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| In mathematics, a '''Beatty sequence''' (or '''homogeneous Beatty sequence''') is the sequence of integers found by taking the [[Floor and ceiling functions|floor]] of the positive [[Multiple (mathematics)|multiples]]
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| of a positive [[irrational number]]. Beatty sequences are named after [[Samuel Beatty (mathematician)|Samuel Beatty]], who wrote about them in 1926.
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| '''Rayleigh's theorem''', named after [[John William Strutt, 3rd Baron Rayleigh|Lord Rayleigh]], states that the [[complement (set theory)|complement]] of a Beatty sequence, consisting of the positive integers that are not in the sequence, is itself a Beatty sequence generated by a different irrational number.
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| Beatty sequences can also be used to generate [[Sturmian word]]s.
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| ==Definition==
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| A positive irrational number <math>r\,</math> generates the Beatty sequence
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| :<math>\mathcal{B}_r = \lfloor r \rfloor, \lfloor 2r \rfloor, \lfloor 3r \rfloor,\ldots = ( \lfloor nr \rfloor)_{n\geq 1}</math>
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| If <math>r > 1 \,,</math> then <math>s = r/(r-1)\,</math> is also a positive irrational number. They naturally satisfy
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| :<math>\frac1r + \frac1s = 1 \,</math>
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| and the sequences
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| :<math>\mathcal{B}_r = ( \lfloor nr \rfloor)_{n\geq 1}</math> and
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| :<math>\mathcal{B}_s = ( \lfloor ns \rfloor)_{n\geq 1}</math>
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| form a ''pair of complementary Beatty sequences''.
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| A more general '''non-homogeneous Beatty sequence''' takes the form
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| :<math>\mathcal{B}_r = \lfloor r+p \rfloor, \lfloor 2r+p \rfloor, \lfloor 3r+p \rfloor,\ldots = ( \lfloor nr+p \rfloor)_{n\geq 1}</math>
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| where <math>p\,</math> is a real number. For <math>p=1\,</math>, the complementary non-homogeneous Beatty sequences can be found by making <math>t = 1/r\,</math> so that
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| :<math>\mathcal{B}_r = ( \lfloor n(r+1) \rfloor)_{n\geq 1}</math> and
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| :<math>\mathcal{B}_t = ( \lfloor n(t+1) \rfloor)_{n\geq 1}</math>
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| form a pair of complementary Beatty sequences. For other values of <math>p\,</math> the procedure for finding complementary sequences is not as straightforward.{{Citation needed|date=July 2010}}
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| ==Examples==
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| For ''r'' = the [[golden ratio|golden mean]], we have ''s'' = ''r'' + 1. In this case, the sequence <math>( \lfloor nr \rfloor)</math>, known as the ''lower Wythoff sequence'', is
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| * [[1 (number)|1]], [[3 (number)|3]], [[4 (number)|4]], [[6 (number)|6]], [[8 (number)|8]], [[9 (number)|9]], [[11 (number)|11]], [[12 (number)|12]], [[14 (number)|14]], [[16 (number)|16]], [[17 (number)|17]], [[19 (number)|19]], [[21 (number)|21]], [[22 (number)|22]], [[24 (number)|24]], [[25 (number)|25]], [[27 (number)|27]], [[29 (number)|29]], ... {{OEIS|id=A000201}}.
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| and the complementary sequence <math>( \lfloor ns \rfloor)</math>, the ''upper Wythoff sequence'', is
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| * [[2 (number)|2]], [[5 (number)|5]], [[7 (number)|7]], [[10 (number)|10]], [[13 (number)|13]], [[15 (number)|15]], [[18 (number)|18]], [[20 (number)|20]], [[23 (number)|23]], [[26 (number)|26]], [[28 (number)|28]], [[31 (number)|31]], [[34 (number)|34]], [[36 (number)|36]], [[39 (number)|39]], [[41 (number)|41]], [[44 (number)|44]], [[47 (number)|47]], ... {{OEIS|id=A001950}}.
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| These sequences define the optimal strategy for [[Wythoff's game]], and are used in the definition of the [[Wythoff array]]
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| As another example, for ''r'' = [[square root of 2|√2]], we have ''s'' = 2 + √2. In this case, the sequences are
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| * 1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, ... {{OEIS|id=A001951}} and
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| * 3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, 47, 51, 54, 58, ... {{OEIS|id=A001952}}.
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| Notice that any number in the first sequence is lacking in the second, and vice versa.
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| == History ==
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| Beatty sequences got their name from the problem posed in the ''[[American Mathematical Monthly]]'' by [[Samuel Beatty (mathematician)|Samuel Beatty]] in 1926.<ref>{{cite journal
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| | author = Beatty, Samuel
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| | author2 = <!-- this comment stops Citation bot adding irrelevant authors here-->
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| | title = Problem 3173
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| | journal = [[American Mathematical Monthly]]
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| | volume = 33
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| | issue = 3
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| | year = 1926
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| | pages = 159
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| | doi = 10.2307/2300153
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| | last3 = Dunkel
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| | first3 = O.
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| | last4 = Pelletier
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| | first4 = A.
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| | last5 = Irwin
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| | first5 = F.
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| | last6 = Riley
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| | first6 = J. L.
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| | last7 = Fitch
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| | first7 = P.
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| | last8 = Yost
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| | first8 = D. M.}}</ref><ref>{{cite journal |title=Solutions to Problem 3173 |author=S. Beatty, A. Ostrowski, J. Hyslop, A. C. Aitken |journal=[[American Mathematical Monthly]] |volume=34 |year=1927 |pages=159–160 |doi=10.2307/2298716 |jstor=2298716 |issue=3}}</ref> It is probably one of the most often cited problems ever posed in the ''Monthly''. However, even earlier, in 1894 such sequences were briefly mentioned by [[John Strutt, 3rd Baron Rayleigh|John W. Strutt (3rd Baron Rayleigh)]] in the second edition of his book ''The Theory of Sound''.<ref name="Strutt1894"/>
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| ==Rayleigh theorem==
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| The '''Rayleigh theorem''' (also known as '''Beatty's theorem''') states that given an irrational number <math>r > 1 \,,</math> there exists <math>s > 1</math> so that the Beatty sequences <math>\mathcal{B}_r</math> and <math>\mathcal{B}_s</math> partition the set of positive integers: each positive integer belongs to exactly one of the two sequences.<ref name="Strutt1894">{{cite book |author=[[John William Strutt, 3rd Baron Rayleigh]] |title=The Theory of Sound |publisher=Macmillan |volume=1 |edition=Second |year=1894 |page=123 |url=http://books.google.com/books?id=EGQSAAAAIAAJ&pg=PA123}}</ref>
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| ===First proof===
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| Given <math>r > 1 \,,</math> let <math>s = r/(r-1)\,</math>. We must show that every positive integer lies in one and only one of the two sequences <math>\mathcal{B}_r</math> and <math>\mathcal{B}_s</math>. We shall do so by considering the ordinal positions occupied by all the fractions ''j''/''r'' and ''k''/''s'' when they are jointly listed in nondecreasing order for positive integers ''j'' and ''k''.
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| To see that no two of the numbers can occupy the same position (as a single number), suppose to the contrary that <math>j/r = k/s</math> for some ''j'' and ''k''. Then ''r''/''s'' = ''j''/''k'', a rational number, but also, <math>r/s = r(1 - 1/r) = r - 1,</math> not a rational number. Therefore no two of the numbers occupy the same position.
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| For any ''j''/''r'', there are ''j'' numbers ''i''/''r'' ≤ ''j''/''r'' and <math> \lfloor js/r \rfloor</math> numbers <math>k/s \le j/r</math>, so that the position of <math>j/r</math> in the list is <math>j + \lfloor js/r \rfloor</math>. The equation <math>1/r + 1/s = 1</math> implies
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| : <math>j + \lfloor js/r \rfloor = j + \lfloor j(s - 1) \rfloor = \lfloor js \rfloor.</math>
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| Likewise, the position of ''k''/''s'' in the list is <math>\lfloor kr \rfloor</math>.
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| Conclusion: every positive integer (that is, every position in the list) is of the form <math>\lfloor nr \rfloor</math> or of the form <math>\lfloor ns \rfloor</math>, but not both. The converse statement is also true: if ''p'' and ''q'' are two real numbers such that every positive integer occurs precisely once in the above list, then ''p'' and ''q'' are irrational and the sum of their reciprocals is 1.
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| ===Second proof===
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| <u>Collisions</u>: Suppose that, contrary to the theorem, there are integers ''j'' > 0 and ''k'' and ''m'' such that
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| :<math>j = \left\lfloor {k \cdot r} \right\rfloor = \left\lfloor {m \cdot s} \right\rfloor \,.</math>
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| This is equivalent to the inequalities
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| :<math>j \le k \cdot r < j + 1 \text{ and } j \le m \cdot s < j + 1. \, </math>
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| For non-zero ''j'', the irrationality of ''r'' and ''s'' is incompatible with equality, so
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| :<math>j < k \cdot r < j + 1 \text{ and } j < m \cdot s < j + 1 \,</math>
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| which lead to
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| :<math>{j \over r} < k < {j + 1 \over r} \text{ and } {j \over s} < m < {j + 1 \over s}. \,</math>
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| Adding these together and using the hypothesis, we get
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| :<math>j < k + m < j + 1 \,</math>
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| which is impossible (one cannot have an integer between two adjacent integers). Thus the supposition must be false.
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| <u>Anti-collisions</u>: Suppose that, contrary to the theorem, there are integers ''j'' > 0 and ''k'' and ''m'' such that
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| :<math>k \cdot r < j \text{ and } j + 1 \le (k + 1) \cdot r \text{ and } m \cdot s < j \text{ and } j + 1 \le (m + 1) \cdot s \,.</math>
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| Since ''j'' + 1 is non-zero and ''r'' and ''s'' are irrational, we can exclude equality, so
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| :<math>k \cdot r < j \text{ and } j + 1 < (k + 1) \cdot r \text{ and } m \cdot s < j \text{ and } j + 1 < (m + 1) \cdot s. \, </math>
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| Then we get
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| :<math>k < {j \over r} \text{ and } {j + 1 \over r} < k + 1 \text{ and } m < {j \over s} \text{ and } {j + 1 \over s} < m + 1 \,</math>
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| Adding corresponding inequalities, we get
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| :<math>k + m < j \text{ and } j + 1 < k + m + 2 \,</math>
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| :<math>k + m < j < k + m + 1 \,</math>
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| which is also impossible. Thus the supposition is false.
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| ==Properties==
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| # <math>m \in \mathcal{B}_r</math> if and only if
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| ::<math>0 \leq 1 - \frac{1}{r} \leq \left[ \frac{m}{r} \right]_1</math>
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| :where <math>[x]_1</math> denotes <math>x \mod 1</math> or the fractional part of <math>x</math> i.e., <math>[x]_1 = x - \lfloor x \rfloor</math>. Furthermore, if <math>m \in \mathcal{B}_r</math> | |
| ::<math>m = \left\lfloor \left( \left\lfloor \frac{m}{r} \right\rfloor + 1 \right) r \right\rfloor </math>
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| :<u>Proof:</u> <math>m = \left\lfloor \frac{m}{r} \right\rfloor r + \left[ \frac{m}{r} \right]_1 r = \left( \left\lfloor \frac{m}{r}\right\rfloor + 1 \right)r - \left( 1 - \left[ \frac{m}{r} \right]_1 \right)r</math>
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| :If <math>\left\lfloor \left( \left\lfloor \frac{m}{r} \right\rfloor + 1 \right) r \right\rfloor = m</math>, then <math>\left\lfloor \left( \frac{m}{r} - \left[ \frac{m}{r} \right]_1 + 1\right) r \right\rfloor = m</math>
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| :Or, <math>\left\lfloor m + \left(1 - \left[ \frac{m}{r} \right]_1 \right)r \right\rfloor = m</math> and thus, <math>0 \leq \left( 1 - \left[ \frac{m}{r} \right]_1 \right)r < 1</math> which is the given condition rearranged.
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| ===Relation with Sturmian sequences===
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| The [[difference operator|first difference]]
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| :<math>\lfloor (n+1)r\rfloor-\lfloor nr\rfloor</math>
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| of the Beatty sequence associated to the irrational number <math>r</math> is a characteristic [[Sturmian word]] over the alphabet <math>\{\lfloor r\rfloor,\lfloor r\rfloor+1\}</math>.
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| == Generalizations ==
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| The [[Lambek–Moser theorem]] generalizes the Rayleigh theorem and shows that more general pairs of sequences defined from an integer function and its inverse have the same property of partitioning the integers.
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| [[J. V. Uspensky|Uspensky's]] theorem states that, if <math>\alpha_1,\ldots,\alpha_n</math> are positive real numbers such that <math>(\lfloor k\alpha_i\rfloor)_{k,i\ge1}</math> contains all positive integers exactly once, then <math>n\le2.</math> That is, there is no equivalent of Rayleigh's theorem to three or more Beatty sequences.<ref>J. V. Uspensky, On a problem arising out of the theory of a certain game, ''Amer. Math. Monthly'' '''34''' (1927), pp. 516–521.</ref><ref>R. L. Graham, [http://www.math.ucsd.edu/~fan/ron/papers/63_01_uspensky.pdf On a theorem of Uspensky], ''Amer. Math. Monthly'' '''70''' (1963), pp. 407–409.</ref>
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| ==References==
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| {{Reflist}}
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| * {{cite journal|last = Holshouser|first = Arthur|coauthors = Reiter, Harold|title = A generalization of Beatty's Theorem|journal = Southwest Journal of Pure and Applied Mathematics|volume = 2|year = 2001|pages = 24–29|url=http://www.math.uncc.edu/preprint/2002/2002_17.pdf}}
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| * {{cite journal
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| | author = Stolarsky, Kenneth
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| | title = Beatty sequences, continued fractions, and certain shift operators
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| | journal = [[Canadian Mathematical Bulletin]]
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| | volume = 19
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| | issue = 4
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| | year = 1976
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| | mr = 0444558
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| | pages = 473–482
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| | doi = 10.4153/CMB-1976-071-6}} Includes many references.
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| ==External links==
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| * {{mathworld|title = Beatty Sequence|urlname = BeattySequence}}
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| * Alexander Bogomolny, [http://www.cut-the-knot.org/proofs/Beatty.shtml Beatty Sequences], [[Cut-the-knot]]
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| [[Category:Integer sequences]]
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| [[Category:Theorems in number theory]] <!-- nb the page [[Beatty's theorem]] redirects to this page -->
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| [[Category:Diophantine approximation]]
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| [[Category:Combinatorics on words]]
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| [[Category:Articles containing proofs]]
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