**6**

votes

**2**answers

213 views

### What methods do we have to understand the spectrum of matrices with restricted entries?

Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"),
What is the largest possible spectral radius of a $n \times n$ matrix ...

**0**

votes

**1**answer

73 views

### What is the maximal number of solutions of $\sum_{i = 1}^n 1/a_i^x - \sum_{i = 1}^m 1/b_i^x = 0$?

What is the maximal number of solutions of the following equation?
$\sum_{i = 1}^n 1/a_i^x - \sum_{i = 1}^m 1/b_i^x = 0$
where $x$ is the unknown and $n, m$, $a_i$'s, $b_i$'s are constant.
It ...

**22**

votes

**1**answer

694 views

### Two conjectures about zero inner products and dissociated sets

The following problems come from something I worked on (with my coauthors) related to proving a new time lower bound for streaming problems. Having worked on these problems for some time with little ...

**0**

votes

**0**answers

125 views

### Generalized arithmetic progressions contained in Bohr sets

Recall that generalized arithmetic progression of dimension $d$ is by definition a set of the form $P = P_1+\dots+P_d$, where $P_j = \{lp_j\ \mid \ |l|\leq l_j\}\subset \mathbb Z$ is an ordinary ...

**88**

votes

**7**answers

4k views

### Is the set $ AA+A $ always at least as large as $ A+A $?

Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A+A|$?
My first instinct is that this is obviously true, and there is a one-line proof which I am foolishly ...

**3**

votes

**0**answers

155 views

### On covering by smooth numbers

Denote $P(n)=\mathsf{greatest}\mbox{ }\mathsf{prime}\mbox{ }\mathsf{factor}\mbox{ }\mathsf{of}\mbox{ }n$.
Denote $S(x,y)=\{n<x: P(n)<y\}$.
Denote $S_t(x,y)=\sum_{i=1}^tS(x,y)$ as $t$-fold ...

**5**

votes

**2**answers

209 views

### Ordered lattice point enumeration

I initially asked this question over at StackOverflow as it has algorithmic flavor to it, but I haven't been getting much traction so I thought I would probe the mathematics community.
Setup: Let ...

**5**

votes

**1**answer

206 views

### Element with unique representation in A+B

Let $A, B \subseteq \mathbb{Z}$ be finite subsets of the integers. Then there exists an element in $A+B$ with a unique representation as a sum of an element in $A$ and an element in $B$, namely ...

**2**

votes

**2**answers

239 views

### Applications of Szemeredi's Theorem

Szemeredi's Theorem is a famous theorem in Additive Combinatorics, Ergodic Theory and maybe some other parts of Mathemtaitcs:
(Szemeredi's Theorem) Let $\Lambda \in \mathbb{Z}$ be a subset of ...

**2**

votes

**0**answers

64 views

### How is the structure of spectrum in cap-sets with no strong increments unrealistic if density is too large?

I am reading this excellent paper by Bateman and Katz on improved bounds on the cap set problem.
Let A be a set in $\mathbb{F}_3^n$ containing no 3-term arithmetic progression and let $A(x)$ denote ...

**2**

votes

**1**answer

119 views

### On a problem about $GF(2)^n$

For $A\subseteq {\mathbb F}_2^n$ let
$$
Q(A)=\{\alpha+\beta\mid \alpha,\beta \in A,\ \alpha\neq\beta \}.
$$
I want to prove or disprove that if $|A|=2^k+1$ for some integer $k$, then
$$
...

**5**

votes

**3**answers

346 views

### Sets of natural numbers with finite intersections and divergent sums of reciprocals

Does there exist an uncountable collection $\Lambda$ of infinite subsets of the set of natural numbers such that (i) any two distinct subsets in the collection have a finite intersection and (ii) the ...

**3**

votes

**0**answers

110 views

### Applications of Freiman's theorem?

What are some interesting applications of Freiman's theorem or, better-yet, its recent generalizations (eg Green-Ruzsa) that could be included in a graduate course in additive combinatorics?
I'm ...

**1**

vote

**1**answer

129 views

### 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 ...

**2**

votes

**1**answer

184 views

### Proof of Pollard's inequality

Let $A, B \subseteq \mathbb{Z}_p$, $p$ prime, $|B| \leq |A|$.
If $N_t$ denotes the number of elements of $\mathbb{Z}_p$ having at least $t$ representations as $a+b$, $a \in A, b \in B$, Pollard's ...

**11**

votes

**2**answers

736 views

### Most dense subset of numbers that avoids arbitrarily long arithmetic progressions

The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression.
I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid
...

**3**

votes

**0**answers

262 views

### Solving a doubly exponential generating function

I am analyzing the average time complexity of some algorithm on some probabilistic model, and I've come to a doubly exponential sequence for which I cannot find corresponding generating function. I ...

**3**

votes

**1**answer

187 views

### Packing bounds for sumsets, or, very discrete balls

Let $D\subset \mathbb{F}_2^n$ with $D=-D$ and $0\in D$. Write $k D$ for the set of all sums of $k$ (not necessarily distinct) elements of $D$. (This is the "ball" in the title.)
Now let $d(g,h)$ be ...

**3**

votes

**2**answers

224 views

### Number of subsum of a given set of integers

I am asking myself for few days the following but I can't find any references, although I am pretty sure this was already studied, because it seems quite natural:
Given n integers $(e_i)$ chosen ...

**9**

votes

**3**answers

294 views

### Polynomial expressions of roots of unity with integer norm

Say a nonconstant polynomial $p(z)$ is $k$-magical if it satisfies the following properties:
$p$ is of the form $$p(z) = a_{k-1} z^{k-1} + a_{k-2} z^{k-2} + \cdots + a_1 z + 1$$ where each $a_i \in ...

**11**

votes

**0**answers

340 views

### How large must $A$ be if $\{1, \ldots, N\} \subseteq A-A$?

Given a positive integer $N$, what is the size of the smallest set of integers $A$ such that, for any integer $1 \leq k \leq N$, we can find two integers $x, y \in A$ such that $x - y = k$? (An ...

**1**

vote

**1**answer

143 views

### asymptotic for restricted partitions

Let $m$ and $n$ be two positive integers and denote by $P(n,m)$ the number of partitions of $n$ into $m$ non-negative integers.
Is there an asymptotic formula for $P(n,m)$ ?? Any reference is ...

**2**

votes

**2**answers

205 views

### Functions representable as a sum of two permutations of Z/nZ

Let $\ f:Z/n\rightarrow Z/n\ $ be a function such that $\ \sum_{i\in Z/n}\,f(i)=0.\ $ Is it true that $\ f\ $ can be represented as a sum of two permutations of Z/nZ?

**3**

votes

**1**answer

165 views

### How to generate $n$ FP32 rationals s.t. no two distinct k-el. subsets have same sum?

First some
Background: I have lots and lots of integer matrices, whose rows are $k$-combinations (without repetitions and sorted) of numbers from the set $S:=\{1,...,n\}$ and needed to be compared ...

**13**

votes

**1**answer

403 views

### Minimal “sumset basis” in the discrete linear space $\mathbb F_2^n$

For a set $C\subseteq \mathbb F_2^n$, let $2C=C+C:=\{\alpha+\beta\colon \alpha,\beta\in C\}$.
I want to find $C$ of the smallest possible size such that $2C=\mathbb F_2^n$. Let $m(n)$ be the size of a ...

**1**

vote

**0**answers

162 views

### Sets of coprime numbers

Consider the set $\{0, 3, 7, 15\}$ of four integers. If you add each of these numbers to a fixed power of 2, then the resulting four numbers are pairwise coprime. For example, $\{4, 7, 11, 19\}$ are ...

**4**

votes

**0**answers

146 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 ...

**1**

vote

**1**answer

232 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

**0**answers

180 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**

votes

**0**answers

74 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

**1**answer

109 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

**0**answers

85 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 ...

**1**

vote

**1**answer

109 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**

votes

**1**answer

62 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**

votes

**1**answer

162 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

**0**answers

154 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 ...

**2**

votes

**2**answers

526 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

**1**answer

505 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

**0**answers

50 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

**1**answer

96 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 $Spec_\alpha(\phi(A))$ ...

**5**

votes

**2**answers

223 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
...

**23**

votes

**1**answer

597 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**

votes

**0**answers

217 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 ...

**13**

votes

**2**answers

871 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

**2**answers

302 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**

votes

**0**answers

215 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 ...

**5**

votes

**1**answer

267 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

**5**answers

628 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

**2**answers

284 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

**0**answers

93 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 ...