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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399 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
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0answers
250 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
3answers
579 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
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3answers
734 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
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1answer
420 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
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3answers
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
1answer
242 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
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1answer
634 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 ...
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0answers
93 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
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2answers
611 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
2answers
268 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
2answers
565 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
1answer
406 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
3answers
742 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
0answers
307 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
1answer
445 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
2answers
472 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
1answer
925 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
3answers
749 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
1answer
535 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
3answers
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) ...
6
votes
1answer
431 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
3answers
777 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
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1answer
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 ...
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3answers
612 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
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0answers
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
1answer
639 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
1answer
372 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
0answers
534 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
2answers
857 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
3answers
371 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
4answers
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
0answers
922 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
1answer
333 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
1answer
280 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
1answer
189 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
2answers
823 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
1answer
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
2answers
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
1answer
345 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
1answer
540 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
2answers
579 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
1answer
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 ...
5
votes
3answers
395 views

Any rigorous way to claim that sums with repeat summands are few?

Let $B \subset \mathbb{Z}^+$. Define $r_{B,h}(n)$ to be the number of ways of writing $n$ as the sum of $h$ elements of $B$ and $R_{B,h}(n)$ the number of ways to write $n$ as the sum of $h$ DISTINCT ...
7
votes
2answers
866 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 ...
3
votes
0answers
307 views

A question about Erdos thin bases

Let $B \subset \mathbb{N}$ be an additive basis of order $h$. Define $r_{B,h}(n)$ to be the number of ways $n$ can be written as a sum of $h$ elements of $B$. In particular, $B$ is a basis of order ...
11
votes
2answers
589 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 ...
6
votes
1answer
348 views

Arbitrarily thin additive bases of the natural numbers

A subset $A$ of $\mathbb{N}$ is called a basis of order $k$ if the set $kA$ = {$a_1 + \cdots + a_k | a_1, \cdots, a_k \in A$} $= \mathbb{N} \setminus C$, where $C$ is a finite set of positive ...
5
votes
1answer
492 views

A question on the singular series and singular integral in Hardy-Littlewood Circle Method

Given a set $A$ of positive integers, set $r_{A,h}(N)$ to be the number of $h$-tuples $(a_1, \cdots, a_h)$ such that $N = a_1 + \cdots + a_h$. Set $f(z) = \sum_{a \in A} z^a$. Then by Cauchy's Theorem ...
10
votes
3answers
734 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 ...