**27**

votes

**4**answers

2k views

### Cliques, Paley graphs and quadratic residues

A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs.
If p=1 mod 4 is a prime, we can define the ...

**23**

votes

**2**answers

2k views

### Number of elements in the set $\{1,\cdots,n\}\times\{1,\cdots,n\}$

Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?

**23**

votes

**3**answers

1k views

### long enough interval of integers to solve a simultaneous congruence

Let $a$, $b$ be two coprime natural numbers. Let $A \subseteq \{0,1,\ldots, a-1\}$ and $B \subseteq \{0,1,\ldots,b-1\}$ be two nonempty sets, which we think of as sets of residues mod $a$ and $b$ ...

**23**

votes

**1**answer

513 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 $S \subset \{0,\dots,n\}$. ...

**22**

votes

**1**answer

922 views

### Arithmetic Progressions of Squares

Fermat may or may not have known that there are 3-term arithmetic progressions of squares (like $1^2, 5^2, 7^2$, and that there are no 4-term APs. Murky history aside, Keith Conrad has two pleasant ...

**21**

votes

**3**answers

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 ...

**21**

votes

**2**answers

957 views

### The Erdős-Turán conjecture or the Erdős' conjecture?

This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics:
Conjecture: If $A\subset \mathbb{N}$ ...

**20**

votes

**1**answer

1k views

### Monochromatic triangles in every two-coloring of the plane?

An old problem (possibly due to Erdős and Graham?): given a triangle $T$ and a two-coloring of the plane, does there necessary exist a monochromatic congruent copy of $T$? Here "monochromatic" means ...

**19**

votes

**4**answers

1k 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 ...

**19**

votes

**3**answers

970 views

### Can nonabelian groups be detected “locally”?

Suppose $m,n\geq 2$ are two integers. Is it true that for every sufficiently large nonabelian group $G$, one can find a set $A\subset G$, with $|A|=n$, so that $|A^m| >\binom{n+m-1}{m}$?
(Edit) ...

**19**

votes

**3**answers

725 views

### Are sets with similar asymptotic behavior as the primes necessarily finite additive bases?

The set of primes $\mathbb{P}$ has many interesting properties in additive number theory and some of the most famous open problems about $\mathbb{P}$ are the well-known Goldbach's strong and weak ...

**19**

votes

**3**answers

517 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 $$\frac{|A+X|}{|X|}\geq \frac{|A+A|}{|A|}?$$
...

**19**

votes

**1**answer

501 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 ...

**17**

votes

**3**answers

773 views

### An “Average” Erdős–Turán conjecture

Right, so the Erdős–Turán conjecture for additive bases (of order 2) says, with the usual notations, that $\sup r_B (n) = \infty$. Let’s look instead at the average number of representations, i.e.: ...

**17**

votes

**4**answers

1k views

### Arithmetic progressions inside polynomial sets

There are at most 3 perfect squares in arithmetic progression (Fermat, Euler). It was shown in [1] that if $n>2$ there are no three term arithmetic progression consisting of nth powers.
Take a ...

**17**

votes

**0**answers

345 views

### probability of zero subset sum

Almost 17 years ago, I asked the following question on USENET, motivated by a method in numerology (I kid you not).
Pick integers $n \ge 2$, $k \ge 1$. Toss $n$ $k$-sided dice. The sides of each die ...

**15**

votes

**2**answers

854 views

### Roth's theorem and Behrend's lower bound

Roth's theorem on 3-term arithmetic progressions (3AP) is concerned with the value of $r_3(N)$, which is defined as the cardinality of the largest subset of the integers between 1 and N with no ...

**15**

votes

**3**answers

1k views

### What is the shortest route to Roth's theorem?

Roth first proved that any subset of the integers with positive density contains a three term arithmetic progression in 1953. Since then, many other proofs have emerged (I can think of eight off the ...

**14**

votes

**1**answer

443 views

### The hypercube: $|A {\stackrel2+} E| \ge |A|$?

I have a good motivation to ask the question below, but since the post is
already a little long, and the problem looks rather natural and appealing
(well, to me, at least), I'd rather go straight to ...

**13**

votes

**1**answer

587 views

### Goldbach-type theorems from dense models?

I'm not a number theorist, so apologies if this is trivial or obvious.
From what I understand of the results of Green-Tao-Ziegler on additive combinatorics in the primes, the main new technical tool ...

**13**

votes

**3**answers

702 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 $\frac{a^n-1}{a-1}$?

**12**

votes

**2**answers

608 views

### Zero-sum partition of an abelian group

This is a question I have been asking myself some 5 years ago. I later got bored by lack of progress, but maybe some additive combinatorialists here know further. I'm not claiming it is conceptual or ...

**12**

votes

**2**answers

577 views

### 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 ...

**12**

votes

**1**answer

613 views

### On the $L^1$-norm of certain exponential sums.

I am stuck with an elementary-looking problem, which does not belong to my usual field of research so I eventually decided to ask it on MO.
Let $S$ be a finite set of integers. For $P$ a subset of ...

**11**

votes

**2**answers

683 views

### Did Erdős publish his proof of the multiplicative version of the Erdős-Turán conjecture?

I read in an article of Erdős ("Extremal problems in number theory") that he had a proof of the multiplicative version of the Erdős-Turán conjecture. The statement of this theorem is
Let $a_1 < ...

**11**

votes

**2**answers

580 views

### Sums of subsets of $\mathbb{Z}/n\mathbb{Z}$

I have encountered a problem that I suspect has been thoroughly studied but I have not been able to find references. Can anyone point me to a published reference dealing with this or a closely related ...

**11**

votes

**7**answers

806 views

### describe subsets of the integers closed under the binary operation Ax+By

Could one describe the subsets of the integers closed under the binary operation Ax+By
where A and B are arbitrary fixed integers ? That is, describe the subsets S
of the integers such that if ...

**11**

votes

**0**answers

345 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 ...

**10**

votes

**3**answers

730 views

### The sum of integers being a bijection

What are the pairs $(P,Q)$ of subsets of $\mathbb N$ for which the map
\begin{eqnarray*}
P\times Q & \rightarrow & {\mathbb N} \\\\
(p,q) & \mapsto & p+q
\end{eqnarray*}
is a bijection ...

**10**

votes

**3**answers

735 views

### Density Ramsey theorems with explicit asymptotics

I wonder what interesting and non-trivial examples of density Ramsey theorems with explicit asymptotics are there?
I'm aware of two examples: Szemerédi's theorem and density Hales-Jewett theorem.
...

**10**

votes

**2**answers

242 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 ...

**10**

votes

**0**answers

378 views

### What are the limits of the Erdős-Rankin method for covering intervals by arithmetic progressions?

To construct gaps between primes which are marginally larger than average, Erdős and Rankin covered an interval $[1,y]$ with arithmetic progressions with prime differences. A nice short exposition is ...

**9**

votes

**1**answer

500 views

### 4 squares almost in an arithmetic progression

Does there exist infinitely many coprime pairs of integers x,d such that x, x+d, x+2d, x+4d are all square numbers?
One example would be 49,169,289,529. This is the only example I have found so far ...

**9**

votes

**2**answers

414 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 ...

**9**

votes

**1**answer

629 views

### Is it known that $(F_p^{\times} \ltimes F_p, F_p)$ is not a relative expander family?

Shalom [edit: originally M. Burger] showed that the pair $(\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2, \mathbb{Z}^2)$ has Relative Property (T) with respect to standard generating sets.
(The ...

**9**

votes

**0**answers

507 views

### Cliques in the Paley graph and a problem of Sarkozy

The following question is motivated by pure curiosity; it is not a part
of any research project and I do not have any applications. The question
comes as an interpolation between two notoriously ...

**8**

votes

**3**answers

574 views

### A simple looking problem in partitions that became increasingly complex

I began with problem which looked simple in the beginning but became increasingly complex as I dug deeper.
Main questions: Find the number of solutions $s(n)$ of the equation
$$
n = \frac{k_1}{1} + ...

**8**

votes

**2**answers

310 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 ...

**8**

votes

**1**answer

284 views

### A Balog-Szemeredi-Gowers-type question

A nice Lemma due to Konyagin asserts that for any subset $B \subset \mathbb{F}_p$ holds
$$
|B.B - B.B + B.B - B.B + B.B - B.B| \geq \frac{1}{2}\min\{p, |B|^2 \},
$$
where the standard notation for the ...

**8**

votes

**1**answer

539 views

### When does $P(a−b)=0$ for $a≠b$ ensure $P(0)=0$? (Continued.)

As a natural (and expectable) extension of my earlier question:
How large must be a set $A\subset F_2^n$ to ensure that if $P$ is a cubic polynomial in $n$ variables over the field $F_2$, ...

**8**

votes

**0**answers

123 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 $r(n) > 0$ ...

**7**

votes

**3**answers

746 views

### 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 ...

**7**

votes

**2**answers

845 views

### 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 ...

**7**

votes

**2**answers

312 views

### Elementary Proof of Basis of Order k

Context
According to the FAQ, questions of the form "the sorts of questions you come across when you're writing or reading articles or graduate level books" are acceptable. This falls into the ...

**7**

votes

**1**answer

276 views

### When does $P(a-b)=0$ for $a\ne b$ ensure $P(0)=0$?

Let $n$ be a positive integer. How large must be a set $A\subset F_2^n$ to ensure that if $P$ is a quadratic polynomial in $n$ variables, vanishing at all non-zero points of the sumset ...

**7**

votes

**1**answer

430 views

### A variation of Minkowski sum

I have to work with the following variation of Minkowski sum:
Let $\mathbb E$ be a Euclidean space and $K$ be a convex set in $\mathbb E\times \mathbb E$.
Set
$$K^+=\{\\,x+y\in\mathbb ...

**7**

votes

**1**answer

573 views

### Homogeneous arithmetic progressions in difference sets

I have a nasty feeling that I ought to be able to answer this question, but I've got other things to think about right now and I'm interested in the answer just so that I can reply to a mathematical ...

**7**

votes

**0**answers

170 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 ...

**7**

votes

**0**answers

100 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 ...

**6**

votes

**1**answer

532 views

### Relationship between Erdos and Falconer distance problems

Given a set $E \subset \mathbb{R}^d$, define the distance set of $E$
$$
\Delta(E) = \lbrace|x-y| : x,y \in E \rbrace,
$$
where $|\cdot |$ is the usual Euclidean distance.
$\bullet$ The ...