Questions tagged [sumsets]

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What is the minimal density of a set A such that A+A = N?

Thinking about the four square theorem and related questions, I found myself wondering: What is the minimal density of a set $A \subset \{0, 1, 2, ... \}$ such that $A + A = \mathbb{N}$? What I know: ...
Zur Luria's user avatar
  • 1,613
5 votes
1 answer
883 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|^{...
Rishabh Kothary's user avatar
15 votes
2 answers
712 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....
Adam Chapman's user avatar
1 vote
1 answer
307 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 ...
RFZ's user avatar
  • 298
2 votes
0 answers
173 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 ...
Talesseed's user avatar
11 votes
2 answers
655 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 $a',a''\in A$, ...
Seva's user avatar
  • 22.8k
0 votes
0 answers
67 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 ...
mukhujje's user avatar
  • 281
5 votes
2 answers
238 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 ...
Zach Hunter's user avatar
  • 3,375
0 votes
0 answers
58 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 ...
Benoît Kloeckner's user avatar
9 votes
0 answers
263 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,\, \...
Alufat's user avatar
  • 805
2 votes
1 answer
257 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 $L^N$ ...
ACL's user avatar
  • 23
3 votes
2 answers
343 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 ...
Shahab's user avatar
  • 433
4 votes
2 answers
229 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 ...
e1c25ec7's user avatar
0 votes
0 answers
122 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 $|A|>\sqrt2q^{5/8}$. ...
Sei's user avatar
  • 11
2 votes
0 answers
112 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 ...
Seva's user avatar
  • 22.8k
0 votes
0 answers
139 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|,...
Seva's user avatar
  • 22.8k
0 votes
1 answer
200 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 ...
RFZ's user avatar
  • 298
16 votes
2 answers
2k 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\...
Hailong Dao's user avatar
  • 30.3k
2 votes
1 answer
401 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 ...
Shahab's user avatar
  • 433
15 votes
1 answer
793 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 ...
Tomasz Popiel's user avatar
1 vote
1 answer
199 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 $0\leq i_j\leq n_j$. ...
Krishnarjun's user avatar
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 $[...
Seva's user avatar
  • 22.8k
6 votes
1 answer
154 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 ...
Seva's user avatar
  • 22.8k
2 votes
1 answer
146 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,...
Shaull's user avatar
  • 203
1 vote
1 answer
223 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?
Yuri's user avatar
  • 11
3 votes
0 answers
93 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) ...
Seva's user avatar
  • 22.8k
3 votes
1 answer
336 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 ...
Vincent Granville's user avatar
0 votes
1 answer
447 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 $\chi$...
Vincent Granville's user avatar
0 votes
0 answers
152 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^{...
Vincent Granville's user avatar
-2 votes
1 answer
669 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 ...
Arbuja's user avatar
  • 1
5 votes
1 answer
172 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\...
user avatar
4 votes
1 answer
256 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 ...
Seva's user avatar
  • 22.8k
1 vote
1 answer
205 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 $p$, ...
Matt Groff's user avatar
1 vote
1 answer
299 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 $g\in A$?
Seva's user avatar
  • 22.8k
12 votes
2 answers
599 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 ...
RyanChan's user avatar
  • 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 ...
Hailong Dao's user avatar
  • 30.3k
19 votes
4 answers
856 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 ...
Hailong Dao's user avatar
  • 30.3k
7 votes
2 answers
529 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 ...
James Ingram's user avatar
3 votes
0 answers
133 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)...
Chain Markov's user avatar
  • 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}...
Turbo's user avatar
  • 13.6k
6 votes
1 answer
260 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 $\tau(S_n)$, ...
Chain Markov's user avatar
  • 2,618
0 votes
0 answers
58 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= ...
Moni's user avatar
  • 11
1 vote
0 answers
202 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. ...
Moni's user avatar
  • 11
6 votes
1 answer
936 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 $\alpha_1,\dots,\alpha_k$. ...
Christoph's user avatar
  • 363
3 votes
1 answer
144 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 ...
TOM's user avatar
  • 2,218
3 votes
1 answer
218 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 $...
Turbo's user avatar
  • 13.6k
3 votes
1 answer
248 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 ...
Yaakov Baruch's user avatar
4 votes
0 answers
149 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\...
Wolfgang's user avatar
  • 13.2k
18 votes
3 answers
1k 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 $S\times S = G$? ...
Scott Aaronson's user avatar
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 ...
Salvo Tringali's user avatar