Tagged Questions

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 ...

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0
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0answers
58 views

Reduction argument from a general vertex set V(G) to a prime power in Prof. Keevash's proof on the Existence of Designs

The proof flow of the paper "On the Existence of Designs" by Prof. Keevash as I understand it is the following: -- Reduction from the general case to $V = \mathbb{F}_{p^a}$ (Lemma 6.3) -- Covering ...
0
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0answers
103 views

every arithmetic progression contains a sequence of $k$ “consecutive” primes for possibly all natural numbers $k$?

I ask the same question here:http://math.stackexchange.com/q/1019404/192097 writing a little better the previous question: it´s true that if we let $a$ and $b$ be coprime integers, then the ...
5
votes
0answers
151 views

The sum of all the elements of every non empty subset of $A$ is not a multiple of $n$

Let $N=\{1,2,\ldots ,n\},n>1$. We wish to construct a set $A\subseteq N$ with the property: The sum of all the elements of every non empty subset of $A$ is not a multiple of $n$. ...
0
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0answers
65 views

Integral representation of the function

I try find integral representation to following function ($d_1<d_2$ and $d_2-d_1=1(\mod 2)$ $f(d_1,d_2)=\frac{(-1)^{\frac{d_2-d_1-1}{2}}}{2^{d_1-1} d_1!}\sum_{k=[\frac{d_1}{2}]+1}^{ d_1} { d_1 ...
1
vote
1answer
91 views

Finding a sufficiently large complete bipartite subgraph using matrix counting

I'm trying to reconstruct the proof using matrix counting that there exists two subsets $A,B$ of $\{1,\cdots,N\}$ with $\#A=\#B$ such that for any $a\in A$ and $b\in B$, $a+b$ is prime, and $\#A=\#B$ ...
6
votes
0answers
73 views

Why have most maximal cliques of Paley graphs odd size?

I ask this question mainly by curiosity. See here for definitions and a plot of the clique numbers of the Paley graphs for the primes $p\equiv 1 \pmod 4$ up to $10000$. Is there an ...
0
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0answers
50 views

Sumset of parallel arithmetic progressions in Euclidean space

Let $A$ be a finite subset of $\mathbb R^n$ such that dim $A=n$ ($A$ is not contained in any lower dimensional hyperplane). It is a known result from Freiman that the size of the sumset $A+A$ is lower ...
1
vote
1answer
102 views

Expression and growth bound for $r_{p^m,k}(n)$

Let's define , $$R_{p^m,k}(n)=\#\{(a_1,\dots,a_k)\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2\le n \ \text{and} \ p^m|\sum_{i=1}^ka_i^2\}$$ what will be growth bound of $R_{p^m,k}(n)$? This can be thought as a ...
0
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1answer
57 views

Monotonicity of the gap of permutated sequence

Let $a$ be an arbitrary sequence and denote by $\mbox{gap}_k(a) = a_{(k)} - a_{(k+1)}$, where $a_{(k)}$ is the $k$th largest component of $a$. Of course, $k+1$ should be no larger than the length of ...
2
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1answer
146 views

Growth of $r_k(n)$

What is the best known growth bound of $r_k(n)$, where $$r_k(n)=\#\{(a_1,\dots,a_k\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2=n\}?$$ Please provide some reference if known. Thanks.
3
votes
0answers
133 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 ...
1
vote
1answer
307 views

Asymptotic formula for the number of ways to write a number as the sum of $k$ triangular numbers

How would one derive an asymptotic formula for the number of representations of a number $n$ as the sum of $k$ numbers of the form $\frac{m(m + 1)}{2}$ I think that one could use the circle method, ...
1
vote
1answer
446 views

Subset of the integers with certain properties

How would one find the maximal $n$ such that there exists an $n$-subset $S$ of $\mathbb{Z}^+$ such that $\forall A\subseteq S, \sum_{a\in A}a$ is either a perfect square or a perfect cube, or can one ...
1
vote
0answers
39 views

$B_k[1]$ sets with smallest possible $m = max B_k[1]$ for given $k$ and $n = |B_k[1]|$ 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 ...
4
votes
1answer
69 views

Spectrum of image of a Freiman Homomorphism

Let $A\subset \mathbb{F}_p^n$ along with a k-Freiman isomorphism $\phi:A \rightarrow \mathbb{F}_p^m$ for $A$. Can we say any meaningful statement about the structure of $Spec_\alpha(\phi(A))$ ...
4
votes
2answers
195 views

Intuition for Freiman dimension

Let $A\subset G$ and $B\subset H$ be subsets of abelian groups. We say that a map $f\colon A\to B$ is a Freiman homomorphism if for all $a_1,\dots, a_4\in A$ one has ...
0
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0answers
73 views

latest classification of difference sets

I am looking for a reference including the last classification of difference sets and almost difference sets. Would you mind letting me know some of them? The recent ones are preferred.
23
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1answer
538 views

integers which are sums of binomial coefficients: $\sum_i {n \choose k_i}$

Let $n$ be an integer. For $S$ a subset of $\{0,\dots,n\}$, define $$m(S) = \sum_{k \in S} {n \choose k}.$$ Let $M_n$ be the set of integers of the form $m(S)$ for all sets $S \subset \{0,\dots,n\}$. ...
2
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0answers
159 views

Can the affine sieve be used to sieve for $k$-free values?

The affine sieve, developed initially by Bourgain-Gamburd-Sarnak in the paper "Affine linear sieve, expanders, and sum-product" published in Inventiones Mathematicae in 2010, deals generally with the ...
8
votes
2answers
356 views

Sum and product estimate over integers, rationals, and reals

My question is the following: is finding a lower bound for $|A+A\cdot A|$ (as a function of $|A|$) where $A$ is any finite subset of the positive integers equivalent to finding the same lower bound ...
5
votes
2answers
266 views

Size of distinct sums in A

Let $G$ be an abelian group. Let $A\subset G$ be a finite set. $\sum_A$ is defined as: $$\left\{\sum_{b\in B}b \mid B\subset A\right\}$$ Is there any result similar to Freiman's Theorem for $\sum_A$? ...
7
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0answers
186 views

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 ...
4
votes
1answer
238 views

Examples of specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} ...
6
votes
5answers
594 views

What makes a set random?

There are many results in number theory, where the existence of some $B \subseteq \mathbb{N}$ with certain properties is proved by a probabilistic argument employing "random sets". One such example ...
10
votes
2answers
266 views

Large sets $X \subseteq \mathbb{Z}_2^n$ with $X+X \ne \mathbb{Z}_2^n$

I've come across a seemingly natural question in additive combinatorics to which I can't find the answer in the literature. I would like to know if, given an $\alpha > 0$ and large $n$, there are ...
3
votes
0answers
86 views

Sets of polynomials with restricted set of products

Let $F = \{f_1, ..., f_n\}$ be a set of distinct monic polynomials (and thus also $f_i^2$ are distinct). Let $F' = \{f_i f_j: i \neq j\}$ be the set of pairwise products of all distinct elements from ...
7
votes
0answers
109 views

Ordering subsets of the cyclic group to give distinct partial sums

Suppose that you are given a set $S$ of $k$ nonzero elements from $\mathbb{Z}_n$. Is it always possible to order the elements of $S$, say $a_1,a_2,\dots,a_k$ in such a way that the partial sums ...
0
votes
1answer
158 views

Collecting terms of a linear expression with nested sums and combinatorics in coefficients

I need to collect the $\Pr(\cdot)$ terms of the following expression: $\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left( 1-\theta \right) }\right) ^{m}}\left[ ...
4
votes
1answer
275 views

Circle method on things other than the integers

The circle method is often used to estimate the number of solutions to the equation $$x_1 + x_2 + ... x_k = N$$ if for all $i$ $x_i\in A\subseteq\mathbb{N}_0$ and some subset of the nonnegative ...
0
votes
0answers
66 views

Cover one finite subset of integers by another one

Let $A$, $B$ be two finite subsets of integers. We denote by $C(A, B)$ the minimum number of shifts of $A$ to cover $B$. More formally, it can be written as $$ C(A, B)=\min\{|S|: S\subseteq ...
3
votes
1answer
151 views

Is every sufficiently dense well mixed set an additive basis?

Let $B \subset \mathbb{N}$ be a set of natural numbers such that $|B \cap [1,N]| \sim N^\gamma$, for some $\gamma > 0$ with the following property: For any pair of positive integers $k,n$ we ...
3
votes
0answers
98 views

Number cubes with consecutive line sums

This is barely of research interest, but I'd classify it as a curiosity with connections to combinatorics. The problem is to place integers in an $n \times n \times n$ array so that all $3n^2$ line ...
1
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0answers
67 views

Stronger condition than being a normal polytope?

A polytope $P$ with integer vertices is called normal if for every $p = \sum_j a_j p_j $ such that $a_j \geq 0$, $\sum_j a_j = k \in \mathbb{N}$, $p_j$ are vertices of $P$ and $p$ is an integer ...
5
votes
1answer
389 views

Balog-Szemeredi-Gowers with dilates of sets

All sets are assumed to be finite subsets of the integers. The additive energy of two sets $E(A,B)$ is defined as the number of solutions to $a+b=a'+b'$ with $a,a'\in A$ and $b,b'\in B$. The ...
2
votes
1answer
338 views

Coin problem with permutations

Let $a,b,c$ be positive integers with gcd$(a,b,c)=1$, and let $\mathbb{N}$ denote the set of nonnegative integers. It is well known that $\mathbb{N} \setminus (a \mathbb{N}+b \mathbb{N} + c ...
20
votes
3answers
556 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 $$\frac{|A+X|}{|X|}\geq \frac{|A+A|}{|A|}?$$ ...
5
votes
1answer
236 views

Normal polytopes - counterexample?

An integral polytope $P$ is normal if all lattice points inside the integer dilation $kP$ can be expressed as $p_1+p_2+\dots+p_k$, where $p_i \in P$ are lattice points. I am looking for an example ...
9
votes
2answers
464 views

Finite field Szemeredi-Trotter theorem with unequal number of points and lines

My question concerns the Szemerédi-Trotter theorem in $\mathbb{F}_q^2$. If we have $m$ points and $n$ lines in $\mathbb{F}_q^2$, then by Cauchy-Schwartz the number of point-line incidences is as most ...
19
votes
4answers
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 ...
8
votes
0answers
129 views

Erdös-Fuchs Theorem for multivariate linear forms

Let $A$ be an infinite set of positive integers, and denote by $r(n)$ the number of solutions to the equation $a+a'=n$, with $a,\, a' \in A$. It is not very difficult to show that if $r(n) > 0$ ...
0
votes
0answers
60 views

$\sum_{a\in A} r(a)$, where $A\subseteq\mathbb{Z}_n$ and $r(a)$ is the number of representations of $a$ as sum of two elements from $A$

Reading Seva's answer to this question, I got lost at the line relating $\sum_{a\in A} r(a)$ to $|A\cap 2A|$. More precisely, restating the problem: Let $A\subseteq\mathbb{Z}_n$ be an set of ...
5
votes
0answers
141 views

Distribution of subset-sums

Let $A$ be a set of $n$ integers uniformly distributed in $\{0,\dots,N-1\}$. Let $S$ be the set of subset-sums modulo $N$ of $A$. Let $f_{n,N}(k)$ be the probability that $|S|=k$. Is there an ...
3
votes
0answers
564 views

How many combinations does Android pattern have? [closed]

Rules- 1) At-least 4 and at-max 9 dots must be connected. 2) There can be no jumps 3) Once a dot is crossed, you can jump over it.
3
votes
0answers
60 views

Distribution of the evaluation (at a non-trivial root of -1) of polynomials with small coefficients

When we studied some cryptographic protocol, we came accross the following problem, which seems linked to the uniformity of the residues of small multiplicative subgroups of $\mathbb{F}_q$. Problem ...
21
votes
3answers
2k views

How many different numbers can be obtained as product of first $n$ natural numbers?

Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set $\{1^{a_1} \cdot ...
3
votes
0answers
83 views

“Pseudo-random” subsets of additive bases

We say that a subset $B \subset \mathbb{N}$ is an (asymptotic) additive basis of order $k$ if the set $kB = B + \cdots + B = \mathbb{N} \setminus C$, where $C$ is a finite set of positive integers. ...
4
votes
1answer
470 views

a conjecture in sum-free sets

Let $ A $ be a set of non-zero integers. Then $A$ contains a sum-free subset $B$ of size $ |B|> \frac{|A|}{3} $ (a result of Erdős). It is conjectured that RHS can be improved to $\frac{|A|}{3} ...
11
votes
0answers
354 views

Correlation of Fourier transforms of characteristic functions

Let $A$ and $B$ ($A\subset B$) be subsets of a finite abelian group $G$. (For the sake of argument, you can take $G$ to be $\mathbb{Z}/p\mathbb{Z}$ for large $p$, say.) Write $1_S$ for the ...
13
votes
3answers
725 views

Density of all n such that 2^n-1 is square free

Is it true that the set $$S:=\{n\in \mathbb N\ |\ 2^n-1 \ \mbox{ is square free}\}$$ has positive density? What can we say when we replace $2^n-1$ with $\frac{a^n-1}{a-1}$?
19
votes
1answer
516 views

Avoiding multiples of $p$

Let $p$ be a prime number and $P=\{1,2,...,p-1\}$ In how many ways we can sum all the elements of $P$ in such a way that we will reach a multiple of $p$ only when we sum the last summand? For ...