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| A '''disjunctive sequence''' is an infinite [[Sequence#Infinite sequences in theoretical computer science|sequence]] (over a finite [[alphabet (computer science)|alphabet]] of [[character (computing)|characters]]) in which every [[String (computer science)#Formal theory|finite string]] appears as a [[substring]]. For instance, the binary [[Champernowne constant|Champernowne sequence]]
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| :<math>0\ 1\ 00\ 01\ 10\ 11\ 000\ 001 \ldots</math>
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| formed by concatenating all binary strings in [[shortlex order]], clearly contains all the binary strings and so is disjunctive. (The spaces above are not significant and are present solely to make clear the boundaries between strings). The [[complexity function]] of a disjunctive sequence ''S'' over an alphabet of size ''k'' is ''p''<sub>''S''</sub>(''n'') = ''k''<sup>''n''</sup>.<ref name=Bug91>Bugeaud (2012) p.91</ref>
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| Any [[normal number|normal sequence]] (a sequence in which each string of equal length appears with equal frequency) is disjunctive, but the [[Conversion (logic)|converse]] is not true. For example, letting 0''<sup>n</sup>'' denote the string of length ''n'' consisting of all 0s, consider the sequence
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| :<math>0\ 0^1\ 1\ 0^2\ 00\ 0^4\ 01\ 0^8\ 10\ 0^{16}\ 11\ 0^{32}\ 000\ 0^{64}\ldots</math>
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| obtained by splicing exponentially long strings of 0s into the [[shortlex order]]ing of all binary strings. Most of this sequence consists of long runs of 0s, and so it is not normal, but it is still disjunctive.
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| ==Examples==
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| The following result<ref>
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| {{citation
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| | last1 = Calude | first1 = C. | author1-link = Cristian S. Calude
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| | last2 = Priese | first2 = L. | author2-link = Lutz Priese
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| | last3 = Staiger | first3 = L. | author3-link = Ludwig Staiger
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| | publisher = University of Auckland, New Zealand
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| | pages = 1–35
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| | title = Disjunctive sequences: An overview
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| | url = http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.1370
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| | year = 1997 }}</ref><ref>
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| {{citation
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| | last1 = Istrate | first1 = G. | author1-link = Gabriel Istrate
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| | last2 = Păun | first2 = Gh. | author2-link = Gheorghe Păun
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| | journal = Discrete Applied Mathematics
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| | pages = 83–86
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| | title = Some combinatorial properties of self-reading sequences
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| | volume = 55
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| | doi = 10.1016/0166-218X(94)90037-X
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| | year = 1994 | zbl=0941.68656 }}</ref> can be used to generate a variety of disjunctive sequences:
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| :If ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ..., is a strictly increasing infinite sequence of positive integers such that [[Limit of a sequence|<tt>lim</tt>]] <sub>''n'' → ∞</sub> (''a''<sub>''n''+1</sub> / ''a''<sub>''n''</sub>) = 1,
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| :then for any positive integer ''m'' and any integer [[Radix|base]] ''b'' ≥ 2, there is an ''a''<sub>''n''</sub> whose expression in base ''b'' starts with the expression of ''m'' in base ''b''.
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| :(Consequently, the infinite sequence obtained by concatenating the base-''b'' expressions for ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ..., is disjunctive over the alphabet {0, 1, ..., ''b''-1}.)
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| Three simple cases illustrate this result:
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| * ''a''<sub>''n''</sub> = ''n''<sup>''k''</sup>, where ''k'' is a fixed positive integer. (In this case, <tt>lim</tt> <sub>''n'' → ∞</sub> (''a''<sub>''n''+1</sub> / ''a''<sub>''n''</sub>) = <tt>lim</tt> <sub>''n'' → ∞</sub> ( (''n''+1)<sup>''k''</sup> / ''n''<sup>''k''</sup> ) = <tt>lim</tt> <sub>''n'' → ∞</sub> (1 + 1/''n'')<sup>''k''</sup> = 1.)
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| : E.g., using base-ten expressions, the sequences
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| :: 123456789101112... (''k'' = 1, [[natural numbers|positive natural number]]s),
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| :: 1491625364964... (''k'' = 2, [[square numbers|squares]]), | |
| :: 182764125216343... (''k'' = 3, [[Cube (algebra)|cube]]s),
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| :: etc.,
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| :are disjunctive on {0,1,2,3,4,5,6,7,8,9}.
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| * ''a''<sub>''n''</sub> = ''p''<sub>''n''</sub>, where ''p''<sub>''n''</sub> is the ''n''<sup>th</sup> [[prime number]]. (In this case, <tt>lim</tt> <sub>''n'' → ∞</sub> (''a''<sub>''n''+1</sub> / ''a''<sub>''n''</sub>) = 1 is a consequence of [[Prime number theorem#Approximations for the nth prime number|''p''<sub>''n''</sub> ~ ''n'' ln ''n'']].)
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| : E.g., the sequences
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| :: 23571113171923... (using base ten),
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| :: 10111011111011110110001 ... (using base two),
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| :: etc.,
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| :are disjunctive on the respective digit sets.
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| * ''a''<sub>''n''</sub> = ''f''(''p''<sub>''n''</sub>), where ''f'' is a non-constant [[polynomial]] with [[real numbers|real]] [[coefficient]]s such that ''f''(''x'') > 0 for all ''x'' > 0.<ref>http://matwbn.icm.edu.pl/ksiazki/aa/aa81/aa8143.pdf</ref> Note that the previous example is the specific case ''f''(''x'') = ''x''.
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| : E.g., using base-ten expressions, the sequences
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| :: 23571113171923...
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| :: 492549121169289... (with ''f''(''x'') = ''x''<sup>2</sup>)
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| :: 184218051821783692... (with ''f''(''x'') = 2''x''<sup>3</sup> - 5''x''<sup>2</sup> + 11''x'' )
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| :: etc.,
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| :are disjunctive on {0,1,2,3,4,5,6,7,8,9}.
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| ==Rich numbers==
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| A '''rich number''' or '''disjunctive number''' is a [[real number]] whose expansion with respect to some base ''b'' is a disjunctive sequence over the alphabet {0,...,''b''−1}. Every [[normal number]] in base ''b'' is disjunctive but not conversely. The real number ''x'' is rich in base ''b'' if and only if the set { ''x b<sup>n</sup>'' mod 1} is [[Dense set|dense]] in the [[unit interval]].<ref name=Bug92>Bugeaud (2012) p.92</ref>
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| A number that is disjunctive to every base is called ''absolutely disjunctive'' or is said to be a ''lexicon''. A set is called "[[Residual set|residual]]" if it contains the intersection of a countable family of open dense sets. The set of absolutely disjunctive reals is a residual.<ref name=CZ1999>Calude & Zamfirescu (1999)</ref> It is conjectured that every real irrational algebraic number is absolutely disjunctive.<ref name=AB414>Adamczewski & Bugeaud (2010) p.414</ref>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{refbegin}}
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| *{{cite book | last1=Adamczewski | first1=Boris | last2=Bugeaud | first2=Yann | chapter=8. Transcendence and diophantine approximation | editor1-last=Berthé | editor1-first=Valérie | editor2-last=Rigo | editor2-first=Michael | title=Combinatorics, automata, and number theory | location=Cambridge | publisher=[[Cambridge University Press]] | series=Encyclopedia of Mathematics and its Applications | volume=135 | page=410–451 | year=2010 | isbn=978-0-521-51597-9 | zbl=pre05879512 }}
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| *{{cite book | last=Bugeaud | first=Yann | title=Distribution modulo one and Diophantine approximation | series=Cambridge Tracts in Mathematics | volume=193 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-0-521-11169-0 | zbl=pre06066616 }}
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| *{{cite journal | last1 = Calude | first1 = C.S.| author-link = Cristian S. Calude | last2 = Zamfirescu | first2 = T.
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| | issue = Supplement | journal = Publicationes Mathematicae Debrecen | pages = 619–623 | title = Most numbers obey no probability laws | volume = 54 | year = 1999 | zbl= }}
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| {{refend}}
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| {{DEFAULTSORT:Disjunctive Sequence}}
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| [[Category:Sequences and series]]
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Emilia Shryock is my title but you can call me something you like. South Dakota is where me and my spouse live. Hiring has been my profession for some time but I've currently utilized for an additional 1. Doing ceramics is what adore doing.
My blog :: home std test