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Results tagged with additive-combinatorics
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user 61536
Questions on the subject additive combinatorics, also known as arithmetic combinatorics, such as questions on: additive bases, sum sets, inverse sum set theorems, sets with small doubling, Sidon sets, Szemerédi's theorem and its ramifications, Gowers uniformity norms, etc. Often combined with the top-level tags nt.number-theory or co.combinatorics. Some additional tags are available for further specialization, including the tags sumsets and sidon-sets.
11
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1
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What is the smallest cardinality of a set A whose difference A-A contains $n$ consequtive in...
Problem. What is the smallest cardinality $d(n)$ of a set $A$ of integer numbers such that the difference set $A-A=\{a-b:a,b\in A\}$ contains $n$ consequtive integer numbers?
It can be shown that $(1 …
- 41.8k
4
votes
1
answer
154
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Difference bases in simple cyclic groups
A subset $B$ of an abelian group $G$ is called a difference-basis if $B-B=G$. For a finite group $G$ by $\Delta(G)$ we denote the smallest cardinality of a difference basis of $G$. Let $C_n=\{z\in\ma …
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12
votes
2
answers
890
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Factorizable groups
Definition. A finite group $G$ is factorizable if for any positive integer numbers $a,b$ with $ab=|G|$ there are subsets $A,B\subset G$ of cardinality $|A|=a$ and $|B|=b$ such that $AB=G$.
Problem 1. …
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11
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0
answers
531
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Cyclic and prime factorizations of finite groups
A tuple $(A_1,\dots,A_n)$ of subsets of a finite group $G$ is called a factorization of $G$ if $G=A_1\cdots A_n$ and $|A_1|\cdots|A_n|=G$.
In Cryptology factorizations of groups are known as logarit …
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7
votes
3
answers
496
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Product-one sets in non-commutative groups
A nonempty subset $D$ of a group $G$ is called
$\bullet$ decomposable if $D\subseteq DD$, that is every element $x\in D$ is can be written as the product $x=yz$ of some elements $y,z\in D$;
$\bullet$ …
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5
votes
Is each finite group multifactorizable?
I've just discovered that the alternating group $A_4$ is not multifactorizable. Namely, it can not be written as the product $A_4=ABC$ of subsets $A,B,C\subset A_4$ of cardinality $|A|=2$, $|B|=3$, an …
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1
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0
answers
72
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Unit-product sets in finite decomposable sets in groups
A non-empty subset $D$ of a group is called decomposable if each element $x\in D$ can be written as the product $x=yz$ for some $y,z\in D$.
Problem. Let $D$ be a finite decomposable subset of a gr …
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3
votes
0
answers
136
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A question on a result of Imre Ruzsa concerning sum-sets
Th main result of this preprint of Imre Ruzsa implies the following
Corollary (Ruzsa): For every $r\in\mathbb N$ there exists a real number $\alpha<1$ and a positive integer $m$ such that for ever …
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0
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Product-one sets in non-commutative groups
This is not an answer, but too long for a comment. Below I write down some conditions (on a group or a decomposable set) guaranteeing that a decomposable set in a group is product-one.
Proposition 1. …
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3
votes
Accepted
Product-one sets in non-commutative groups
GAP shows that the groups SmallGroup(27,3), SmallGroup(27,4), SmallGroup(36,11), SmallGroup(39,1) SmallGroup(48,3) do contain many 5-element decomposable sets, which are not product-one. So, the lower …
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2
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Large product-1-free sets in finite groups
Realizing the idea of @NickGill we shall confirm the lower bound for solvable groups with five exceptions of the groups $G$ isomorphic to the groups $C_3,C_5,C_3\times C_3, D_{10}$ and $(C_3\times C_ …
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3
votes
3
answers
473
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Large product-1-free sets in finite groups
$\DeclareMathOperator\SmallGroup{SmallGroup}$Definition. A subset $A$ of a group $G$ is called product-1-free if for any sequence of pairwise distinct elements $a_1,\dots,a_n$ of $A$ the product $a_1\ …
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Large product-1-free sets in finite groups
For finite solvable groups $G$ we have the following lower bound for the number $f_1(G)$.
Theorem. Let $G$ be a finite solvable group of cardinality $|G|=\prod_{k=1}p_k$ for some prime numbers $p_1,\d …
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11
votes
3
answers
563
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Is each finite group multifactorizable?
Definition. A finite group $G$ is called multifactorizable if for any positive integer numbers $a_1,\dots,a_n$ with $a_1\cdots a_n=|G|$ there are subsets $A_1,\dots,A_n\subset G$ such that $A_1\cdots …
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5
votes
0
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114
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$m$-thick sets with small $n$-fold sumsets in finite cyclic groups
Problem. Is it true that for every positive integers $n,m$ there exists a subset $A_{n,m}$ of a finite cyclic group $G$ having the following two properties:
$(\Sigma_n)$ the $n$-fold sum $A_{n,m}^{+ …
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