Questions tagged [sumsets]

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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 ...
Rodrigo's user avatar
  • 1,235
23 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: ...
Zur Luria's user avatar
  • 1,613
20 votes
3 answers
1k views

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 ...
Eric Naslund's user avatar
  • 11.2k
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
18 votes
4 answers
2k views

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 \...
Simd's user avatar
  • 3,195
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
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
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
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
15 votes
1 answer
705 views

The hypercube: $|A {\stackrel2+} E| \ge |A|$?

I have a good motivation to ask the question below, but since the post is already a little long, and the problem looks rather natural and appealing (well, to me, at least), I'd rather go straight to ...
Seva's user avatar
  • 22.8k
14 votes
1 answer
615 views

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 ...
pointer's user avatar
  • 197
13 votes
1 answer
569 views

Size of a certain sumset in $\mathbb{Z}/p^2\mathbb{Z}$

We are interested in estimating the size of a certain sumset in $\mathbb{Z}/p^2\mathbb{Z}$. Let $p$ be an odd prime, $g$ a primitive root modulo $p^2$, and $A=\langle g^p\rangle$ the unit subgroup of ...
Bob Lutz's user avatar
  • 165
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
12 votes
1 answer
303 views

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 ...
Arseniy Akopyan's user avatar
11 votes
1 answer
602 views

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 ...
Rodrigo's user avatar
  • 1,235
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
11 votes
0 answers
813 views

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 ...
Seva's user avatar
  • 22.8k
10 votes
1 answer
541 views

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\...
T. Amdeberhan's user avatar
10 votes
2 answers
626 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 $...
Seva's user avatar
  • 22.8k
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
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 $|A^n| \le \frac{|AA|^n}{|A|^n}|A|$...
zeb's user avatar
  • 8,523
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
8 votes
2 answers
597 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\...
T. Amdeberhan's user avatar
8 votes
1 answer
322 views

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 ...
shadow10's user avatar
  • 1,090
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 ...
George Shakan's user avatar
8 votes
2 answers
777 views

A sum-product estimate in Z/p^2Z

We are interested in a sum-product type estimate. Let $p$ be an odd prime, and let $A$ be the order $p-1$ subgroup of $(\mathbb{Z}/p^2\mathbb{Z})^\times$. That is, let $A = \langle g^p \rangle$, where ...
Bob Lutz's user avatar
  • 165
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
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
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
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
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
5 votes
2 answers
531 views

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 ...
mermeladeK's user avatar
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
5 votes
1 answer
433 views

A problem related with 'Postage stamp problem'

A friend of mine taught me this question. I found that it is related with 'Postage stamp problem' (though it does not seem to be same). Let $m,a_1\lt a_2\lt \cdots\lt a_n$ be natural numbers. Now let ...
mathlove's user avatar
  • 4,727
5 votes
1 answer
272 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(...
user68529's user avatar
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
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
5 votes
2 answers
493 views

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 \...
Maciej Skorski's user avatar
4 votes
2 answers
404 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 ...
deadcat's user avatar
  • 41
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
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
4 votes
1 answer
153 views

$B_k[1]$ sets with smallest possible $m = \max B_k[1]$ for given $k$ and $n = \lvert B_k[1]\rvert$ elements

Sidon sets are sets $A \subset \mathbb{N}$ such that for all $a_j,b_j \in A$ holds $$a_1+a_2=b_1+b_2 \iff \{a_1,a_2\}=\{b_1,b_2\}.$$ Thus if you know the sum of two elements, you know which elements ...
Shannon's user avatar
  • 71
4 votes
1 answer
830 views

Intersecting Hamming spheres: is $|A\stackrel k+E|\ge|A|$?

Since my original posting some ten days ago, I discovered an amazing example which changed significantly my perception of the problem. Accordingly, the whole post got re-written now. The most general ...
Seva's user avatar
  • 22.8k
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
4 votes
0 answers
126 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 ...
user93878's user avatar
4 votes
0 answers
215 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 ...
Seva's user avatar
  • 22.8k
3 votes
3 answers
724 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 (|\...
Mark Lewko's user avatar
  • 11.7k
3 votes
3 answers
477 views

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 :...
Shannon's user avatar
  • 71
3 votes
2 answers
416 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 ...
Doc Brown's user avatar
  • 133
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