Questions tagged [sumset]
For questions regarding sumsets such as $A+B$, the set of all sums of one element from $A$ and the other from $B$.
109
questions
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votes
1
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Subsets of $\mathbb Z/n\mathbb Z$ disjoint with some of its shifts
Are there any descriptions of all subsets $X$ of $\mathbb Z/n\mathbb Z$ with the following property: there exists $a\ne 0$ in $\mathbb Z/n\mathbb Z$ such that $X$ is disjoint with $X + a = \{x + a \...
- 245
2
votes
1
answer
48
views
Subsets of $\mathbb Z/n\mathbb Z$ that remain disjoint with themselves under shifts
Are there any descriptions of all subsets $X$ of $\mathbb Z/n\mathbb Z$ such that for any $a\ne 0$ in $\mathbb Z/n\mathbb Z$, $X$ is disjoint with $X + a = \{x + a \pmod n\mid x \in X\}$?
- 245
2
votes
1
answer
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views
Prove that if $|A+A| \leq K|A|$ then $2A - 2A$ is a $K^{16}$-approximate group.
Let $A$ be a finite subset of an abelian group, $G$ (call the operation addition). We say $A$ is a $K$-approximate group if:
1) $e_G \in A$
2) $A^{-1} = \{ a^{-1} \mid a \in A \} = A$
3) $\exists X ...
- 2,683
2
votes
0
answers
21
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Lower-bounding the density of 3A in terms of that of 2A
Let $A\subset\mathbb{N}$ and $2A=A+A=\{a+b \lvert a,b\in A\}$ and $3A=2A+A$. I wonder how small the density of $3A$ can be, knowing that the density of $2A$ is, say, $\beta >0$, but not knowing ...
- 21
5
votes
2
answers
497
views
Finding elements such that none add to a perfect square
Bob asks us to find an infinite set $S$ of positive integers such that the sum of any finite number of distinct elements of S is not a perfect square.
Can Bob's request be fulfilled?
I can find some ...
- 87
0
votes
0
answers
19
views
Nr of monthly eggs that have converted to chickens based on nr of months
I get 1 egg per month every month, which convert to chickens in time t (e.g 9 or 15 months) from when I got the egg with probability P. Given a specific month, e.g 97, how many chickens do I have?
- 101
1
vote
1
answer
38
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Retrieve a series knowing all its convergent infinite powersums
We would like to identify $s_n$ (non-increasing series) once we know, assuming are all of them convergent:
$$S_k=\sum_{n>=0}{s_n^k}$$
Known for all k values.
As example $\zeta(2k)$ should ...
- 623
4
votes
0
answers
149
views
Where can I find this article by I. Ruzsa?
Title says it all. Have tried googling and my college library, but no success so far.
I. Ruzsa, On the cardinality of $A + A$ and $A − A$. In Combinatorics (Keszthely, 1976), Coll. Math. Soc. Bolyai ...
- 753
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1
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If the set $A$ is open in $X$, is the set $\{x+y : x\in A \}$ also open for a given $y \in X$ under any metric space?
I think it can be shown for the Euclidean metric in an $\mathbb{R}^n$ set, since $\|x-y\|=\|(x+a)-(y+a)\|$ but is it true for any metric space? Let's say the discrete metric space?
- 93
16
votes
1
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Is the sum (difference) of Borel set with itself a Borel set?
Let $d \in \mathbb{N}$, $A \subset \mathbb{R}^d$, be a Borel set. Consider Minkowski sums
$$
\mathbb{S}(A) = A + A = \{x + y:\; x,y\in A \}
$$
$$
\mathbb{D}(A) = A - A = \{x - y:\; x,y\in A\}
$$
Must ...
- 3,666
1
vote
1
answer
33
views
number of sums in $\mathbb{Z}_{p^r}$ which are coprime to $p^r$
We look at the ring of integers modulo a prime power, say $p^r$ and $r>1$. Eulers totient formula says that there are $p^r-p^{r-1}$ elements in this ring $\mathbb{Z}_{p^r}$ that are coprime to $p^r$...
- 369
4
votes
2
answers
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Sum of two subspaces is a subspace
I am wondering if someone can check my proof that the sum of two subspaces is a subspace:
1) First show that $0 \in W_1 + W_2$:
Since $W_1, W_2$ are subspaces, we know that $0 \in W_1, W_2$.
So if $...
- 619
0
votes
3
answers
137
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Supremum of Sumset (Proof Writing)
Given $A,b\subseteq\mathbb{R}$, define the set $A+B=\lbrace a + b | a\in A, b\in B\rbrace$. I would like to prove that $\sup(A+B)=\sup(A)+\sup(B)$, but in a specific way.
Here is what I have done so ...
- 2,125
0
votes
0
answers
65
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How many consecutive numbers in a sumset?
Let $A=\{a_1,a_2,\dots,a_n\,\vert\,a_1\lt a_2\lt\cdots\lt a_n\}$ be a finite subset of $\Bbb N$ with sumset $$A+A=\{a_i+a_j\,\vert\, a_i,a_j\in A\}$$ What is the longest possible chain of consecutive ...
- 3,226
3
votes
3
answers
284
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$\lim_{n\to\infty} \sum_{k=1}^n \frac{k!}{n!}$
I'm presented with the limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k!}{n!}$
I've got a hunch that it diverges to infinity but I wasn't able the prove that the sum is superior to a series diverging ...
- 115