Questions tagged [additive-combinatorics]
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.
100
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12
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2
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867
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Arithmetic progressions modulo $p$ under the squaring map
I feel that the following problem should be known, but I'm not sure where to look for it.
Fix a real constant $\frac{1}{2} \ge \epsilon > 0$. For varying primes $p$, Let $A_p$ denote the set of ...
user631
11
votes
0
answers
526
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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 ...
- 40.9k
11
votes
2
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402
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Extension of Dickson's theorem on integers of the form $a^2+b^2+2c^2$
Theorems V in this paper of L.E. Dickson states that the following two sets are equal. $$E=\{a^2+b^2+2c^2 \ | \ a,b,c \in \mathbb{Z}\} \ \text{ and } \ F=\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \...
- 25.9k
10
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4
answers
1k
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Binomial coefficient in Andrews' partition book
First of all, I think MathOverflow is a very great community to discuss math, either basic or advanced, and I'm glad to participate here. It's my first post, so I'm sorry if i did anything wrong, and ...
- 103
9
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0
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296
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An abstract zero-sum problem
I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
- 1,169
8
votes
1
answer
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Properties of Zero Line-Sum Matrices
By a Zero Line-Sum (ZLS) matrix I mean matrices with the property, that each row sum and each column sum equals zero:
$$A\in\mathbb{R}^{m\times n}:\ \sum_{i=1}^{n}a_{ij}=\sum_{j=1}^{m}a_{ij}=0$$
...
- 12.7k
8
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1
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369
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Question about estimating random symmetric sums modulo p
Let $n > 0$ be a positive integer (large) and $p > 2$ a fixed prime number. What is the probability that $$\sum_{ 1 \leq i < j \leq n} a_ia_j = 0 \mod p$$ where $a_1, a_2, \dots a_n$ are ...
- 1,020
8
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0
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299
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A strong sum-product "for translates" in finite fields
In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...
- 11.2k
8
votes
0
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449
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Small maximal sets with no 3-AP?
Everyone knows about the problem of finding the largest set in 1,2,...,N which contains no arithmetic progression of length 3.....what happens if we look at the set which is the
smallest maximal ...
- 703
7
votes
2
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614
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Decomposition of a natural number as sum of positive integers
Let $n \in \mathbb{N}$ be a positive natural number and denote by $f(n)$ the number of decompositions of $n$ of the form $n = a+b+c+d$ where $a,b,c,d > 0$ are also positive natural numbers such ...
- 8,852
7
votes
1
answer
192
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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 $[...
- 22.8k
7
votes
2
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2k
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Recent results on the Gauss circle problem?
Possible Duplicate:
What is the status of the Gauss Circle Problem?
The Gauss circle problem is the following: Let $N(r)$ denote the number of solutions in integer pairs $(i,j)$ to the inequality ...
- 25.5k
7
votes
1
answer
561
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Upper bound for size of subsets of a finite group that contains a sum-full set
Problem
I'm looking for an upper bound for the number $k(G)$ of a finite group $G$, defined as follow:
Let $\mathcal{F}_k$ be the family of subsets of $G$ with size $k$, and we
define $k(G)$ be ...
- 1,250
7
votes
0
answers
241
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Distribution of trivial subset sums
Suppose I have a set $S$ of $n$ integers picked independently, uniformly at random from $[-L, L].$ Let $z(S)$ be the number of subsets of $S$ which sum to zero. The question is: what is the ...
- 95.6k
7
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1
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A permutation problem
Here I ask a question on permutations of $n$ distinct real numbers.
QUESTION: Let $a_1,a_2,\ldots,a_n\ (n>1)$ be (pairwise) distinct real numbers. Is there a permutation $b_1,\ldots,b_n$ of $a_1,\...
- 14.5k
7
votes
1
answer
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Additive combinatorics and large Fourier coefficients
Elon Lindenstrauss explains in his talk at the MSRI in Fall 2008 (the relevant comment is at minute 41 of the video) that the set of large Fourier coefficients of a probability measure $\mu$ on the ...
- 25.3k
6
votes
1
answer
151
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Are the extremal points of a certain set of functions $\mathcal P(\mathbf N) \to \bf R$ weakly additive?
Let an upper density (on $\mathbf N$) be a (set) function $f: \mathcal P(\mathbf N) \to \mathbf R$ such that, for all $X, Y \subseteq \bf N$ and $h,k \in \mathbf N^+$, the following hold:
(F1) $f(\...
- 9,636
6
votes
1
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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
5
votes
1
answer
227
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Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$
Let $\mathcal D$ be the set of all (finitely) additive probability measures $\mu^\ast: \mathcal P(\mathbf N^+) \to [0,\infty[$ such that $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X \...
- 9,636
5
votes
1
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328
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Large gaps in Singer planar difference sets?
By a classical result of Singer (1938), for a prime number $p$ the cyclic group $C_n$ of order $n=1+p+p^2$ contains a subset $D$ of cardinality $|D|=1+p$ such that $DD^{-1}=C_n$. Such set $D$ is ...
- 40.9k
5
votes
1
answer
254
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Divisibility labeling on a boolean lattice and nonzero Euler totient
Let $B_n$ be the subset lattice of $\{1,2, \dots , n \}$, also called the boolean lattice of rank $n$.
A labeling $f: B_n \to \mathbb{N}_{\ge 1}$ is called acceptable if $\forall a,b \in B_n$:
...
- 25.9k
4
votes
2
answers
405
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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
1
answer
197
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Bound of size $X\subset \mathbb{Z}/N\mathbb{Z}$ which satisfies $X+X=\mathbb{Z}/N\mathbb{Z}$
(Sorry for my poor english skill..)
Let $N$ be a large integer and the set $X$ be the subset of $\mathbb{Z}/N\mathbb{Z}$. For two sets $A$ and $B$, we define
\begin{equation}
A+B:=\{a+b : a\in A, b\...
- 661
4
votes
1
answer
287
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Set of small numbers with distinct $k$-sums
Let $A$ be a set of $n$ positive integers with distinct $k$-sums. In other words, if $a_1\le\cdots\le a_k$ and $b_1\le\cdots\le b_k$ are elements of $A$ such that $a_1+\cdots+a_k=b_1+\cdots+b_k$, then ...
- 41
4
votes
1
answer
301
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Difference set of difference set
I am a hobby computer scientist and searching for an algorithm to construct a set of n numbers (integers) with certain properties.
Property 1 / Step 1
All pairwise differences of the elements should ...
- 73
4
votes
2
answers
381
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An extremal combinatorics problem
Given two integers integers $0<b<p$ and a real $\alpha\in(0,1)$ what is the largest $m$ we have such that in the interval $(p^\alpha,p-p^\alpha)$ there are $m$ integers $a_1<\dots<a_m$ ...
- 13.7k
3
votes
0
answers
90
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Dimension of a kernel of a cocycle map
Inspired by a previous question (Dimension of a kernel of a linear map) and some of the answers I was given I thought wheter I can generalize the question to the following:
Compute the kernel (or at ...
- 577
3
votes
1
answer
342
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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 ...
- 3,088
3
votes
0
answers
142
views
Flat polynomials with factors of big height
Let $p(x)$ be a polynomial of degree $n$ with all coefficients in $\{-1,0,1\}$ (such polynomials are sometimes called flat). I am wondering how big the coefficients of a factor of $p$ can be. Call ...
- 13.2k
3
votes
1
answer
499
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Karolyi's theorem for finite groups and its extensions
Suppose that $\mathbb A = (A, +)$ is a (possibly non-commutative) group, and denote by $p(\mathbb A)$ the minimum of $|S|$ as $S$ ranges in the set of non-trivial subgroups of $\mathbb A$, with the ...
- 9,636
3
votes
1
answer
144
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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 ...
- 2,248
3
votes
3
answers
724
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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 (|\...
- 11.8k
3
votes
2
answers
473
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Two equivalent statements about primes
Regarding to our hypothesis in https://math.stackexchange.com/questions/1918406/a-hypothesis-about-the-conjecture-every-even-number-is-the-difference-of-two-p , we guess that the following statements ...
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3
votes
1
answer
309
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Problem related to Frobenius coin problem
Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$.
Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ good if,
for any $r,s,u,...
- 13.7k
2
votes
1
answer
594
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On 'Improved Bounds for the Sunflower Lemma' [Alweiss, Lovett, Wu, Zhang]
I have been reading the paper 'Improved Bounds for the Sunflower Lemma' (Ann. of Math., Vol. 194(3), pp. 795-815), and have not managed to understand the following:
I would like a formalization for ...
user469866
2
votes
1
answer
567
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On the upper Banach density of the set of positive integers whose base-$b$ representation misses at least one prescribed digit
Let $b$ be a fixed integer $\ge 2$ and $A$ a proper subset of $\{0, \ldots, b-1\}$. Then define $X$ to be the set of all positive integers whose base-$b$ representation consists only of digits from $A$...
- 9,636
2
votes
1
answer
156
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Prime divisors of $\prod(a_i-a_j)$
For a prime $p$, let $\varphi_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the group of order $p$.
Given an integer $n\ge 3$, what is the smallest ...
- 22.8k
2
votes
1
answer
109
views
Capset problem but considering differences with bounded support
For any dimension $D\ge 1$, we define the homomorphism $\phi: \Bbb{Z}^D\to (\Bbb{Z}/3\Bbb{Z})^D; \xi\mapsto \xi+3\Bbb{Z}^D$.
Given a set $A \subset \{0,1,2\}^D$, we define $S_A$ to be the set of $v \...
- 3,413
2
votes
0
answers
137
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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 ...
- 9,636
2
votes
0
answers
142
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A set in Z/nZ which contains two elements, one of which is a small multiple of the other
While doing my research, I came across yet another problem in $\mathbb Z/n\mathbb Z$ (see my previous question on a related matter here).
Let $n$ be a prime and let $k$ be an integer, $1 \leq k \leq ...
- 1,573
1
vote
1
answer
191
views
Maximum number of vectors with bounds on inner products
Suppose there are 2n vectors $\{m_1,m_2,...,m_n\}$ and $\{\mu_1,\mu_2,...,\mu_n\}$. All vectors are in k-dimensional Euclid space $R^k$. The vectors satisfy:
$$m_i \mu_j \leq 0, \quad \forall i\neq j $...
- 23
1
vote
1
answer
176
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Pollard's inequality modulo a composite number
Here is question similar to another one asked today (Proof of Pollard's inequality).
When working in $\mathbb{Z}/n\mathbb{Z}$ for composite $n$ Cauchy-Davenport and Pollard's inequalities may not ...
- 83
1
vote
1
answer
462
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Some divisibility constraints in Frobenius coin problem
Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$.
Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ excellent if linear form ...
- 13.7k
1
vote
1
answer
181
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
1
vote
1
answer
112
views
Invariant complementary sets modulo $p$
Let $p \ge 11$ be a prime number, $k,n$ be positive integers such that $n|gcd(p-1,k-1)$ and $p > k > n \ge 5$. Let $s \in \mathbb Z_p$ such that $ord_p(s) = n$. Is it possible that the sets $A = ...
- 169
1
vote
0
answers
139
views
Counting Hamiltonian cycles in graph and finding a coefficient of polynomial
Exact result is #P-Hard, so we are looking for bounds.
Let $G$ be simple graph or simple digraph and $A$ its
adjacency matrix. $A$ is $n \times n$ with entries only zeros or ones.
Let $K=\mathbb{Z}[...
- 24.2k
1
vote
1
answer
272
views
Does the asymptotic formula for Partitions into parts <c exist?
A partition of $n$ is a weakly decreasing tuple of numbers $(\lambda_1,\lambda_2,\lambda_3,....\lambda_k)$ whose sum is $n.$ A natural problem studied is counting partitions whose summands $\lambda_i$...
- 1,286
0
votes
1
answer
88
views
Maximum number of vectors with bounds on inner products (follow up question)
This is a follow-up question from my previous question.
Suppose there are (2n+1) vectors $\{m_1,m_2,...,m_n\}$, $\{p_1,p_2,...,p_n\}$ and $p^*$ in $R^{k+1}$. $m_i$ are weakly positive vectors. $p_i$ ...
- 23
0
votes
0
answers
100
views
Number of $b$-separated Sidon sets with pairwise difference set intersection bounds
Given two integers integers $0<b<p$ and a real $\alpha\in(0,1)$ call a set of $m$ integers $a_1<\dots<a_m$ in the interval $(p^\alpha,p-p^\alpha)$ to be $b$-separated Sidon if:
$a_i-a_j\...
- 13.7k
0
votes
1
answer
455
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 $\chi$...
- 3,088