All Questions
Tagged with sumsets additive-combinatorics
63
questions
22
votes
2
answers
3k
views
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:
...
- 1,574
5
votes
1
answer
816
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|^{...
15
votes
2
answers
696
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....
- 153
0
votes
1
answer
246
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 ...
- 288
2
votes
0
answers
163
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
652
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 $a',a''\in A$, ...
- 22.6k
0
votes
0
answers
64
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
225
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 ...
- 2,874
0
votes
0
answers
56
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 ...
- 14.1k
9
votes
0
answers
262
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
4
votes
2
answers
226
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
118
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
111
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 ...
- 22.6k
0
votes
0
answers
119
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|,...
- 22.6k
0
votes
1
answer
198
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 ...
- 288
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\...
- 30.2k
15
votes
1
answer
789
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
184
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$.
...
- 507
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 $[...
- 22.6k
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 ...
- 22.6k
3
votes
0
answers
92
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) ...
- 22.6k
3
votes
1
answer
324
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 ...
- 3,058
0
votes
1
answer
439
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}).$$
Here $\chi$...
- 3,058
0
votes
0
answers
152
views
General asymptotic result in additive combinatorics (sums of sets)
Let $S_1,\cdots,S_k$ be $k$ infinite sets of positive integers. Let $N_i(z)$ be the numbers of elements in $S_i$ that are less or equal to $z$. Let us further assume that
$$N_i(S) \sim \frac{a_i z^{...
- 3,058
5
votes
1
answer
152
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
247
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 ...
- 22.6k
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 $p$, ...
- 221
1
vote
1
answer
298
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 $g\in A$?
- 22.6k
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
851
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
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)...
- 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
3
votes
1
answer
143
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
247
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
18
votes
3
answers
992
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 $S\times S = G$? ...
- 9,663
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 ...
- 9,446
3
votes
0
answers
64
views
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$$|X(n) + X(n)| &...
- 31
3
votes
1
answer
240
views
Limit measuring failure of sum-set cancellability
Suppose $A$, $B$ are finite sets of positive integers.
Let $$\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$ ...
- 265
1
vote
0
answers
116
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?
4
votes
2
answers
402
views
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 ...
- 41
4
votes
0
answers
125
views
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 ...
- 41
2
votes
1
answer
188
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 ...
- 167
1
vote
1
answer
179
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 ~...
- 167
8
votes
1
answer
720
views
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 ...
- 2,283
3
votes
2
answers
279
views
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,...
- 1,483
3
votes
3
answers
722
views
Is the sumset or the sumset of the square set always large?
Let A be a finite subset of $\mathbb{N}$, $\mathbb{R}$, or a sufficiently small subset of $\mathbb{F}_{p}$.
Do we have a lower bound of the form $|A|^{1+\delta}$ on the following quantity:
$$\max (|\...
- 11.7k
10
votes
2
answers
431
views
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 $|A^n| \le \frac{|AA|^n}{|A|^n}|A|$...
- 8,503
10
votes
2
answers
622
views
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 $...
- 22.6k
5
votes
1
answer
271
views
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(...
- 83