All Questions
17 questions
59
votes
2
answers
4k
views
For a finite set A of positive reals, prove that the set A + A - A contains at least as many positive as negative elements
I am currently working on a proof that would need to use the following theorem that I cannot prove:
"Let $A$ be a finite set of positive real numbers. Then, the set $A + A - A$ contains at least ...
- 643
19
votes
4
answers
867
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 ...
- 30.6k
15
votes
2
answers
752
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
11
votes
0
answers
830
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 ...
- 23k
10
votes
1
answer
547
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\...
- 43.2k
10
votes
2
answers
641
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 $...
- 23k
9
votes
0
answers
265
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
7
votes
0
answers
177
views
Sumsets that contains many squares, Improvement on the bound
I'm being troubled by this problem on AoPS:
https://artofproblemsolving.com/community/c6h1998237p13955033
I searched for any literature related to it such as
Nguyen, Hoi H., and Van H. Vu., Squares ...
- 73
5
votes
1
answer
218
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
5
votes
0
answers
185
views
Gaps in sumsets and difference sets
a) Let $S\subset \{1,2,\dotsc,N\}$ be a fairly thick set (with at least $N^{1-\epsilon}$ elements, say). Suppose that the intersection of, say,
$$3 S - 3 S = \{a_1+a_2+a_3-(a_4+a_5+a_6): a_1,\dotsc,...
- 20.2k
4
votes
2
answers
427
views
How big must the sumset $A+A$ be if $A$ satisfies no translation-invariant equations of low height?
Suppose $A$ is a finite subset of an abelian group. If there is no solution to $ma+nb=(m+n)c$ with $0\leq m,n\leq M$, can we bound $|A+A|$ from below? I am interested if one can obtain bounds much ...
- 41
4
votes
0
answers
150
views
Dividing a finite arithmetic progression into two sets of same sum: always the same asymptotics?
This is inspired by the recent question How many solutions $\pm1\pm2\pm3…\pm n=0$.
The oeis entries A063865 linked to this question and A292476/A156700 for the related one "How many solutions $\pm1\...
- 13.4k
3
votes
3
answers
749
views
Is the sumset or the sumset of the square set always large?
Let A be a finite subset of $\mathbb{N}$, $\mathbb{R}$, or a sufficiently small subset of $\mathbb{F}_{p}$.
Do we have a lower bound of the form $|A|^{1+\delta}$ on the following quantity:
$$\max (|\...
- 13k
3
votes
3
answers
498
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 $A :...
- 71
3
votes
2
answers
365
views
Sumsets with the property "$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 ...
- 429
3
votes
1
answer
220
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 $...
- 13.9k
1
vote
1
answer
218
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