Questions tagged [sumsets]
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108 questions
12
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The $r$-dimensional volume of the Minkowski sum of $n$ ($n\geq r$) line sets
Let $n$ line sets be $\mathcal{S}_i=\{a\mathbf{h}_i:0 \le a \le 1\}$, for $1 \le i \le n$, where $\{\mathbf{h}_1,\cdots,\mathbf{h}_n\}$ is a vector group of rank $r$ in the $r$-dimensional Euclidean ...
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1
answer
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Freiman inequality for projective space?
This question is suggested by some results in a paper I am writing. I would like to write it down there but want to make sure that it is not known or at least MO-hard.
Freiman's inequality states ...
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8
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2
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596
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Do Minkowski sums have anything like calculus?
Is there anything resembling differential calculus over the space of (nicely behaved) regions in $\mathbb{R}^d$, where addition is interpreted in terms of Minkowski sums? For example, it is known ...
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19
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3
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Decomposing a finite group as a product of subsets
My friend Wim van Dam asked me the following question:
For every finite group $G$, does there exist a subset $S\subset G$ such that $\left|S\right| = O(\sqrt{\left|G\right|})$ and $S\times S = G$? ...
- 47.9k
3
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0
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Is there some sort of formula for $t(S_n)$?
Let $G$ be a finite group. Define $t(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > t(G)$ and $\langle X \rangle = G$, then $XXX = G$.
Is there some sort of formula for $t(S_n)...
- 2,618
2
votes
0
answers
39
views
Weighted unrestricted Golomb rulers?
A set of integers
${\displaystyle A=\{a_{1},a_{2},...,a_{m}\}\quad a_{1}<a_{2}<...<a_{m}} $
is a Golomb ruler if and only if
${\displaystyle \forall i,j,k,l\in \left\{1,2,...,m\right\},a_{i}...
- 13.9k
6
votes
1
answer
262
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Is there some sort of formula for $\tau(S_n)$?
Let $G$ be a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$.
Is there some sort of formula for $\tau(S_n)$, ...
- 44.4k
3
votes
1
answer
210
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On a problem about $GF(2)^n$
For $A\subseteq {\mathbb F}_2^n$ let
$$
Q(A)=\{\alpha+\beta\mid \alpha,\beta \in A,\ \alpha\neq\beta \}.
$$
I want to prove or disprove that if $|A|=2^k+1$ for some integer $k$, then
$$
|Q(A)|\ge2^{k+...
- 4,713
0
votes
0
answers
72
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Rewriting a set of integers to get rid of repetition but keeping subset sum ordering
Say, I have a set of 6 +ve integers sorted in ascending order:
$A = \{2,4,4,4,5,7\}$
Now to make it easier to deal with (Minimum one starts with 1) I deducted one from all of them:
$\therefore B= ...
- 21
2
votes
0
answers
230
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Generating Subsets of a Multiset in Ascending Order of the Sums of the Elements of the Subset
I am trying to come up with an algorithm where you can generate combination from a set in a order such that their sums are in increasing order. This set has to be a multiset i.e. repetition allowed.
...
- 39.9k
7
votes
1
answer
1k
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Minkowski sum of polytopes from their facet normals and volumes
By Minkowski's work in the early 1900s, every polytope $P\subset\mathbb R^n$ is determined up to translation by its unit facet normals $u_1,\dots,u_k$ and facet volumes $\alpha_1,\dots,\alpha_k$. ...
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3
votes
1
answer
146
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Bounding the size of certain sumsets in the plane
Let $A$ be a finite set in $\mathbb{R}^2$ of $k^2$ elements and consider a set $B=\{x_1,x_2,x_3,x_4\}$ such that the points in $B$ are in general position (no three points on a line).
Question 1: Is ...
- 23k
3
votes
1
answer
220
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On particular sumset properties of permanent?
Denote $\mathcal R_2[n]=\mathcal R[n] + \mathcal R[n]$ to be sumset of integers in $\mathcal R[n]$ where $\mathcal R[n]$ to be set of permanents possible with permanents of $n\times n$ matrices with $...
CommunityBot
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3
votes
1
answer
635
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Additive set with small sum set and large difference set
I have a question!
Can someone explain how (the intuition, method?) one can try to construct an additive set of cardinality $N$ with a small sum set (around $N$) and a very large difference set (say,...
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3
votes
1
answer
251
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Are there unique additive decompositions of the reals?
Given $b\in \mathbb{R}_{>1}$ is there $U\subseteq\mathbb{R}_{\ge 0}$ such that $U+bU=\mathbb{R}_{\ge 0}$ and $(U-U)\cap b(U-U)=\{0\}$ (or equivalently: $u+bv=u'+bv' \implies u=u', v=v'$)?
Here is ...
- 23.7k
4
votes
0
answers
150
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Dividing a finite arithmetic progression into two sets of same sum: always the same asymptotics?
This is inspired by the recent question How many solutions $\pm1\pm2\pm3…\pm n=0$.
The oeis entries A063865 linked to this question and A292476/A156700 for the related one "How many solutions $\pm1\...
- 13.4k
2
votes
1
answer
205
views
When does the equality hold in Dias da Silva - Hamidoune Theorem?
Let $p$ be prime number and let $A$ be a $k$-elements subset of $\mathbb{Z}/p\mathbb{Z}$. Dias da Silva - Hamidoune Theorem states that $|h^{\hat{}}A| \geq \min(p, hk -h^2 + 1)$, where $h$ is an ...
- 23k
2
votes
0
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137
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The set of lengths of $nX$ gets larger and larger for every non-zero, non-empty, finite $X \subseteq \mathbf N$ with $0 \in X$
Let $H$ be a multiplicatively written monoid with identity $1_H$. Given $x \in H$, we take ${\sf L}_H(x) := \{0\}$ if $x = 1_H$; otherwise, ${\sf L}_H(x)$ is the set of all $k \in \mathbf N^+$ for ...
- 10.5k
8
votes
2
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618
views
sum-sets in a finite field
Let $\mathbb{F}_p$ be a finite field, $A=\{a_1,\dots,a_k\}\subset\mathbb{F}_p^*$ a $k$-element set, for $k<p$. $\mathfrak{S}_k=$permutation gp.
Question. Is it true there is always a $\pi\in\...
CommunityBot
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3
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0
answers
65
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What's known about $X$ when $|X(n) + X(n)| < kn$, $n \in \mathbb{N}$, absolute constant $k$?
Let $X$ be an infinite sequence of integers$$x_1 < x_2 < x_3 < \ldots,$$and let $X(n)$ be the set$$\{x_1, x_2, \ldots, x_n\}.$$
Question. What is known about $X$ when we have$$|X(n) + X(n)| &...
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3
votes
1
answer
246
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Limit measuring failure of sum-set cancellability
Suppose $A$, $B$ are finite sets of positive integers.
Let $$\mathcal{S}_n = \{C \subset [1,n] \, : \, A+C = B+C \}, $$ and denote $a_n = |\mathcal{S}_n|$.
Note that for any $X \in \mathcal{S}_n$ ...
- 114k
1
vote
0
answers
120
views
Lower bound for sumset in discrete cube
Suppose $A\subset\{0,1\}^d$ for some $d\geq 1$. Then how large must $A+A=\{a+b:a,b\in A\}$ be?
- 23k
10
votes
1
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547
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what is the status of this problem? an equivalent formulation?
R. Guy, Unsolved problems in number theory, 3rd edition, Springer, 2004.
In this book, on page 167-168, Problem C5, Sums determining members of a set, discusses a question Leo Moser asked: suppose $X\...
CommunityBot
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4
votes
2
answers
427
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How big must the sumset $A+A$ be if $A$ satisfies no translation-invariant equations of low height?
Suppose $A$ is a finite subset of an abelian group. If there is no solution to $ma+nb=(m+n)c$ with $0\leq m,n\leq M$, can we bound $|A+A|$ from below? I am interested if one can obtain bounds much ...
- 23k
4
votes
0
answers
127
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Restricted addition analogue of Freiman's $(3n-4)$-theorem
There is a well-known theorem of Freiman saying that if $A$ is a finite set of integers with $|2A| \le 3|A|-4$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms. Is ...
1
vote
1
answer
189
views
Sumset achieving extreme upper bound [closed]
It is trivial that $|A_1 + \cdots + A_h| \leq |A_1|\cdots |A_h|$, where $h \geq 2$ and $A_i \subseteq \mathbb{Z}$ are nonempty finite sets and $A_1 + \cdots + A_h :=\{a_1 + \cdots + a_h : a_i \in A_i ~...
- 13k
1
vote
1
answer
206
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Some question on haar measure for sumsets of closed subsets of profinite groups
Let $H$ be a profinite group with the haar measure $\mu_H$. Let $H_1$ and $H_2$ be closed subgroups of $H$. $H_1$ and $H_2$ have their own haar measures $\mu_{H_1}$ and $\mu_{H_2}$ respectively.
...
- 23k
10
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2
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445
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Iterated sumset inequalities in cancellative semigroups
This question is motivated by the following well-known theorems:
Thm (Plünnecke): If $A$ is a finite nonempty subset of an abelian group, then for every $n$ we have $|A^n| \le \frac{|AA|^n}{|A|^n}|A|$...
CommunityBot
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5
votes
2
answers
550
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Can you simplify (or approximate) $\sum_{n=0}^{N-1} \binom{N-1}n \frac{(-1)^n}{n+1} e^{-\frac{n}{2(n+1)}\lambda}$?
Let $\binom x y$ be the binomial coefficient. I am trying to get a better understanding of the sum
$$
f(N,\lambda)=\sum_{n=0}^{N-1}\binom{N-1}n\frac{(-1)^n}{n+1} e^{-\frac{n}{2(n+1)}\lambda}
$$
as a ...
- 1,228
8
votes
1
answer
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Does $|A+A|$ concentrate near its mean?
Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My ...
- 14.2k
2
votes
4
answers
653
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Show that the Minkowski sum of two triangles in 3D is the union of Minkowski sums of each triangle along the other's edges?
I'd like to show (or disprove) the claim that the Minkowski sum of two triangles with vertices in $\mathbb{R}^3$, $A+B$, is equal to the union of the unions of the Minkowski sums of $A$ along all ...
- 96.4k
3
votes
2
answers
442
views
Sumsets with distinct numbers, upper bound for maximum element
Let $A$ be a finite set of positive natural numbers with $n$ elements, $|A|=n$, with the property that all sums of two (not necessarily different) elements are distinct, or in the usual notation for ...
- 109k
3
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2
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316
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Partition regular systems: do they have solution in (very dense) set of integers?
A partition regular system is a linear system of equations of the form $A\cdot x=0$, which satisfies a Ramsey-type result (namely, that for each $r>0$ whenever we colour the integers in $r$ classes,...
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0
votes
1
answer
239
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Set of number with unique sums of elements [closed]
Is it possible to construct a set of numbers of arbitrary size such that any calculation involving addition and subtraction, on any combination of those numbers, produces a unique result?
For example,...
- 556
0
votes
1
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206
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determine elements of a sum given set
Is it possible to efficiently find the elements that make up a sum $S$ given a set of number sets which determine the sum elements? It seems like an NP-Complete problem, though I might miss something.
...
- 25.4k
2
votes
1
answer
554
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Asymptotic of a sum involving binomial coefficients
Good evening, I'm trying to find an asymptotic of this sum:
$$\sum_{j=0}^n (-1)^j {n \choose j} (n - j)^n = n^n - {n \choose 1} (n - 1)^n + {n \choose 2} (n - 2)^n + ... + (-1)^n {n \choose n} (n - ...
- 188k
10
votes
2
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641
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Sumsets and dilates: does $|A+\lambda A|<|A+A|$ ever hold?
The following problem is somehow hidden in this recently asked question, but I believe that it deserves to be asked explicitly.
Is it true that for any finite set $A$ of real numbers, and any real $...
CommunityBot
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14
votes
1
answer
633
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Minimal "sumset basis" in the discrete linear space $\mathbb F_2^n$
For a set $C\subseteq \mathbb F_2^n$, let $2C=C+C:=\{\alpha+\beta\colon \alpha,\beta\in C\}$.
I want to find $C$ of the smallest possible size such that $2C=\mathbb F_2^n$. Let $m(n)$ be the size of a ...
- 10.9k
5
votes
1
answer
279
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Element with unique representation in A+B
Let $A, B \subseteq \mathbb{Z}$ be finite subsets of the integers. Then there exists an element in $A+B$ with a unique representation as a sum of an element in $A$ and an element in $B$, namely $\max(...
- 23k
12
votes
1
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353
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Number of orders of $k$-sums of $n$-numbers
Suppose we have a $n$-element set $S$. Denote the set of its $k$-element subsets by $K$ ($|K|=\binom{n}{k}$).
If the elements of $S$ are real numbers then to each $k$-element subset we can associate ...
- 50.8k
4
votes
0
answers
216
views
Subgroup cliques in the Paley graph
It is a famous open problem to estimate non-trivially, for a prime $p\equiv 1\pmod 4$, the largest size of a subset $A\subset{\mathbb F}_p$ such that the difference of any two elements of $A$ is a ...
CommunityBot
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3
votes
3
answers
498
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How to find an integer set, s.t. the sums of at most 3 elements are all distinct?
How to find a set $A \subset \mathbb{N}$ such that any sum of at most three Elements $a_i \in A$ is different if at least one element in the sum is different.
Example with $|A|=3$: Out of the set $A :...
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8
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1
answer
326
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Sumsets and a bound
Let $q$ be a positive integer. Is it true there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of reals:
$$\displaystyle |A+qA|\ge (q+1)|A|-C_q\qquad (1)$$
I ...
- 7,023
26
votes
1
answer
786
views
Distribution of $a^2+\alpha b^2$
It is well known that size of the set of positive integers up to $n$ that can be written as $a^2+b^2$ is asymptotic to $C \frac{n}{\sqrt{\log n}}$. Here I'm interested mostly in the weaker fact that ...
- 43.7k
20
votes
3
answers
1k
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A sumset inequality
A friend asked me the following problem:
Is it true that for every $X\subset A\subset \mathbb{Z}$, where $A$ is finite and $X$ is non-empty, that $$\frac{|A+X|}{|X|}\geq \frac{|A+A|}{|A|}?$$
Here ...
- 3,028
18
votes
4
answers
2k
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Number of vectors so that no two subset sums are equal
Consider all $10$-tuple vectors each element of which is either $1$ or $0$. It is very easy to select a set $v_1,\dots,v_{10}= S$ of $10$ such vectors so that no two distinct subsets of vectors $S_1 \...
CommunityBot
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11
votes
0
answers
830
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Cliques in the Paley graph and a problem of Sarkozy
The following question is motivated by pure curiosity; it is not a part
of any research project and I do not have any applications. The question
comes as an interpolation between two notoriously ...
CommunityBot
- 1
11
votes
1
answer
617
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Lower bounds for $|A+A|$ if $A$ contains only perfect squares
Let $A$ a set with $|A|=n$ that contains only perfect squares of integers.
What lower bounds can we give for $|A+A|$?
I think the lower bound $\gg \frac{n^2}{\sqrt{log \,n}}$ holds (this would be ...
CommunityBot
- 1
2
votes
0
answers
189
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Doubling for Sumset of the same set
Let $G$ ($G=\mathbb{Z}^n_2$ for my case) be a additive group and $A$ be a subset of $G$. For any set $S\subseteq G$ define its doubling as $$\sigma (S)=\dfrac{|S+S|}{|S|}$$
Suppose $A$ has small ...
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5
votes
2
answers
517
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Anticoncentration of the convolution of two characteristic functions
Edit: This is a question related to my other post, stated in a much more concrete way I think.
I am interested in anything (ideas, references) related to the following problem:
Suppose that $A \...
- 696