Talk:Schnirelmann density

Waring's problem
In the section on Waring's problem we find this statement:


 * $$ r_N^k(n) \leftrightarrow n \in N\mathfrak{G}^k, $$

I think this should be $$ r_N^k(n) > 0 \leftrightarrow n \in N\mathfrak{G}^k, $$, assuming that $$ N\mathfrak{G}^k $$ means the N-fold set sum of $$\mathfrak{G}^k$$. As it stands I cannot make sense of the statement, as $$r_N^k(n)$$ is a number, not a statement. Molinari 00:56, 19 October 2005 (UTC)


 * I agree - it only makes sense for a statement and not a number, so I have changed it. Madmath789 07:39, 28 June 2006 (UTC)

more help needed
Hello! The sharp symbol "#" is used here but not listed in in the Mathematical notation article.

Where should one find an explanation for that symbol (I ignore its meaning), and where this question must be posted for a quick answer ?

Thnks, --DLL 20:00, 27 June 2006 (UTC)


 * The symbol "#" is used in front of a set to indicate the number of elements in the set, so #{2,4,6,8} = 4. In the article, $$\#(A \cap \{1, 2, \ldots n\})$$ is the number of elements of A which are in the set {1, 2, 3, ..., n}. Hope that helps! Madmath789 21:00, 27 June 2006 (UTC)


 * Thank you! The table used in Table of mathematical symbols is quite complex and I do not want to break it. I'll add a comment for some help there with your indications. --DLL 17:38, 28 June 2006 (UTC)


 * That's the first time I've ever seen #. Why not go with the standard $$|A \cap \{1, 2, \ldots n\}|$$ set cardinality operator here? TomJF 06:02, 5 September 2006 (UTC)

Numbered theorems?
The section on Mann's theorem refers to Theorem 1 and Theorem 1.1, but there are no numbered theorems in this article. Which theorems are they? Ntsimp 21:30, 4 October 2007 (UTC)

Positive vs. non-negative
Schnirelmann density should properly refer to sets of non-negative integers. Otherwise, the sumset of two subsets of the natural numbers always has Schnirelmann density zero (because it does not contain 1). Unfortunately, I'm not sure where in the definitions this should be fixed. —Preceding unsigned comment added by 71.121.232.16 (talk) 01:38, 10 July 2008 (UTC)
 * Cojocaru and Murty define the concept for sequences of non-negative integers but the definition only involves counting the number of positive integers in the sequence. This is a little obscure but better than Schnirelmann's original paper which has to define the sumset awkwardly instead. Richard Pinch (talk) 06:10, 5 August 2008 (UTC)

Changes made on 14th October 2010
Hi,

I was the person making the changes on the 14th October. I do not understand why this was then reverted without discussion, which as I understand it is against Wikipedia policy. My changes were made to address the many problems in this article, which as it stands does Wikipedia a disservice. I have no intention of going in and starting an edit or revert war but something does need to be done.

Here is a list of at least some of the problems I attempted to address. Note that previous contributors to this discussion page, going back several years, had some similar concerns but nothing had been done about it.

There are two important definitions which have to be understand for this discussion. First N is used in the current article. This is referred to the Wikipedia entry on the natural numbers. There they are described as being the set {1,2,3,...} but it is also observed that some authors include 0. From the wording I infer that the Wikipedia preference is for {1,2,3,...} and that is what I will suppose for the discussion. My changes were intended to make this clear independent of the readers assumptions about N. Second the symbol \oplus is used and the reader is referred to the Wikipedia article on sumset. There the symbol \oplus is not used but A+B is defined to be the set of numbers of the form a+b with a in A and b in B. From my long experiece in this area this is the normal definition and mostly + is used in place of \oplus. This is the definition of \oplus I will assume for the discussion and is the one most readers will assume, I believe. I would prefer + as this is the standard useage but do not feel strongly about that. With these definitions most of the theorems in the article are false. The point is that if A and B are subsets of N then the least element of A\oplus B is at least 2 and the Schnirelman density of A\oplus B will always be 0! The usual way around this is to allow sets of integers rather than subsets of N, but to only count the positive elements in the defintion of Schnirelman density, and in each of the theorems to assume that 0 is in at least one of the sets. My changes attempted to address this problem and to bring the article into line with standard useage in the area. There is a possible alternative definition of A\oplus B as the set of everything of the form {a+b: a in A, b in B} union A union B which would work for most of the theorems. Unfortunately this is not the standard way and anyway one couldn't use it in Schnirelman's original theorem as, in its original form at least, this is concerned with {a+b: a in A, b in B} union B. There is also an apparent inconsistency in the section on essential components where + is used in place of \oplus.

The problem of not including 0 is also a concern for the paragraph on Lagrange's theorem since as stated the smallest number represented is 4.

The section on Waring's problem also needs a correction. Of course it is in general false that R_N^k(n)=n^{N/k}. The left hand side is an integer but n^{N/k} is not usually an integer. Indeed it is not even a good approximation. The volume of the N-dimensional region described there is actually n^{N/k} multiplied by a ratio of gamma factors - in fact a generalised beta integral and the factor is smaller than 1 when N>1. Presumably the point is to indicate the order of magnitude and probably the simplest upper bound to illustrate this is (1+n^{1/k})^N.

There are two classic expositions of Schnirelman density. One by Mann is listed, but would be hard reading for someone trying to learn more. Mann does things in great generality and concentrates on explicit estimates rather than the consequences for Schnirelman density. On the other hand the relevant chapter in Halberstam and Roth's book Sequences is a masterly exposition and might still perhaps be the resource of choice for anyone going significantly further.

I am happy to discuss further the changes I proposed on this discussion page. Anyone wishing to do so, please alert me at

rvaughan@math.psu.edu

Bob Vaughan

Rvaughan2000 (talk) 20:58, 16 October 2010 (UTC)


 * I noticed the same problem yesterday and I corrected it (at least that part) changing the definition of $$A\oplus B$$ (In his paper Mann basically defines C that way), without realising there had been discussions on that. I didn't read the whole article, so I'm not sure if there are other problems but I'm sure Vaughan prefectly knows what changes need to be made, so I'd restore his version.--Sandrobt (talk) 17:22, 18 October 2010 (UTC)
 * Ok, I read everything you wrote just now, and I totally agree, so if there are no oppositions I'll restore Vaughan version. The only thing I don't get is why the definition of $$A\oplus B=(A\cup\{0\})+(B\cup\{0\})$$ doesn't work, but anyway since you said this is not the standard notation is surely better to avoid it.--Sandrobt (talk) 17:38, 18 October 2010 (UTC)

Excellent. When I have a moment I will adjust the part on Waring's problem and add the reference to Halberstam and Roth. Schnirelman's orginal theorem is asymmetric in that he basically proves the bound for $$A+(B\cup\{0\})$$ (and then he needs 1 to be in A) not as stated. However the result as stated is correct, of course, just not what was proved, but in order to avoid too much confusion it is perhaps best left as it now is.

There is also a simple elementary proof of the alpha+beta theorem in Niven, Zuckerman and Montgomery and I will add a reference to that.

Bob Vaughan

146.186.134.75 (talk) 14:47, 20 October 2010 (UTC)

Examples, please
Could someone provide a few examples. The only ones explicitly mentioned are the sets
 * A \ 1 = {2, 3, 4, 5, ...}, and
 * A \ 2 = {1, 3, 4, 5, ...}.

How about showing more examples, such as
 * AE = {2, 4, 6, 8, 10, ...}
 * AO = {1, 3, 5, 7, 9, ...}
 * Asq = {1, 4, 9, 16, 25, ...}
 * A10 = {1, 10, 100, 1000, ...}

— Loadmaster (talk) 01:53, 7 March 2011 (UTC)

add a little description
add a little description written on *--Enyokoyama (talk) 08:52, 22 February 2014 (UTC)