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
2
votes
2
answers
60
views
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 ...
- 971
1
vote
0
answers
41
views
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)^...
1
vote
1
answer
55
views
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 ...
- 11
0
votes
0
answers
67
views
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 \...
- 115
0
votes
2
answers
61
views
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? ...
- 117
1
vote
1
answer
100
views
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(...
- 115
0
votes
0
answers
16
views
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 ...
0
votes
1
answer
118
views
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/...
- 185
0
votes
1
answer
79
views
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 ...
- 10.4k
1
vote
0
answers
34
views
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 $...
- 4,057
4
votes
1
answer
167
views
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 ...
- 402
0
votes
1
answer
81
views
$\mathbb N-\{1\}\subseteq\mid S+S\mid$ 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: ...
- 1,190
0
votes
1
answer
53
views
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 ...
- 123
0
votes
0
answers
41
views
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
0
votes
0
answers
64
views
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 ...
- 115
0
votes
1
answer
90
views
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 $$|\{S : (S \subseteq [0,n]) \land (k \not \in S+S)\}| = 2^{|n-...
- 115
0
votes
0
answers
86
views
Instead of "sum-free" sets, consider sets where $S\subset S+S$. This is trivially satisfied when $0 \in S$. Is there a subset of $S$ with sum $0$?
As an example, here is a set $S$ with $S\subset S+S$ and $|S|=8$ and having subsets of four elements whose sum is zero, but no smaller subsets have this property: $S=\{-14,-13,-11,-7,1,2,4,8\}$. This ...
- 11
0
votes
2
answers
63
views
Sumsets : optimality of two basic inequalities
One can prove that for $A$ and $B$ subsets of $\mathbb{Z}$ one has
$$
|A|+|B|-1 \leq |A+B| \leq |A|\times |B|
$$
where $A+B = \left\{ a+b,a\in A \text{ and } b \in B \right\}$ and, for $X$ a finite ...
- 4,067
2
votes
0
answers
58
views
Which sets of $n-1$ non-multiples of $n$ can't make a multiple of $n$ using $+,-$?
This is a follow up to my previous question (see linked question).
In short, there it is shown that if $n$ is prime, then any set can make it.
I want to characterize sets $\mathbb A_n$ of multisets $...
- 12.5k
7
votes
1
answer
154
views
For any $n-1$ non-zero elements of $\mathbb Z/n\mathbb Z$, we can make zero using $+,-$ if and only if $n$ is prime
Inspired by Just using $+$ ,$-$, $\times$ using any given n-1 integers, can we make a number divisible by n?, I wanted to first try to answer a simpler version of the problem, that considers only two ...
- 12.5k
1
vote
1
answer
383
views
Minkowski sum of the intersection of a closed and an open set with a compact set
Consider $\mathbb R^n$ with the usual topology and the Borel sigma-algebra. Let $A$ be open and $B$ be closed sets, respectively, in $\mathbb R^n$. Let $C$ be a compact set. Is the set $(A \cap B) \...
- 1,091
1
vote
0
answers
77
views
Minkowski sum of disks in 3D
Suppose we have a set of disks in $\Bbb R^3$. These disks are neither parallel nor perpendicular to each other. In general, is it possible to formulate (or write an equation for) the object ...
- 43
0
votes
1
answer
36
views
Banach spaces, $\lVert\cdot\rVert_X$ and $\lVert\cdot\rVert_{X+X}$ are equivalent.
Let $(X,\lVert\cdot\rVert_X)$ and $(Y,\lVert\cdot\rVert_Y)$ be Banach spaces. Then as I understand, $X+Y$ endowed with $$\lVert v\rVert_{X+Y}=\inf\limits_{a+b=v\\ a\in X\\b\in Y}\lVert a\rVert_X+\...
- 4,256
0
votes
0
answers
56
views
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 ...
- 149
1
vote
0
answers
62
views
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.
- 11
11
votes
1
answer
374
views
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 ...
- 3,645
2
votes
2
answers
1k
views
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 ...
- 108k
3
votes
1
answer
176
views
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 ...
- 622
1
vote
2
answers
373
views
$X$ 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 ...
- 11
0
votes
1
answer
71
views
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|\}...
- 169