicians studied periodic continued fractions before considering the more general concept of restricted partial quotients.
To make the following theorems precise we will consider CF(M), the set of restricted continued fractions whose values lie in the open interval (0, 1) and whose partial denominators are bounded by a positive integer M – that is, By making an argument parallel to the one used to construct the Cantor set two interesting results can be obtained.
Zaremba has conjectured the existence of an absolute constant A, such that the rationals with partial quotients restricted by A contain at least one for every (positive integer) denominator.
[5] Jean Bourgain and Alex Kontorovich have shown that A can be chosen so that the conclusion holds for a set of denominators of density 1.