Beatty sequence

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

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

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

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

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

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

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

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

The first difference

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

Generalizations

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

  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. John William Strutt, 3rd Baron Rayleigh (1894). The Theory of Sound. Vol. 1 (Second ed.). Macmillan. p. 123.
  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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