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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Regarding scaling in sumsets
Let $A$ be a finite subset of $\mathbb{N}$. We denote the set $\{a_1 +a_2: a_1, a_2\in A\}$ as $2A$. We call the quantity $\sigma[A]:= |2A|/|A|$ as the doubling constant of $A$, and this constant can ...
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A Question on the Brunn-Minkowski inequality
It is a direct consequence of the Brunn-Minkowski inequality that
\begin{equation}
|A\oplus B| - \Big(\sqrt{|A|}+\sqrt{|B|}\Big)^2 \geq |A\oplus\tilde{B}| - \Big(\sqrt{|A|}+\sqrt{|\tilde{B}|}\Big)^...
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Affine Combinations and Span
I was reading a bit of convex analysis and came across this problem.
Let $S$ be convex. Let $A$ be the set of finite affine combinations of points in $S$ (i.e. finite linear combinations whose weights ...
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What is the maximum range of a convex finite additive 2-basis of cardinality k?
Conjecture:
Given any $d \in \mathbb{Z}_{\geq 2}$ and $k=2d-2$, we have \begin{align*}
\max \{ n : (\exists &f \in \{ \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0} \})\\ &[((\forall i \in \...
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Whether $\sup(\sum\limits_{i=1}^{\infty}X_i)$ is equal to $\sum\limits_{i=1}^{\infty}(\sup X_i)$
We have $\sup(A+B)=\sup(A)+\sup(B) $.Thus, we have $\sup(\sum\limits_{i=1}^{n}X_i)=\sum\limits_{i=1}^{n}(\sup X_{i})$ for every finite integer $n\in\mathbb{N}$. However, what about the set sequence? ...
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How many subsets $S$ of integer interval $[0,n]$ such that $n, n-1 \not \in S+S$?
Conjecture:
Given any $n \in \mathbb{Z}_{\geq 0}$, we have $$|\{S : (S \subseteq [0,n]) \land (n, n-1 \not \in S+S)\}| = F(n+2),$$ where $F$, the sequence of Fibonacci numbers, is given by $F(j) = F(...
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Convexity of minkowski space when two triangles collide
I am working on a Python program to show the collision of two triangles by using Minkowski difference. I am subtracting each point from one triangle from every other point on the other triangle. The ...
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Is this Minkowski Sum result correct?
Is this Minkowski Sum result correct?
I expected a filled shape as it happens when the two polygons don't overlap (longer translation vector).
Full discussion here: https://github.com/AngusJohnson/...
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Is the sum of a closed unit ball and a closed set itself closed?
I am reading here that in a Banach space, the sum of the closed unit ball with a closed bounded convex set might fail to be closed itself. It seems there is a counterexample if and only if the ...
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Asymptotic behavior of a set of nonnegative integers whose sumset with itself is the nonnegative integers
Let $S$ be a set of nonnegative integers such that the sumset $S+S$ is the nonnegative integers. If it can, what is the fastest growing function $f$ such that the number of elements of $S$ less than $...
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General formula for $\prod_{i<j} (a_i + b_j)$
I want to expand the product of a sum into a sum of products
$$
\prod_{i<j}^n (a_i + b_j) = \sum_{\text{sets } A,B} ~ \prod_{i\in A} a_i \prod_{j\in B} b_j.
$$
With the result from this post ...
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$\mathbb N-\{1\}\subseteq\mid S+S\mid$ such that $d\left(S\right)=0$
I am dealing with the set of integers $A=\{x:x\neq i+j+2ij, x\in\mathbb N, i\in\mathbb N, j\in\mathbb N\}$. I am trying to show that $\mathbb N-\{1\}\subseteq\mid A+A\mid$, $\mid A+A\mid=\{a_i+a_j: ...
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Name for a shape consisting of the union of all spheres of a given radius centered at all points in a triangular (or tetrahedral) region?
I'm struggling with finding a name for the following object:
Suppose we have a set of points. For example a triangle, including the area inside the triangle, its edges and its vertices. We then ...
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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 ...
user931598
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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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How many subsets $S$ of integer interval $[0,n]$ such that $k \not \in S+S$?
After a bit of experimentation, I thought of the following conjecture:
Given any $n \in \mathbb{Z}_{\geq 0}$ and $k \in [0,2n]$, we have $$|\{S : (S \subseteq [0,n]) \land (k \not \in S+S)\}| = 2^{|n-...
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Instead of "sum-free" sets, consider sets where $S\subset S+S$. This is trivially satisfied when $0 \in S$. Is there a subset of $S$ with sum $0$?
As an example, here is a set $S$ with $S\subset S+S$ and $|S|=8$ and having subsets of four elements whose sum is zero, but no smaller subsets have this property: $S=\{-14,-13,-11,-7,1,2,4,8\}$. This ...
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Sumsets : optimality of two basic inequalities
One can prove that for $A$ and $B$ subsets of $\mathbb{Z}$ one has
$$
|A|+|B|-1 \leq |A+B| \leq |A|\times |B|
$$
where $A+B = \left\{ a+b,a\in A \text{ and } b \in B \right\}$ and, for $X$ a finite ...
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Which sets of $n-1$ non-multiples of $n$ can't make a multiple of $n$ using $+,-$?
This is a follow up to my previous question (see linked question).
In short, there it is shown that if $n$ is prime, then any set can make it.
I want to characterize sets $\mathbb A_n$ of multisets $...
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For any $n-1$ non-zero elements of $\mathbb Z/n\mathbb Z$, we can make zero using $+,-$ if and only if $n$ is prime
Inspired by Just using $+$ ,$-$, $\times$ using any given n-1 integers, can we make a number divisible by n?, I wanted to first try to answer a simpler version of the problem, that considers only two ...
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Minkowski sum of the intersection of a closed and an open set with a compact set
Consider $\mathbb R^n$ with the usual topology and the Borel sigma-algebra. Let $A$ be open and $B$ be closed sets, respectively, in $\mathbb R^n$. Let $C$ be a compact set. Is the set $(A \cap B) \...
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Minkowski sum of disks in 3D
Suppose we have a set of disks in $\Bbb R^3$. These disks are neither parallel nor perpendicular to each other. In general, is it possible to formulate (or write an equation for) the object ...
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Banach spaces, $\lVert\cdot\rVert_X$ and $\lVert\cdot\rVert_{X+X}$ are equivalent.
Let $(X,\lVert\cdot\rVert_X)$ and $(Y,\lVert\cdot\rVert_Y)$ be Banach spaces. Then as I understand, $X+Y$ endowed with $$\lVert v\rVert_{X+Y}=\inf\limits_{a+b=v\\ a\in X\\b\in Y}\lVert a\rVert_X+\...
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Dimension of sumset
Suppose $X$ and $Y$ are $d$-dimensional set, subsets of $\mathbf{R}^N (N>>d)$
(More precisely, my case is : $X =Y= \{vec(xy^T)|x \in R^{d_1}, y \in R^{d_2}, \|x\|\leq 1, \|y\|\leq 1\}, N=d_1 d_2 ...
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A finite Fibonacci sum
Is there a closed form for
$$
A(n)=\sum_{k=1}^n\binom{n}{k}\frac{F_k}k
$$
A closed form that is not in terms of two hypergeometric functions.
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If an infinite set $S$ of positive integers is equidistributed, is $S+S$ also equidistributed?
By $S+S$, I mean $\{x+y,$ with $x,y \in S\}$. By equidistributed, I mean equidistributed in residue classes, as defined here (the definition is very intuitive, and examples of such equidistributed ...
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Let s be a set of five positive integers at most 9. Prove that the sums of the elements in all the non empty subsets of s cannot be distinct.?
Let s be the set of five positive integer the maximum of which is at most 9 prove that the sums of the elements in all the non empty subset of as cannot be distinct?
Note:
I know this is similar to ...
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Determine the structure of all finite sets $A$ of integers such that $|A| = k$ and $|2A| = 2k + 1$.
An exercise in Nathanson's text: Additive Number Theory, Inverse problems and the geometry of sumsets is the following (Excercise 16, P.No.37):
Determine the structure of all finite sets $A$ of ...
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$X$ open, $X+Y$ also open
Question: Let:
$$X,Y \subset\mathbb{R}$$
and:
$$X+Y= \{x + y : x\in X, y \in Y\} $$
Show that if $X$ is open, then $X+Y$ is also open.
I'm not sure where to start can someone help me it would be ...
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Minimal size of a sumset over $\mathbb{F}_p$
Let $A, B \subseteq \mathbb{F}_p$ ($p$ a prime). How to show that $|A+B| \ge \min\{p, |A|+|B|-1\}$?
Since $\mathbb{F}_p$ has only $p$ elements, $\forall S \subseteq \mathbb{F}_p, |S| \ge \min\{p, |S|\}...
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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 $A$, ...
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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?
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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 ...
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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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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= ...
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Existence of a nilpotent subgroup $N \leq G$ of step $\leq n$ such that a finite $A$ is in $K^{O_n(1)}$ left cosets of $N$
Some extra details left out of the title:
Given a group $G$, a symmetric subset $A \subset G$ containing $1$ is called a $K$-approximate group if $|A^2| = |\{ab \mid a,b \in A\}| \leq K|A|$
We are ...
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Show that there exists $i\in \lbrace 1, 2, 3 \rbrace $ s.t. there exists $a, b\in A_i $ s.t. $a+b\in B $.
Let $A=\lbrace 1, 2, 3,..., 2019\rbrace= A_1\cup A_2\cup A_3$, where $A_1\cap A_2=A_2\cap A_3= A_1\cap A_3=\emptyset $ and $B=\lbrace 672, 1008, 1344, 1680, 2016\rbrace $.
Show that there exists $i\...
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Coset Progression is Freiman Isomorphic to Bohr Set
For an abelian group $G$, $H$ a finite subgroup of $G$, $x_1, \dots, x_r \in G$ and $L_1, \dots, L_r \in \mathbb N$, let:
$P(x ; L) = P(x_1, \dots, x_r ; L_1, \dots, L_r) = \{l_1x_1 + \dots + l_rx_r ...
- 2,683
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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 \...
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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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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
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0
answers
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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 ...
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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 ...
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