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.
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.
that is greater than one generates the Beatty sequence
naturally satisfy the equation
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.
, the complementary Beatty sequence is generated by
, the upper Wythoff sequence, is These sequences define the optimal strategy for Wythoff's game, and are used in the definition of the Wythoff array.
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.
partition the set of positive integers: each positive integer belongs to exactly one of the two sequences.
We must show that every positive integer lies in one and only one of the two sequences
We shall do so by considering the ordinal positions occupied by all the fractions
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
Therefore, no two of the numbers occupy the same position.
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.
Collisions: Suppose that, contrary to the theorem, there are integers j > 0 and k and m such that
For non-zero j, the irrationality of r and s is incompatible with equality, so
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
denotes the fractional part of
of the Beatty sequence associated with the irrational number
is a characteristic Sturmian word over the alphabet
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
For example, the white and black keys of a piano keyboard are distributed as such sequences for
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.
are positive real numbers such that
That is, there is no equivalent of Rayleigh's theorem for three or more Beatty sequences.