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Results tagged with sumsets
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user 9924
6
votes
Accepted
Element with unique representation in A+B
(3) It is known that for any finite integer set $A$ and any integer $h\ge 2$, one has the following inequality relating the sizes of the $(h-1)$-fold and the $h$-fold sumsets of $A$:
$$ \frac{|hA …
- 22.6k
0
votes
0
answers
73
views
Additive energy and uniquely representable elements
Suppose that $A$ is a finite, nonempty set in an abelian group. If there is a group element with a unique representation as $a-b$ with $a,b\in A$, then none of $A-A$ and $2A$ are small:
$$ \min\{|A-A| …
- 22.6k
14
votes
Sets that are not sum of subsets
Two more links: Large sets in finite fields are sumsets by Alon, and Problem 4.11 here. …
- 22.6k
7
votes
A problem related with 'Postage stamp problem'
To my understanding, you are asking about the smallest size of a subset $A\subset[0,m]$ such that $[0,m]\subset 2A$ (where $2A:=\{a'+a''\colon a',a''\in A\}$ is the doubling of $A$). This is OEIS sequ …
- 22.6k
8
votes
Additive set with small sum set and large difference set
A great work on this has been done by Imre Ruzsa; see, for instance, his paper "Many differences, few sums" in Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 51 (2008), 27–38 (2009).
As a very brief a …
- 22.6k
3
votes
A sumset inequality
This is a very incomplete answer at this stage, but it establishes some connections worth recording.
A well-known lemma by Petridis says that if $A$ and $B$ are finite, non-empty subsets of an abeli …
- 22.6k
12
votes
Accepted
Decomposing a finite group as a product of subsets
It was brought to my attention by Noga Alon that my previous answer (which I keep to avoid any confusion) was in fact incorrect: the Rohrbach conjecture got solved completely by Finkelstein, Kleitman, …
- 22.6k
20
votes
Number of vectors so that no two subset sums are equal
Following marshall's comment below, I (sadly) had to completely re-write my original answer.
A famous open conjecture of Paul Erdos, first stated about 80 years ago, is that if all subset sums of an …
- 22.6k
14
votes
Decomposing a finite group as a product of subsets
This problem, first raised in 1937 by H. Rohrbach, has been considered, for instance, in the paper "On $h$-bases and $h$-decompositions of the finite solvable and alternating groups" (J. Number theory …
- 22.6k
4
votes
Accepted
Bounding the size of certain sumsets in the plane
As shown by Gardner and Gronchi ("A Brunn-Minkowski Inequality for the Integer Lattice", equality (8) / Theorem 6.6), if $A,B\subset\mathbb R^n$ are finite sets such that $B$ has full dimension, then
…
- 22.6k
6
votes
1
answer
151
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 elemen …
- 22.6k
4
votes
0
answers
211
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 squ …
- 22.6k
1
vote
1
answer
292
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
3
votes
0
answers
89
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) subgrou …
- 22.6k
7
votes
1
answer
186
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