Integers formed by rounding down the integer multiples of a positive irrational number
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
![{\displaystyle {\mathcal {B}}_{r}={\bigl \{}\lfloor r\rfloor ,\lfloor 2r\rfloor ,\lfloor 3r\rfloor ,\ldots {\bigr \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/822046b92e5479660217fcf0d69d3ca330fdf032)
The two irrational numbers
![{\displaystyle r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
and
![{\displaystyle s=r/(r-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd9d227c9f8d65cbcaa1e5f58f10e51163fc403)
naturally satisfy the equation
![{\displaystyle 1/r+1/s=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/265697a65a7019f95495b4e4383947f4bcb3948b)
. The two Beatty sequences
![{\displaystyle {\mathcal {B}}_{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f43be8c3e275ba20820ca853d754407b515238cb)
and
![{\displaystyle {\mathcal {B}}_{s}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d76e50cc4e55a8805bb80075bbddb3e9b1ff665b)
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
![{\displaystyle j+\lfloor js/r\rfloor =j+\lfloor j(s-1)\rfloor =\lfloor js\rfloor .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce353f0e6c164a83913fbb01a7261c23f08473b9)
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
![{\displaystyle j=\left\lfloor {k\cdot r}\right\rfloor =\left\lfloor {m\cdot s}\right\rfloor \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef7358de0db61c08072093cf058a1c8fb4d78de1)
This is equivalent to the inequalities
![{\displaystyle j\leq k\cdot r<j+1{\text{ and }}j\leq m\cdot s<j+1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf8733196bee3c3137c1495e452cc7815289b2d)
For non-zero j, the irrationality of r and s is incompatible with equality, so
![{\displaystyle j<k\cdot r<j+1{\text{ and }}j<m\cdot s<j+1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec0cfdeea8171ce1e0c93ce4b18528ade4aa198b)
which leads to
![{\displaystyle {j \over r}<k<{j+1 \over r}{\text{ and }}{j \over s}<m<{j+1 \over s}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/baed4778475f414c7611c5ae11b29b0a2463f04d)
Adding these together and using the hypothesis, we get
![{\displaystyle j<k+m<j+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09922e209ad63746f32a3710e57d2e09a866c8b6)
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
![{\displaystyle k\cdot r<j{\text{ and }}j+1\leq (k+1)\cdot r{\text{ and }}m\cdot s<j{\text{ and }}j+1\leq (m+1)\cdot s\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b573de991b3f0552d519b1ada6075e128ea19180)
Since j + 1 is non-zero and r and s are irrational, we can exclude equality, so
![{\displaystyle 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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51ef5369c6bea5efa84a4df124027affea0e2704)
Then we get
![{\displaystyle 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76d35819de870a917399a3a6dccef006bb8931cb)
Adding corresponding inequalities, we get
![{\displaystyle k+m<j{\text{ and }}j+1<k+m+2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06f0e55999aee7e77fdd14fb0a08194f0dac67d2)
![{\displaystyle k+m<j<k+m+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f907561b6f5509589f00bdcaf3da3bef9f0bcbc8)
which is also impossible. Thus the supposition is false.
Properties
A number
belongs to the Beatty sequence
if and only if
![{\displaystyle 1-{\frac {1}{r}}<\left[{\frac {m}{r}}\right]_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/189cbb7035d1de043001f96392ebf406596fb41f)
where
![{\displaystyle [x]_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aed2a4c5d3c5b46012f671e4226dfd6ac574b87e)
denotes the fractional part of
![{\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
i.e.,
![{\displaystyle [x]_{1}=x-\lfloor x\rfloor }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5414db00b1ccbafd37539129bd588397d7f0438d)
.
Proof:
Furthermore,
.
Proof:
Relation with Sturmian sequences
The first difference
![{\displaystyle \lfloor (n+1)r\rfloor -\lfloor nr\rfloor }](https://wikimedia.org/api/rest_v1/media/math/render/svg/038d09ff4251cec878056b28da53f5e63c5d3fe3)
of the Beatty sequence associated with the irrational number
![{\displaystyle r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
is a characteristic
Sturmian word over the alphabet
![{\displaystyle \{\lfloor r\rfloor ,\lfloor r\rfloor +1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15d5b610084c70d8842b3ee0a48f920afb07f02b)
.
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
- ^ Beatty, Samuel (1926). "Problem 3173". American Mathematical Monthly. 33 (3): 159. doi:10.2307/2300153. JSTOR 2300153.
- ^ 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.
- ^ 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) - ^ J. V. Uspensky, On a problem arising out of the theory of a certain game, Amer. Math. Monthly 34 (1927), pp. 516–521.
- ^ R. L. Graham, On a theorem of Uspensky, Amer. Math. Monthly 70 (1963), pp. 407–409.
Further reading
- Holshouser, Arthur; Reiter, Harold (2001). "A generalization of Beatty's Theorem". Southwest Journal of Pure and Applied Mathematics. 2: 24–29. Archived from the original on 2014-04-19.
- Stolarsky, Kenneth (1976). "Beatty sequences, continued fractions, and certain shift operators". Canadian Mathematical Bulletin. 19 (4): 473–482. doi:10.4153/CMB-1976-071-6. MR 0444558. Includes many references.
External links