# 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 $B$.

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### Is this true for a sumset?

We let $A,B\subseteq \mathbb{Z}$ such that $|A|=|B|=n$. I am trying to show that $|A+B|\ge 2n-4$ for large $n$ where we define $A+B=\{a+b:a\in A, b\in B\}$. If this is not true, I'd like to see a ...
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### Generating restricted finite additive $2$-bases from doubly-eager bit-strings

A bit-string is any finite sequence of $1$s and $0$s. For example, $1011011$, $1011010$, and $000110$ are bit-strings. In this post, I will refer to bit-strings as strings, to be concise. I now ...
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### Set $A$ with $|2A| \geq 100|A|$ but $|3A| < 1000|A|$

Let $kA$ denote the sumset $\{ a_1 + \cdots + a_k \mid a_i \in A \}$. I want to show that $|2A| \geq 100|A|$ does not imply $|3A| \geq 1000|A|$. [I know this to be true experimentally, but am ...
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### Sum-free sets in finite groups

Suppose $G$ is a group, $S \subset G$. Let’s call $S$ sum-free iff $\forall a, b \in S$ we have $ab \notin S$. Do there exist such $\epsilon > 0$, such that every sufficiently large finite group $G$...
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### Two uncountable subsets of real numbers without any interval and two relations

Are there two uncountable subsets $A, B$ of real numbers such that: (1) $(A-A)\cap (B-B)=\{ 0\}$, (2) $(A-A)+B=\mathbb{R}$ or $(B-B)+A=\mathbb{R}$ ? We know that if one of them contains an interval,...
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### Sidon sets in finite groups

Suppose $G$ is a group, $S \subset G$. Let’s call $S$ a Sidon subset iff $\forall$ quadruples $(a, b, c, d)$ of distinct elements of $S$ we have $ab \neq cd$ (named after Simon Sidon who studied such ...
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### Maximal size of bounded “sparse” sets of natural numbers

Let’s call $A \subset \mathbb{N}$ sparse iff for all quadruples of distinct numbers $(a, b, c, d)$ from $A$ it is true, that $a + b \neq c + d$. What is the maximal possible size of a sparse set $A$, ...
1 vote
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### How small can "spanning sumsets" of $[n]$ be?

Let $[n]$ denote the natural numbers $1$ through $n$. Let's say a subset $X \subset [n]$ is a spanning sumset if $\{x+y: x,y \in X\} = [n] \setminus \{1\}$. I'm interested in studying spanning sumsets ...
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### Minimum number of integers $a_1,....,a_m$ needed to express $2,...,n$ as $a_i + a_j$

I am interested in the following problem. An arbitrary integer $n \geq 2$ is given. Find the minimum integer $m \geq 1$ such that there exist integers $0\lt a_1\lt a_2\lt \cdots \lt a_m$ satisfying ...
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### Following up with a previous question on $\sup(A)+\sup(B) = \sup(A + B)$

The question link is here: Prove that $Sup(A + B) = Sup(A) + Sup(B)$ Can someone look at the answer given and explain why epsilon is introduced and how that whole second part works?
1 vote
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### Prove that $Sup(A + B) = Sup(A) + Sup(B)$

Earlier on in the book it showed that to prove $a = b$ it is often best to show that $a \leq b$ and that $b \leq a$. This is the way I want to go about the proof. I am sure there is an easier way but ...
1 vote
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### Does there exist such a non-trivial semigroup $S$ ($|S| > 1$), that $S \cong Add(S)$?

Suppose $S$ is a semigroup. Define $Add(S)$ as the set of all finite subsets of $S$, equipped with the operation of pairwise “addition” ($\forall A, B \in Add(S)$, $AB = \{ab| a \in A, b \in B\}$). It ...
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### What is the probability that $\exists N \in \mathbb{N}$ such that $\forall n > N$, $2n \in C + C$?

Suppose $C$ is a random subset of $\mathbb{N}\setminus\{1, 2\}$, such that $\forall n \in \mathbb{N}\setminus\{1, 2\}$, $P(n \in C) = \frac{1}{\ln(n)}$ and the events of different numbers belonging to ...
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### 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 $X + a = \{x + a \pmod n\mid x \in X\}$?
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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 ...
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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 ...
### $\lim_{n\to\infty} \sum_{k=1}^n \frac{k!}{n!}$
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 ...