Talk:Beatty sequence

Merge with Beatty's theorem
Can't rate this yet. It needs to merge or link with Beatty's theorem. Geometry guy 14:28, 13 May 2007 (UTC)
 * I went ahead and did the merge — it didn't seem like there was any need for discussion on whether the two should be merged, since there was such significant overlap of content. —David Eppstein 15:14, 13 May 2007 (UTC)

I agree - thanks for doing the work. And thanks also for correcting and adding to some of my other ratings as well! Geometry guy 16:22, 13 May 2007 (UTC)

Start with simple definition of sequence itself?
From what I've read, a Beatty sequence is the spectrum sequence for any irrational number base. This starts out describing his theorem and sort of implies that the base of any Beatty sequence must satisfy 1/r + 1/s = 1. Even if this is the case, this article should probably start out with straightforward definition of the sequence and then go into the theorem. Nonenmac (talk) 01:35, 18 September 2008 (UTC)

Correctness of non-homogeneous Beatty sequence?
Currently the main article says:


 * A more general non-homogeneous Beatty sequence takes the form


 * $$\mathcal{B}_r = \lfloor r+p \rfloor, \lfloor 2r+p \rfloor, \lfloor 3r+p \rfloor,... = ( \lfloor nr+p \rfloor)_{n\geq 1}$$


 * where $$p\,$$ is a real number. For $$p=1\,$$, the complementary non-homogeneous Beatty sequences can be found by making $$t = 1/r\,$$ so that


 * $$\mathcal{B}_r = ( \lfloor n(r+1) \rfloor)_{n\geq 1}$$ and
 * $$\mathcal{B}_t = ( \lfloor n(t+1) \rfloor)_{n\geq 1}$$

Shouldn't be the sequence rather this one?:
 * $$\mathcal{B}_r = \lfloor r+p \rfloor, \lfloor 2(r+p) \rfloor, \lfloor 3(r+p) \rfloor,... = ( \lfloor n(r+p) \rfloor)_{n\geq 1}$$

So for p=1 we have indeed


 * $$\mathcal{B}_r = ( \lfloor n(r+1) \rfloor)_{n\geq 1}$$ and
 * $$\mathcal{B}_t = ( \lfloor n(t+1) \rfloor)_{n\geq 1}$$

PhilippeTeuwen (talk) 14:31, 21 December 2008 (UTC)

"Complementary"?
I'm very much bothered by the phrase "complement of a (or complementary) Beatty sequence". In the current version of the page we say that the complement of a sequence is the set of numbers not in that sequence. Then we give a theorem to the effect that given a sequence, every integer is either in the sequence or in its complement. It's not exactly a deep theorem as stated, and it's not what's meant.

Of course we could agree to define the complement of the sequence for r as being the sequence for s where
 * $$\frac1r + \frac1s = 1. \,$$

In that case, Rayleigh's theorem actually has some content, but the mention of "complement" in the introductory paragraph becomes confusing. Searching through the literature, I haven't yet found a mention of "complementary Beatty sequences" outside Wikipedia. MathSciNet appears to know nothing about this usage of the word "complement".

I'd prefer to rewrite the whole article without using the word "complement" at all. Rayleigh's theorem could be stated in the form "Given irrational numbers r and s satisfying
 * $$\frac1r + \frac1s = 1, \,$$

the Beatty sequences for r and s partition the positive integers." Then there's no tautology and no room for confusion. Jowa fan (talk) 05:30, 1 November 2010 (UTC)
 * You seem to be missing the point of the complementary sequence results. The point is that the set-theoretic complement of a Beatty sequence is always, itself, a Beatty sequence (exactly as the lede says). To see that this is non-tautological, consider e.g. the case of an infinite arithmetic progression of integers: it is not true that the complement of an arithmetic progression is itself an arithmetic progression. For instance, the progression ...-6,-3,0,3,6,... has as its complement the set ...-5,-4,-2,-1,1,2,4,5,... which is not an arithmetic progression. "Complementary" is simply the adjectival form of the word complement. —David Eppstein (talk) 05:52, 1 November 2010 (UTC)
 * No, I did figure out what the point is, after some effort. What I'm telling you is that the article as it's currently structured makes this difficult for the reader.  Whichever way I look at it, I can't see how the sentence "a pair of complementary Beatty sequences partition the set of positive integers" fails to be a tautology.  If complementary Beatty sequences do exist, they can't fail to have this property.  The given proof of the theorem is in fact a proof of existence of complementary Beatty sequences.  The problem is that the lead paragraph already tells us that the complement of a Beatty sequence is another Beatty sequence.  (And mentioning it casually in this manner could leave the reader with the impression that it's supposed to be an obvious fact, not a theorem requiring proof).  Then the "definition" gives an explicit way of finding the complement of a Beatty sequence.  So that when we get to the paragraph headed "Rayleigh theorem", we're told nothing new.  I think the exposition can be improved. Jowa fan (talk) 01:49, 2 November 2010 (UTC)
 * The lead section is supposed to be a summary of the rest of the article; see MOS:LEAD. So since the most important fact about Beatty sequences is the fact that a complement of one of these sequences is another one of these sequences, it should be mentioned briefly in the lead section, as it is. But if you think there's a way of rewriting the rest of the article so that this fact appears less tautological when it's expressed again in more detail, you're welcome to try. —David Eppstein (talk) 06:40, 2 November 2010 (UTC)
 * Ah, it makes sense now! I hadn't thought carefully about the function of the lead section--in particular the requirement that it should stand alone as a summary. Given this context, your point of view is a lot clearer to me.  Thanks for taking the time to explain.  I still feel that some revision is necessary, but will give it further thought before editing again. Jowa fan (talk) 08:25, 2 November 2010 (UTC)

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Theorem needs to be indicated clearly
The section Definition contains this passage:

"The two Beatty sequences $$\mathcal{B}_r$$ and $$\mathcal{B}_s$$ 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."

it is a theorem that "Every positive integer belongs to exactly one of these two sequences".

But the fact that it is a theorem is left unmentioned, and unproved.

I hope someone knowledgeable on this subject can fill in what is missing here. 2601:200:C000:1A0:1CEE:3A1F:3D17:EF53 (talk) 16:04, 8 October 2022 (UTC)