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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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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