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 ...
- 971
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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 \...
- 115
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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(...
- 115
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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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