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| In [[mathematics]], and more particularly in the analytic theory of [[continued fraction|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
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| :<math>x = [a_0;a_1,a_2,\dots] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + \ddots}}}} = a_0 + \underset{i=1}{\overset{\infty}{K}} \frac{1}{a_i},\,</math>
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| and there is some positive integer ''M'' such that all the (integral) partial denominators ''a<sub>i</sub>'' are less than or equal to ''M''.<ref>{{cite book|last = Rockett|first = Andrew M.|coauthors = Szüsz, Peter|title = Continued Fractions|publisher = World Scientific|date = 1992|isbn = 981-02-1052-3}}</ref><ref>For a fuller explanation of the K notation used here, please see [[Generalized continued fraction#Notation|this article]].</ref> | |
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| ==Periodic continued fractions==
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| A regular [[periodic continued fraction]] consists of a finite initial block of partial denominators followed by a repeating block; if
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| :<math>
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| \zeta = [a_0;a_1,a_2,\dots,a_k,\overline{a_{k+1},a_{k+2},\dots,a_{k+m}}],\,
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| </math>
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| then ζ is a [[quadratic irrational]] 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''<sub>0</sub> through ''a''<sub>''k''+''m''</sub>. Historically, mathematicians studied periodic continued fractions before considering the more general concept of restricted partial quotients.
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| ==Restricted CFs and the Cantor set==
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| The [[Cantor set]] is a set ''C'' of [[measure zero]] from which a complete [[interval (mathematics)|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''. | |
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| 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 [[convergent (continued fraction)|convergent]]s is maximized.
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| 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,
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| :<math>
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| \mathrm{CF}(M) = \{[0;a_1,a_2,a_3,\dots]: 1 \leq a_i \leq M \}.\,
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| </math> | |
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| By making an argument parallel to the one used to construct the Cantor set two interesting results can be obtained.
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| * 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
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| ::<math> | |
| (2\times[0;\overline{M,1}], 2\times[0;\overline{1,M}]) =
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| \left(\frac{1}{M} \left[\sqrt{M^2 + 4M} - M \right], \sqrt{M^2 + 4M} - M \right).
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| </math>
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| *A simple argument shows that <math>{\scriptstyle[0;\overline{1,M}]-[0;\overline{M,1}]\ge\frac{1}{2}}</math> holds when ''M'' ≥ 4, and this in turn implies that if ''M'' ≥ 4, every real number can be represented in the form ''n'' + CF<sub>1</sub> + CF<sub>2</sub>, where ''n'' is an integer, and CF<sub>1</sub> and CF<sub>2</sub> are elements of CF(''M'').<ref>{{cite journal|authorlink=Marshall Hall (mathematician)|last = Hall|first = Marshall|title = On the Sum and Product of Continued Fractions|journal = The Annals of Mathematics|volume = 48|issue = 4|date = October 1947|pages = 966–993|doi = 10.2307/1969389|publisher=The Annals of Mathematics, Vol. 48, No. 4|jstor=1969389}}</ref>
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| ==See also== | |
| *[[Markov spectrum]]
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| ==References==
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| {{reflist}}
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| [[Category:Continued fractions]]
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| [[Category:Diophantine approximation]]
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