In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926.

Rayleigh's theorem, named after Lord Rayleigh, states that the 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.

Beatty sequences can also be used to generate Sturmian words.

Definition

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Any irrational number   that is greater than one generates the Beatty sequence   The two irrational numbers   and   naturally satisfy the equation  . The two Beatty sequences   and   that they generate form a pair of complementary Beatty sequences. Here, "complementary" means that every positive integer belongs to exactly one of these two sequences.

Examples

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When   is the golden ratio  , the complementary Beatty sequence is generated by  . In this case, the sequence  , known as the lower Wythoff sequence, is

1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, ... (sequence A000201 in the OEIS),

and the complementary sequence  , the upper Wythoff sequence, is

2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, ... (sequence A001950 in the OEIS).

These sequences define the optimal strategy for Wythoff's game, and are used in the definition of the Wythoff array.

As another example, for the square root of 2,  ,  . In this case, the sequences are

1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, ... (sequence A001951 in the OEIS), and
3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, 47, 51, 54, 58, ... (sequence A001952 in the OEIS).

For   and  , the sequences are

3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, ... (sequence A022844 in the OEIS), and
1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, ... (sequence A054386 in the OEIS).

Any number in the first sequence is absent in the second, and vice versa.

History

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Beatty sequences got their name from the problem posed in The American Mathematical Monthly by Samuel Beatty in 1926.[1][2] 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 Lord Rayleigh in the second edition of his book The Theory of Sound.[3]

Rayleigh theorem

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Rayleigh's theorem (also known as Beatty's theorem) states that given an irrational number   there exists   so that the Beatty sequences   and   partition the set of positive integers: each positive integer belongs to exactly one of the two sequences.[3]

First proof

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Given   let  . We must show that every positive integer lies in one and only one of the two sequences   and  . We shall do so by considering the ordinal positions occupied by all the fractions   and   when they are jointly listed in nondecreasing order for positive integers j and k.

To see that no two of the numbers can occupy the same position (as a single number), suppose to the contrary that   for some j and k. Then   =  , a rational number, but also,   not a rational number. Therefore, no two of the numbers occupy the same position.

For any  , there are   positive integers   such that   and   positive integers   such that  , so that the position of   in the list is  . The equation   implies  

Likewise, the position of   in the list is  .

Conclusion: every positive integer (that is, every position in the list) is of the form   or of the form  , 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.

Second proof

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Collisions: Suppose that, contrary to the theorem, there are integers j > 0 and k and m such that   This is equivalent to the inequalities  

For non-zero j, the irrationality of r and s is incompatible with equality, so   which leads to  

Adding these together and using the hypothesis, we get   which is impossible (one cannot have an integer between two adjacent integers). Thus the supposition must be false.

Anti-collisions: Suppose that, contrary to the theorem, there are integers j > 0 and k and m such that  

Since j + 1 is non-zero and r and s are irrational, we can exclude equality, so  

Then we get  

Adding corresponding inequalities, we get    

which is also impossible. Thus the supposition is false.

Properties

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A number   belongs to the Beatty sequence   if and only if   where   denotes the fractional part of   i.e.,  .

Proof:            

Furthermore,  .

Proof:          

Relation with Sturmian sequences

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The first difference   of the Beatty sequence associated with the irrational number   is a characteristic Sturmian word over the alphabet  .

Generalizations

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If slightly modified, the Rayleigh's theorem can be generalized to positive real numbers (not necessarily irrational) and negative integers as well: if positive real numbers   and   satisfy  , the sequences   and   form a partition of integers. For example, the white and black keys of a piano keyboard are distributed as such sequences for   and  .

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.

Uspensky's theorem states that, if   are positive real numbers such that   contains all positive integers exactly once, then   That is, there is no equivalent of Rayleigh's theorem for three or more Beatty sequences.[4][5]

References

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  1. ^ Beatty, Samuel (1926). "Problem 3173". American Mathematical Monthly. 33 (3): 159. doi:10.2307/2300153. JSTOR 2300153.
  2. ^ S. Beatty; A. Ostrowski; J. Hyslop; A. C. Aitken (1927). "Solutions to Problem 3173". American Mathematical Monthly. 34 (3): 159–160. doi:10.2307/2298716. JSTOR 2298716.
  3. ^ a b John William Strutt, 3rd Baron Rayleigh (1894). The Theory of Sound. Vol. 1 (Second ed.). Macmillan. p. 123.{{cite book}}: CS1 maint: numeric names: authors list (link)
  4. ^ J. V. Uspensky, On a problem arising out of the theory of a certain game, Amer. Math. Monthly 34 (1927), pp. 516–521.
  5. ^ R. L. Graham, On a theorem of Uspensky, Amer. Math. Monthly 70 (1963), pp. 407–409.

Further reading

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