Questions tagged [sumsets]
The sumsets tag has no usage guidance.
94
questions
5
votes
1
answer
749
views
Estimate of Minkowski sum
Let A $\subset [0:2]^n$, where $[0:2]=\{0,1,2\}$, then define $2A= \{ a+b\mid a,b \in A \}$. I wanted to know the best known lower-bound estimates for $|2A|$.
I intuitively expect that $|2A| \geq |A|^{...
14
votes
2
answers
675
views
Subsets of $(\mathbb{Z}/p)^{\times n}$
There seems to be some combinatorial fact that every subset $A$ of $G=(\mathbb{Z}/p)^{\times n}$ of cardinality $\frac{p^n-1}{p-1}+1$ containing $\vec{0}$ satisfies $(p-1)A=G$. ($p$ is a prime number....
- 143
0
votes
1
answer
196
views
Khovanskii's theorem on iterated sumsets
I was watching Gowers video lectures "Introduction to Additive Combinatorics" (my question is about the statement he made at the 21st minute) and came across wonderful theorem due to ...
- 266
2
votes
0
answers
160
views
Component-wise sums of permutations
Given a set $S$ containing all possible permutations of a vector $v = (1, 2, 3, ..., n-1, n)$, find the size of the set $P$, where $P$ is defined as the set of possible component-wise sums obtained by ...
- 21
11
votes
2
answers
647
views
$\mathbb Z/p\mathbb Z=A\cup(A-A)$?
$\newcommand{\Z}{\mathbb Z/p\mathbb Z}$
Can one partition a group of prime order as $A\cup(A-A)$ where $A$ is a subset of the group, $A-A$ is the set of all differences $a'-a''$ with $a',a''\in A$, ...
- 22.4k
0
votes
0
answers
63
views
Bounds on these numbers
Let $[n]$ be the set of natural numbers $1,2,3 \cdots n$ and $k$ be a natural number. Define $S(n,k) = \# \{ A \subset [n] \mid \displaystyle\sum_{i \in A} i =k \}$. My question is; Are there any ...
- 281
5
votes
2
answers
213
views
Progressions in sumset or complement
Fix $\epsilon>0$.
For all large $N$, does there exist $A\subset [N]:=\{1,\dots,N\}$ such that both $A+A$ and $A^c:=[N]\setminus A$ lack arithmetic progressions of length $N^\epsilon$?
I am aware ...
- 2,734
0
votes
0
answers
54
views
How large must "weak Besicovitch" subsets of groups be?
Consider a group $G$; let call $A\subset G$ a weak Besicovitch subset whenever every element of $G$ can be written under the form $gh^{-1}$, where $g,h\in A$.
General question: how large must a weak ...
- 14k
9
votes
0
answers
261
views
If $A+A+A$ contains the extremes, does it contain the middle?
Let $b \ge 1$ and $A\subseteq [0,b]$ be a set of integers (all intervals will be of integers).
Write $hA := \underbrace{A + \ldots + A}_{h\text{ summands}} = \{ \sum_{i=1}^h a_i ~|~a_i \in A,\, \...
- 825
2
votes
1
answer
226
views
Different sum combinations of $L$ identical lists of consecutive natural numbers
Given $L$ variables $k_i$ where each $k_{i} \in \{1, 2, 3, \ldots, N\}$ I want to obtain how many different sums $k_{1}+k_{2}+\cdots+k_{L}$ are generated and the value of these sums.
There are $L^N$ ...
- 23
3
votes
2
answers
328
views
Sumsets with the property "$A+B=C$ implies $A=C-B$"
Let $(G,+)$ be an abelian group and $A$, $B$ and $C$ be finite subsets of $G$ with $A+B=C$. One may conclude that $A\subset C-B$. However, $A$ need not be equal to $C-B$. What is a necessary and ...
- 421
4
votes
2
answers
223
views
Existence of m infinite subsets in an arbitrary group such that all products of one element from each (in order) are distinct
Is it true that for every infinite group $G$ and every $m\in\mathbb{N}$ there are infinite subsets $A_0,\dots,A_{m-1}$ such that all the products $a_0\cdot\dots\cdot a_{m-1}$ with $a_i\in A_i$ are ...
- 43
0
votes
0
answers
117
views
An exercise about sum-product estimate
I am struggling with 1.11 exercise from the George Shakan "Discrete Fourier Transform".
Let $A \subset \mathbb{Z}/q\mathbb{Z}$ be any set not containing zero with $|A|>\sqrt2q^{5/8}$. ...
- 11
2
votes
0
answers
110
views
Restricted sumsets - the origins?
The sumset of the subsets $A$ and $B$ of an additively written group is defined by $A+B:=\{a+b\colon a\in A,\ b\in B\}$. The basic idea to add sets has been around since Cauchy at least.
Erdős and ...
- 22.4k
0
votes
0
answers
87
views
Additive energy and uniquely representable elements
Suppose that $A$ is a finite, nonempty set in an abelian group. If there is a group element with a unique representation as $a-b$ with $a,b\in A$, then none of $A-A$ and $2A$ are small:
$$ \min\{|A-A|,...
- 22.4k
0
votes
1
answer
197
views
Controlling iterated sum sets of "most" of $A+B$
I am reading Tao-Vu book on Additive combinatorics and came across the following lemma. I know that it is better to ask this question on MathStack but I asked few questions before and no one answered ...
- 266
16
votes
2
answers
1k
views
Sets that are not sum of subsets
Let $\mathcal P$ be the set of finite subsets of $\mathbb Z_{\geq 0}$ , each of them contains $0$. We say that $A \in \mathcal P$ is indecomposable if it is not $B+C$ (the sum set of $B,C$) with $B,C\...
- 30.2k
2
votes
1
answer
397
views
Rank of sumsets in matroids
Assume that $G$ is a (finite) abelian group and $M$ is a matroid whose ground set is $G$. Let $X$ and $Y$ be subsets of $G$, and $H$ is the stabilizer of $X+Y$. That is $X+Y+H=X+Y$. We denote the rank ...
- 421
15
votes
1
answer
781
views
Explicit constant in Green/Tao's version of Freiman's Theorem?
Green and Tao's version of Freiman's theorem over finite fields (doi:10.1017/S0963548309009821) is as follows:
If $A$ is a set in $\mathbb{F}_2^n$ for which $|A+A| \leqslant K|A|$, then $A$ is ...
- 153
1
vote
1
answer
173
views
An intriguing inverse sumset problem
Start with a natural number $k$, and choose natural numbers $K=\{n_1,\ldots,n_k\}$ which are pairwise distinct. For each $1\leq j\leq k$, choose another integer $i_j$ such that $0\leq i_j\leq n_j$.
...
- 497
7
votes
1
answer
192
views
Trisecting $3$-fold sumsets, II: is the middle part ever thin?
This is a refined version of the question I asked yesterday.
Let $A$ be a finite set of integers with the smallest element $0$ and the largest element $l$. The sumset $C:=3A$ resides in the interval $[...
- 22.4k
6
votes
1
answer
153
views
Trisecting $3$-fold sumsets: is the middle part always thick?
Here is a truly minimalistic and seemingly basic question which should have a simple solution (I hope it does).
Let $A$ be a finite set of integers with the smallest element $0$ and the largest ...
- 22.4k
2
votes
1
answer
131
views
Equal subset-sums of bounded vectors
Let $S\subseteq \{0,\ldots,n\}^d$ be a set of $d$-dimensional vectors of with bounded, natural, coordinates.
We are given that
$$v'+v_1+\ldots+v_t=u'+u_1+\ldots+u_s$$
where $v_1,\ldots,v_t,u_1,\ldots,...
- 203
1
vote
1
answer
210
views
Probability of getting two subsets with the same sum
Let $A=\{1,...,n\}$. Two subsets of $A$, not necessarily distinct, chosen uniformly at random. What is the probability that both subsets have the same sum? Alternatively, is there a known upper bound?
- 11
3
votes
0
answers
91
views
Origins of the ``baby Freiman'' theorem
It is a basic folklore fact from the area of additive combinatorics that a subset $A$ of an abelian group satisfies $|2A|<\frac32\,|A|$ if and only if $A$ is contained in a coset of a (finite) ...
- 22.4k
3
votes
1
answer
309
views
Prime gap distribution in residue classes and Goldbach-type conjectures
Update on 7/20/2020: It appears that conjecture A is not correct, you need more conditions for it to be true. See here (an answer to a previous MO question).
The general problem that I try to solve is ...
- 3,036
0
votes
1
answer
429
views
Congruential equidistribution, prime numbers, and Goldbach conjecture
Let $S$ be an infinite set of positive integers, $N_S(z)$ be the number of elements of $S$ less than or equal to $z$, and let
$$D_S(z, n, p)= \sum_{k\in S,k\leq z}\chi(k\equiv p\bmod{n}).$$
Here $\chi$...
- 3,036
0
votes
0
answers
151
views
General asymptotic result in additive combinatorics (sums of sets)
Let $S_1,\cdots,S_k$ be $k$ infinite sets of positive integers. Let $N_i(z)$ be the numbers of elements in $S_i$ that are less or equal to $z$. Let us further assume that
$$N_i(S) \sim \frac{a_i z^{...
- 3,036
-2
votes
1
answer
638
views
Prove or disprove this integral of a function, defined on a countable set with infinite limit points, converges to zero [closed]
Edit: I got rid of my old definitions. Everything should be clear now
Since no one has answered my question on MSE, I’m hoping to get an answer here. I apologize if you dislike my writing. I am an ...
- 1
5
votes
1
answer
149
views
Computational version of inverse sumset question
Let $p$ be prime and $\mathbb{F}_p$ the finite field with $p$ elements. Suppose we have a set $B\subseteq \mathbb{F}_p$ satisfying $|B|<p^{\alpha}$ for some $0<\alpha<1$ and there exists $A\...
user156885
3
votes
1
answer
239
views
Unique representation and sumsets
Let $A$ be a finite, nonempty subset of an abelian group, and let $2A:=\{a+b\colon a,b\in A\}$ and $A-A:=\{a-b\colon a,b\in A\}$ denote the sumset and the difference set of $A$, respectively.
If ...
- 22.4k
1
vote
1
answer
200
views
Average size of iterated sumset modulo $p-1$,
Given a prime $p$, what is the average size of the iterated sumset, $|kA|$, modulo $p-1$, with $p$ a prime, and $k$ given, with $A$ chosen at random?
You can pick any type of prime you like for $p$, ...
- 221
1
vote
1
answer
296
views
Does $g+A\subseteq A+A$ imply $g\in A$?
Suppose that $A$ is a subset of a (large) finite cyclic group such that $|A|=5$ and $|A+A|=12$. Given that $g$ is a group element with $g+A\subseteq A+A$, can one conclude that $g\in A$?
- 22.4k
12
votes
2
answers
556
views
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 ...
- 550
10
votes
1
answer
283
views
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 ...
- 30.2k
19
votes
4
answers
820
views
Size of sets with complete double
Let $[n]$ denote the set $\{0,1,...,n\}$. A subset $S\subseteq [n]$ is said to have complete double if $S+S=[2n]$. Let $m(n)$ be the smallest size of a subset of $[n]$ with complete double. My ...
- 30.2k
7
votes
2
answers
479
views
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 ...
- 71
3
votes
0
answers
132
views
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}...
- 13.5k
6
votes
1
answer
259
views
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 $\tau(S_n)$, ...
- 2,618
0
votes
0
answers
54
views
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= ...
- 11
1
vote
0
answers
192
views
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.
...
- 11
6
votes
1
answer
807
views
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 $\alpha_1,\dots,\alpha_k$. ...
- 363
3
votes
1
answer
142
views
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 ...
- 2,208
3
votes
1
answer
215
views
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 $...
- 13.5k
3
votes
1
answer
243
views
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 ...
- 4,663
4
votes
0
answers
146
views
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\...
- 13k
18
votes
3
answers
963
views
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 $S\times S = G$? ...
- 9,623
2
votes
0
answers
137
views
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 ...
- 9,284
8
votes
2
answers
579
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\...
- 41k