**4**

votes

**3**answers

313 views

### L^2-Flattening Lemma

I am reading a significant paper of Bourgain and Gamburd on "Uniform expansion bounds for Cayley graphs of $\mathrm{SL}_2(\Bbb{F}_p)$". They have proven a proposition which in their new papers they ...

**3**

votes

**2**answers

800 views

### There exists B subset A, |B| = log n, A \cap 2*B = \emptyset

This is exercise 1.8.1 of Additive Combinatorics.
Problem
Given $A \in Z^+$ a set of $n$ different integers. Prove that there exists $B\subset A$, $|B| = \Omega ( \log n )$, s.t. $A \cap 2*B = ...

**3**

votes

**0**answers

201 views

### Convex subsets of sumsets

There's a famous old problem in additive number theory which asks whether, for any $\epsilon > 0$, if $B$ is a 2-basis for the set $A =$ {$1^2,2^2,...,n^2$}, i.e.: a set for which $B+B \supseteq ...

**2**

votes

**0**answers

107 views

### Lower bound on P(Y > 3/2 E(Y)) where Y = # of triangles in graph

Context
This is exercise 1.7.1 of Tao/Vu's Additive Combinatorics, generally considered a graduate level math textbook.
My question is not "How do I solve this exercise?"
My question is "What is ...

**3**

votes

**0**answers

115 views

### Other interesting examples of pseudo-random measures

Here we use the notion of pseudo-randomness (or variant of) used in the proof of the Green-Tao theorem. In particular, we say a measure $\nu : \mathbb{Z}_N \rightarrow \mathbb{R}^+$ is pseudo-random ...

**7**

votes

**2**answers

317 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 ...

**1**

vote

**1**answer

222 views

### Technique: Compactness => (Finite -> Reals)

Context
I'm studying a classical results of Erdos and Lovasz, on colorings of the real line.
The theorem to be proved is as follows:
Let $m, k$ be two positive integers satisfying:
...

**10**

votes

**0**answers

426 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 ...

**0**

votes

**0**answers

255 views

### Kakeya problem and arithmetic progressions

Here's something I also posted on Stackexchange recently. It's very related to the Kakeya problem, yet I fail to see why this is true. It goes like this:
Let $r > 2$ be an integer parameter. Let ...

**8**

votes

**3**answers

581 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} + ...

**20**

votes

**3**answers

743 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 ...

**4**

votes

**1**answer

426 views

### Ruzsa-type inequalities for additive energy

As the subject says, I have some questions about some Ruzsa-type inequalities for additive-energy. It would be wonderful if anyone could shed some light.
First, the definition: let $E(A,B)$ be the ...

**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 $b$ ...

**3**

votes

**1**answer

244 views

### Bounds on pseudosquares

Suppose $n\in\mathbb{Z}^+$ is a nonsquare in $\mathbb{Z}$ but is a square mod $2^2,3^2,4^2,5^2,\ldots,k^2.$ How small can $n$ be?
On the ERH, there are no small pseudosquares: ...

**9**

votes

**1**answer

648 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 ...

**1**

vote

**0**answers

99 views

### Examples of using additive structure to analyze size of a set mod some ideal?

There are examples of proofs that analyze the structure of some set $S$ of integers by looking at the size of the set mod some positive integer $n$ (for example, the $3k-3$ theorem.)
Are there ...

**12**

votes

**2**answers

616 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 ...

**4**

votes

**2**answers

275 views

### Conditions for an analogue of Cauchy-Davenport for simple groups

What conditions would be sufficient for a generalization of Cauchy-Davenport for simple groups? I can see two possible difficulties with a generalization for general groups:
The sets could both be ...

**4**

votes

**2**answers

573 views

### How local the property of “being a partition” is?

Note: The problem is solved! See EDIT below.
The following question about integer partitions arose from a purely "practical" question: Does there exist better dynamic programming algorithms for the ...

**4**

votes

**1**answer

412 views

### Intersecting Hamming spheres: is $|A\stackrel k+E|\ge|A|$?

Since my original posting some ten days ago, I discovered an amazing
example which changed significantly my perception of the problem.
Accordingly, the whole post got re-written now.
The most ...

**10**

votes

**3**answers

752 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.
...

**6**

votes

**0**answers

309 views

### 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 ...

**14**

votes

**1**answer

453 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 ...

**4**

votes

**2**answers

478 views

### A weighted sum of non-negative integers

Suppse $n > m$ are positive integers. Then it is an elementary exercise to show that the number of $m$-tuples of non-negative integers $(x_1, \cdots, x_m)$ such that $x_1 + \cdots + x_m = n$ (order ...

**22**

votes

**1**answer

937 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 ...

**7**

votes

**3**answers

757 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 ...

**6**

votes

**1**answer

537 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 ...

**19**

votes

**3**answers

972 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) ...

**6**

votes

**1**answer

433 views

### Minimum cardinality of a difference set in $R^n$

Cross-posted from http://math.stackexchange.com/questions/65195/minimum-cardinality-of-a-difference-set-in-mathbb-rn.
Given a finite set $S$ of $m$ points in $\mathbb R^n$ that do not all lie in the ...

**17**

votes

**3**answers

778 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.: ...

**3**

votes

**1**answer

364 views

### Optimize / simple Set Covering Problem

Let $k,m\in\mathbb{N}$ be given. Let $M:=\{0,... , m-1\}$. How to find a subset $T\subset M$, $|T|=k$ such that $|T+T|$ is maximal, where $T+T=\{ (a+b)\mathbin\%m \mid a\in T,b\in T \}$ (“%” means ...

**2**

votes

**3**answers

613 views

### Arithmetic progressions of length 3 in subset of Z_n of size n^d

Let $A\subset\mathbb{Z}/n\mathbb{Z}$ such that: $|A|>n^{d}$ ($0< d <1$).
Let $C=\{(x,y,2y-x)\in A\times A \times A\}$ be the set of $3$-term arithmetic progressions within $A$.
[The ...

**2**

votes

**0**answers

246 views

### Prime divisors of the difference set

Fix $c\in(0,1)$, and let $N$ be a (large) positive integer. Given a set $A=\{0=a_1<\dots<a_n=N\}$ of density $\alpha:=n/N>c$ with $\gcd(A)=1$, I want to find a prime dividing as few ...

**4**

votes

**1**answer

646 views

### Size of Sum Sets

Let $\{a_{k}\}$ be an increasing sequence of positive real numbers. Let us define $A_{n}:=\{a_1,\ldots,a_{n}\}$ and consider the set
$$
S_{n}^{(2)}=\Big \{(x_1,x_2,x_3,x_4)\in ...

**5**

votes

**1**answer

375 views

### Fourier analysis, orthogonality, and Plancherel for finite abelian groups

I am reading an outstanding paper by Bateman and Katz, improving the best known bounds on the cap set problem (Roth's theorem over $\mathbb{F}_3^N$).
The paper contains some technical lemmas for ...

**5**

votes

**0**answers

536 views

### Expectation of Gowers norm

This was a problem that came up during a course on Analytic Combinatorics that I had taken this semester. Here's the problem:
Let $\mathbf{F}$ be the set of boolean functions, $f: \mathbb{F}_2^n ...

**15**

votes

**2**answers

872 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 ...

**5**

votes

**3**answers

373 views

### Structure of nonaveraging sets of integers

A set of integers is said to be nonaveraging if it contains no three-term arithmetic progression. I call a nonaveraging subset of $\lbrace 1,2, \ldots ,n \rbrace$ optimal when it has maximal ...

**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 ...

**2**

votes

**0**answers

930 views

### Fun question in additive combinatorics

It is easy to see that for a finite set of integers $A$ of cardinality $n$, the cardinality of the sumset $A+A$ satisfies
$$
2n-1\leq |A+A|\leq \frac{n(n+1)}{2}.
$$
The lower bound is essentially ...

**6**

votes

**1**answer

336 views

### Bounds on the size of sets not containing a given finite pattern

Recall the following version of Szemerédi's Theorem: let $r_k(N)$ be the largest cardinality of a subset of $[N]:=\{1,\ldots, N\}$ which does not contain an arithmetic progression of length $k$. Then, ...

**4**

votes

**1**answer

282 views

### Thin subbases for the primes?

Hi all,
My question concerns a general problem concern the Erdos-Turan conjecture on additive bases; that of finding thin subbases in a given basis. For a given $A \subset \mathbb{N}$, define ...

**4**

votes

**1**answer

190 views

### Methods to determine whether a given set is the sum of other sets

In Tao and Vu's Additive Combinatorics, it is mentioned that "if we replace the additive set $A$ by a larger set, such as $A+B$, $A+A+A$, or $2A - 2A$, then one can locate significantly larger ...

**3**

votes

**2**answers

839 views

### Choice of normalization of the finite Fourier transform

Let $f : \mathbb{Z}/N\mathbb{Z} \rightarrow \mathbb{C}$ be a function. Is its Fourier transform typically defined by
$$\widehat{f}(\xi) = \sum_{x \in \mathbb{Z}/N \mathbb{Z}} f(x) e^{- 2 \pi i x \xi ...

**1**

vote

**1**answer

119 views

### In what cases are the counting function and representation functions strongly related?

Let $A \subset \mathbb{N}$ be a subset of the natural numbers and $h > 0$ be a natural number. Let $a(n) = |A \cap [1,n]|$ be the counting function of $A$, and let $r_{A,h}(n)$ be the number of ...

**5**

votes

**2**answers

283 views

### Known additive bases with irregular counting function

For $A \subset \mathbb{N}$ and positive integer $h > 0$, define $r_{A,h}(n)$ to be the number of ways to write $n$ as the sum of $h$ (not necessarily distinct) elements of $A$. We say $A$ is an ...

**4**

votes

**1**answer

347 views

### How large can a non-sumset be?

The theory of sumsets $A+B$ where $A$ and $B$ are finite subsets of an additive group $Z$ is extensively studied in additive combinatorics: finding long arithmetic progressions inside them, finding ...

**8**

votes

**1**answer

542 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 $F_2$, ...

**12**

votes

**2**answers

580 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 ...

**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 ...