Restricted Partial Quotients
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 :x = _0;a_1,a_2,\dots= a_0 + \cfrac = a_0 + \underset \frac,\, and there is some positive integer ''M'' such that all the (
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
) partial denominators ''ai'' are less than or equal to ''M''.


Periodic continued fractions

A regular
periodic continued fraction In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form : x = a_0 + \cfrac where the initial block of ''k'' + 1 partial denominators is followed by a block 'a'k''+1, ''a'k''+2,.. ...
consists of a finite initial block of partial denominators followed by a repeating block; if : \zeta = _0;a_1,a_2,\dots,a_k,\overline\, then ζ is a
quadratic irrational In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible ...
number, and its representation as a regular continued fraction is periodic. Clearly any regular periodic continued fraction consists of restricted partial quotients, since none of the partial denominators can be greater than the largest of ''a''0 through ''a''''k''+''m''. Historically, mathematicians studied periodic continued fractions before considering the more general concept of restricted partial quotients.


Restricted CFs and the Cantor set

The
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Thr ...
is a set ''C'' of
measure zero In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null s ...
from which a complete interval of real numbers can be constructed by simple addition – that is, any real number from the interval can be expressed as the sum of exactly two elements of the set ''C''. The usual proof of the existence of the Cantor set is based on the idea of punching a "hole" in the middle of an interval, then punching holes in the remaining sub-intervals, and repeating this process ''ad infinitum''. The process of adding one more partial quotient to a finite continued fraction is in many ways analogous to this process of "punching a hole" in an interval of real numbers. The size of the "hole" is inversely proportional to the next partial denominator chosen – if the next partial denominator is 1, the gap between successive convergents is maximized. 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, : \mathrm(M) = \.\, By making an argument parallel to the one used to construct the Cantor set two interesting results can be obtained. * If ''M'' ≥ 4, then any real number in an interval can be constructed as the sum of two elements from CF(''M''), where the interval is given by :: (2\times ;\overline 2\times ;\overline = \left(\frac \left sqrt - M \right \sqrt - M \right). *A simple argument shows that holds when ''M'' ≥ 4, and this in turn implies that if ''M'' ≥ 4, every real number can be represented in the form ''n'' + CF1 + CF2, where ''n'' is an integer, and CF1 and CF2 are elements of CF(''M'').


Zaremba's conjecture

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. The choice ''A'' = 5 is compatible with the numerical evidence. Further conjectures reduce that value, in the case of all sufficiently large denominators.
Jean Bourgain Jean, Baron Bourgain (; – ) was a Belgian mathematician. He was awarded the Fields Medal in 1994 in recognition of his work on several core topics of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, ergodic t ...
and Alex Kontorovich have shown that ''A'' can be chosen so that the conclusion holds for a set of denominators of density 1.


See also

*
Markov spectrum In mathematics, the Markov spectrum devised by Andrey Markov is a complicated set of real numbers arising in Markov Diophantine equation and also in the theory of Diophantine approximation. Quadratic form characterization Consider a quadratic f ...


References

{{reflist Continued fractions Diophantine approximation