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 can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$
If $A,B$ non empty, upper bounded sets and $A+B=\{a+b\mid a\in A, b\in B\}$, how can I prove that $\sup(A+B)=\sup A+\sup B$?
- 711
25
votes
4
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Number of vectors so that no two subset sums are equal
Consider all $10$-tuple vectors each element of which is either $1$ or $0$. It is very easy to select a set $v_1,\dots,v_{10}= S$ of $10$ such vectors so that no two distinct subsets of vectors $S_1 \...
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answers
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Sum of closed and compact set in a TVS
I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space ...
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Is the Minkowski sum of a compact set and a closed set necessarily closed?
Why is it that if we have a compact set $X$ and a closed set $Y$ then the Minkowski sum $X+Y$ is necessarily closed?
Sorry for keep asking questions about the Minkowski sum, I am trying to figure out ...
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Is the sum (difference) of Borel set with itself a Borel set?
Let $d \in \mathbb{N}$, $A \subset \mathbb{R}^d$, be a Borel set. Consider Minkowski sums
$$
\mathbb{S}(A) = A + A = \{x + y:\; x,y\in A \}
$$
$$
\mathbb{D}(A) = A - A = \{x - y:\; x,y\in A\}
$$
Must ...
- 3,666
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answer
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Two sets $X,Y \subset [0,1]$ such that $X+Y=[0,2]$
A set $X\subset \mathbb{R}$ is called nice if, for every $\epsilon > 0$, there are a
positive integer $k$ and some bounded intervals $I_1,I_2,...,I_k$ such that
$X \subset I_1 \cup I_2 \cup \...
- 1,256
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votes
2
answers
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Measure of the Cantor set plus the Cantor set
The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets:
If $C$ ...
anonymous
14
votes
3
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The Minkowski sum of two convex sets is convex
Let $A$ and $B$ be two convex subsets in $\mathbb{R}^n$.
Define a set $C$ given by
$$C = A + B = \{a + b : a \in A \mbox{ and } b \in B\}.$$
Is $C$ a convex set?
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If $C$ is the Cantor set, then $C+C=[0,2]$.
Question : Prove that $C+C=\{x+y\mid x,y\in C\}=[0,2]$, using the following steps:
We will show that $C\subseteq [0,2]$ and $[0,2]\subseteq C$.
a) Show that for an arbitrary $n\in\mathbb{...
user345851
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answer
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If $A$ and $B$ are compact, then so is $A+B$.
This is an exercise in Chapter 1 from Rudin's Functional Analysis.
Prove the following:
Let $X$ be a topological vector space. If $A$ and $B$ are compact subsets of $X$, so is $A+B$.
My guess: ...
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Prove that the sum of two compact sets in $\mathbb R^n$ is compact.
Given two sets $S_1$ and $S_2$ in $\mathbb R^n$ define their sum by
$$S_1+S_2=\{x\in\mathbb R^n; x=x_1+x_2, x_1\in S_1, x_2\in S_2\}.$$
Prove that if $S_1$ and $S_2$ are compact, $S_1+S_2$ is ...
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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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Is the measure of the sum equal to the sum of the measures?
Let $A,B$ be subsets in $\mathbb{R}$. Is it true that
$$m(A+B)=m(A)+m(B)?$$
Provided that the sum is measurable.
I think it should not be true, but could not find a counterexample.
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In how many different from a set of numbers can a fixed sum be achieved?
I have a set of number, and I want to know in how many ways from that set with each number being used zero, once or more times can a certain sum if at all, be achieved. The order doesn't matter. For ...
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An inequality on sumsets.
Given two finite sets $A,B\subset \mathbb{R}$, can we assert the inequality $$|A+B|^2\ge |A+A|\cdot|B+B|?$$
I tried to construct an injective function from $(A+A)\times (B+B)$ to $(A+B)^2$ but failed ...
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