# 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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### A Question on the Brunn-Minkowski inequality

It is a direct consequence of the Brunn-Minkowski inequality that |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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### 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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### 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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### 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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### 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 ...
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 ...