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.
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.
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'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]
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.
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]