# Questions tagged [sumsets]

The sumsets tag has no usage guidance.

81
questions

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### Restricted sumsets - the origins?

The sumset of the subsets $A$ and $B$ of an additively written group is defined by $A+B:=\{a+b\colon a\in A,\ b\in B\}$. The basic idea to add sets has been around since Cauchy at least.
Erdős and ...

0
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0
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61
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### Additive energy and uniquely representable elements

Suppose that $A$ is a finite, nonempty set in an abelian group. If there is a group element with a unique representation as $a-b$ with $a,b\in A$, then none of $A-A$ and $2A$ are small:
$$ \min\{|A-A|,...

0
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1
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180
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### Controlling iterated sum sets of "most" of $A+B$

I am reading Tao-Vu book on Additive combinatorics and came across the following lemma. I know that it is better to ask this question on MathStack but I asked few questions before and no one answered ...

15
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2
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### Sets that are not sum of subsets

Let $\mathcal P$ be the set of finite subsets of $\mathbb Z_{\geq 0}$ , each of them contains $0$. We say that $A \in \mathcal P$ is indecomposable if it is not $B+C$ (the sum set of $B,C$) with $B,C\...

2
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1
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381
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### Rank of sumsets in matroids

Assume that $G$ is a (finite) abelian group and $M$ is a matroid whose ground set is $G$. Let $X$ and $Y$ be subsets of $G$, and $H$ is the stabilizer of $X+Y$. That is $X+Y+H=X+Y$. We denote the rank ...

15
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1
answer

737
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### Explicit constant in Green/Tao's version of Freiman's Theorem?

Green and Tao's version of Freiman's theorem over finite fields (doi:10.1017/S0963548309009821) is as follows:
If $A$ is a set in $\mathbb{F}_2^n$ for which $|A+A| \leqslant K|A|$, then $A$ is ...

1
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1
answer

135
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### An intriguing inverse sumset problem

Start with a natural number $k$, and choose natural numbers $K=\{n_1,\ldots,n_k\}$ which are pairwise distinct. For each $1\leq j\leq k$, choose another integer $i_j$ such that $0\leq i_j\leq n_j$.
...

6
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1
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127
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### Trisecting $3$-fold sumsets, II: is the middle part ever thin?

This is a refined version of the question I asked yesterday.
Let $A$ be a finite set of integers with the smallest element $0$ and the largest element $l$. The sumset $C:=3A$ resides in the interval $[...

6
votes

1
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146
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### Trisecting $3$-fold sumsets: is the middle part always thick?

Here is a truly minimalistic and seemingly basic question which should have a simple solution (I hope it does).
Let $A$ be a finite set of integers with the smallest element $0$ and the largest ...

2
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1
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94
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### Equal subset-sums of bounded vectors

Let $S\subseteq \{0,\ldots,n\}^d$ be a set of $d$-dimensional vectors of with bounded, natural, coordinates.
We are given that
$$v'+v_1+\ldots+v_t=u'+u_1+\ldots+u_s$$
where $v_1,\ldots,v_t,u_1,\ldots,...

1
vote

1
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188
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### Probability of getting two subsets with the same sum

Let $A=\{1,...,n\}$. Two subsets of $A$, not necessarily distinct, chosen uniformly at random. What is the probability that both subsets have the same sum? Alternatively, is there a known upper bound?

2
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### Origins of the ``baby Freiman'' theorem

It is a basic folklore fact from the area of additive combinatorics that a subset $A$ of an abelian group satisfies $|2A|<\frac32\,|A|$ if and only if $A$ is contained in a coset of a (finite) ...

3
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1
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272
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### Prime gap distribution in residue classes and Goldbach-type conjectures

Update on 7/20/2020: It appears that conjecture A is not correct, you need more conditions for it to be true. See here (an answer to a previous MO question).
The general problem that I try to solve is ...

0
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1
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376
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### Congruential equidistribution, prime numbers, and Goldbach conjecture

Let $S$ be an infinite set of positive integers, $N_S(z)$ be the number of elements of $S$ less than or equal to $z$, and let
$$D_S(z, n, p)= \sum_{k\in S,k\leq z}\chi(k\equiv p\bmod{n}).$$
Here $\chi$...

0
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0
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122
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### General asymptotic result in additive combinatorics (sums of sets)

Let $S_1,\cdots,S_k$ be $k$ infinite sets of positive integers. Let $N_i(z)$ be the numbers of elements in $S_i$ that are less or equal to $z$. Let us further assume that
$$N_i(S) \sim \frac{a_i z^{...

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1
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563
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### Prove or disprove this integral of a function, defined on a countable set with infinite limit points, converges to zero [closed]

Edit: I got rid of my old definitions. Everything should be clear now
Since no one has answered my question on MSE, I’m hoping to get an answer here. I apologize if you dislike my writing. I am an ...

5
votes

1
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135
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### Computational version of inverse sumset question

Let $p$ be prime and $\mathbb{F}_p$ the finite field with $p$ elements. Suppose we have a set $B\subseteq \mathbb{F}_p$ satisfying $|B|<p^{\alpha}$ for some $0<\alpha<1$ and there exists $A\...

3
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1
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208
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### Unique representation and sumsets

Let $A$ be a finite, nonempty subset of an abelian group, and let $2A:=\{a+b\colon a,b\in A\}$ and $A-A:=\{a-b\colon a,b\in A\}$ denote the sumset and the difference set of $A$, respectively.
If ...

1
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1
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173
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### Average size of iterated sumset modulo $p-1$,

Given a prime $p$, what is the average size of the iterated sumset, $|kA|$, modulo $p-1$, with $p$ a prime, and $k$ given, with $A$ chosen at random?
You can pick any type of prime you like for $p$, ...

1
vote

1
answer

287
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### Does $g+A\subseteq A+A$ imply $g\in A$?

Suppose that $A$ is a subset of a (large) finite cyclic group such that $|A|=5$ and $|A+A|=12$. Given that $g$ is a group element with $g+A\subseteq A+A$, can one conclude that $g\in A$?

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2
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372
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### The $r$-dimensional volume of the Minkowski sum of $n$ ($n\geq r$) line sets

Let $n$ line sets be $\mathcal{S}_i=\{a\mathbf{h}_i:0 \le a \le 1\}$, for $1 \le i \le n$, where $\{\mathbf{h}_1,\cdots,\mathbf{h}_n\}$ is a vector group of rank $r$ in the $r$-dimensional Euclidean ...

10
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1
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279
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### Freiman inequality for projective space?

This question is suggested by some results in a paper I am writing. I would like to write it down there but want to make sure that it is not known or at least MO-hard.
Freiman's inequality states ...

19
votes

4
answers

789
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### Size of sets with complete double

Let $[n]$ denote the set $\{0,1,...,n\}$. A subset $S\subseteq [n]$ is said to have complete double if $S+S=[2n]$. Let $m(n)$ be the smallest size of a subset of $[n]$ with complete double. My ...

7
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2
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410
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### Do Minkowski sums have anything like calculus?

Is there anything resembling differential calculus over the space of (nicely behaved) regions in $\mathbb{R}^d$, where addition is interpreted in terms of Minkowski sums? For example, it is known ...

3
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0
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129
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### Is there some sort of formula for $t(S_n)$?

Let $G$ be a finite group. Define $t(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > t(G)$ and $\langle X \rangle = G$, then $XXX = G$.
Is there some sort of formula for $t(S_n)...

2
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0
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37
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### Weighted unrestricted Golomb rulers?

A set of integers
${\displaystyle A=\{a_{1},a_{2},...,a_{m}\}\quad a_{1}<a_{2}<...<a_{m}} $
is a Golomb ruler if and only if
${\displaystyle \forall i,j,k,l\in \left\{1,2,...,m\right\},a_{i}...

6
votes

1
answer

247
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### Is there some sort of formula for $\tau(S_n)$?

Let $G$ be a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$.
Is there some sort of formula for $\tau(S_n)$, ...

0
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0
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49
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### Rewriting a set of integers to get rid of repetition but keeping subset sum ordering

Say, I have a set of 6 +ve integers sorted in ascending order:
$A = \{2,4,4,4,5,7\}$
Now to make it easier to deal with (Minimum one starts with 1) I deducted one from all of them:
$\therefore B= ...

1
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0
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159
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### Generating Subsets of a Multiset in Ascending Order of the Sums of the Elements of the Subset

I am trying to come up with an algorithm where you can generate combination from a set in a order such that their sums are in increasing order. This set has to be a multiset i.e. repetition allowed.
...

5
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1
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591
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### Minkowski sum of polytopes from their facet normals and volumes

By Minkowski's work in the early 1900s, every polytope $P\subset\mathbb R^n$ is determined up to translation by its unit facet normals $u_1,\dots,u_k$ and facet volumes $\alpha_1,\dots,\alpha_k$. ...

3
votes

1
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135
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### Bounding the size of certain sumsets in the plane

Let $A$ be a finite set in $\mathbb{R}^2$ of $k^2$ elements and consider a set $B=\{x_1,x_2,x_3,x_4\}$ such that the points in $B$ are in general position (no three points on a line).
Question 1: Is ...

3
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1
answer

210
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### On particular sumset properties of permanent?

Denote $\mathcal R_2[n]=\mathcal R[n] + \mathcal R[n]$ to be sumset of integers in $\mathcal R[n]$ where $\mathcal R[n]$ to be set of permanents possible with permanents of $n\times n$ matrices with $...

3
votes

1
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240
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### Are there unique additive decompositions of the reals?

Given $b\in \mathbb{R}_{>1}$ is there $U\subseteq\mathbb{R}_{\ge 0}$ such that $U+bU=\mathbb{R}_{\ge 0}$ and $(U-U)\cap b(U-U)=\{0\}$ (or equivalently: $u+bv=u'+bv' \implies u=u', v=v'$)?
Here is ...

4
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0
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### Dividing a finite arithmetic progression into two sets of same sum: always the same asymptotics?

This is inspired by the recent question How many solutions $\pm1\pm2\pm3…\pm n=0$.
The oeis entries A063865 linked to this question and A292476/A156700 for the related one "How many solutions $\pm1\...

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3
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898
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### Decomposing a finite group as a product of subsets

My friend Wim van Dam asked me the following question:
For every finite group $G$, does there exist a subset $S\subset G$ such that $\left|S\right| = O(\sqrt{\left|G\right|})$ and $S\times S = G$? ...

2
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0
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### The set of lengths of $nX$ gets larger and larger for every non-zero, non-empty, finite $X \subseteq \mathbf N$ with $0 \in X$

Let $H$ be a multiplicatively written monoid with identity $1_H$. Given $x \in H$, we take ${\sf L}_H(x) := \{0\}$ if $x = 1_H$; otherwise, ${\sf L}_H(x)$ is the set of all $k \in \mathbf N^+$ for ...

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### sum-sets in a finite field

Let $\mathbb{F}_p$ be a finite field, $A=\{a_1,\dots,a_k\}\subset\mathbb{F}_p^*$ a $k$-element set, for $k<p$. $\mathfrak{S}_k=$permutation gp.
Question. Is it true there is always a $\pi\in\...

3
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0
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### What's known about $X$ when $|X(n) + X(n)| < kn$, $n \in \mathbb{N}$, absolute constant $k$?

Let $X$ be an infinite sequence of integers$$x_1 < x_2 < x_3 < \ldots,$$and let $X(n)$ be the set$$\{x_1, x_2, \ldots, x_n\}.$$
Question. What is known about $X$ when we have$$|X(n) + X(n)| &...

3
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1
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### Limit measuring failure of sum-set cancellability

Suppose $A$, $B$ are finite sets of positive integers.
Let $$\mathcal{S}_n = \{C \subset [1,n] \, : \, A+C = B+C \}, $$ and denote $a_n = |\mathcal{S}_n|$.
Note that for any $X \in \mathcal{S}_n$ ...

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0
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### Lower bound for sumset in discrete cube

Suppose $A\subset\{0,1\}^d$ for some $d\geq 1$. Then how large must $A+A=\{a+b:a,b\in A\}$ be?

10
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### what is the status of this problem? an equivalent formulation?

R. Guy, Unsolved problems in number theory, 3rd edition, Springer, 2004.
In this book, on page 167-168, Problem C5, Sums determining members of a set, discusses a question Leo Moser asked: suppose $X\...

4
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2
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### How big must the sumset $A+A$ be if $A$ satisfies no translation-invariant equations of low height?

Suppose $A$ is a finite subset of an abelian group. If there is no solution to $ma+nb=(m+n)c$ with $0\leq m,n\leq M$, can we bound $|A+A|$ from below? I am interested if one can obtain bounds much ...

4
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0
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### Restricted addition analogue of Freiman's $(3n-4)$-theorem

There is a well-known theorem of Freiman saying that if $A$ is a finite set of integers with $|2A| \le 3|A|-4$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms. Is ...

2
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1
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### When does the equality hold in Dias da Silva - Hamidoune Theorem?

Let $p$ be prime number and let $A$ be a $k$-elements subset of $\mathbb{Z}/p\mathbb{Z}$. Dias da Silva - Hamidoune Theorem states that $|h^{\hat{}}A| \geq \min(p, hk -h^2 + 1)$, where $h$ is an ...

1
vote

1
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164
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### Sumset achieving extreme upper bound [closed]

It is trivial that $|A_1 + \cdots + A_h| \leq |A_1|\cdots |A_h|$, where $h \geq 2$ and $A_i \subseteq \mathbb{Z}$ are nonempty finite sets and $A_1 + \cdots + A_h :=\{a_1 + \cdots + a_h : a_i \in A_i ~...

1
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1
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184
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### Some question on haar measure for sumsets of closed subsets of profinite groups

Let $H$ be a profinite group with the haar measure $\mu_H$. Let $H_1$ and $H_2$ be closed subgroups of $H$. $H_1$ and $H_2$ have their own haar measures $\mu_{H_1}$ and $\mu_{H_2}$ respectively.
...

8
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1
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### Does $|A+A|$ concentrate near its mean?

Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My ...

2
votes

4
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544
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### Show that the Minkowski sum of two triangles in 3D is the union of Minkowski sums of each triangle along the other's edges?

I'd like to show (or disprove) the claim that the Minkowski sum of two triangles with vertices in $\mathbb{R}^3$, $A+B$, is equal to the union of the unions of the Minkowski sums of $A$ along all ...

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2
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240
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### Partition regular systems: do they have solution in (very dense) set of integers?

A partition regular system is a linear system of equations of the form $A\cdot x=0$, which satisfies a Ramsey-type result (namely, that for each $r>0$ whenever we colour the integers in $r$ classes,...

3
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2
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389
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### Sumsets with distinct numbers, upper bound for maximum element

Let $A$ be a finite set of positive natural numbers with $n$ elements, $|A|=n$, with the property that all sums of two (not necessarily different) elements are distinct, or in the usual notation for ...