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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5
votes
2answers
284 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
349 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
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
2answers
589 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
277 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
398 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
905 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
310 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
606 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
355 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
516 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
748 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 ...
5
votes
2answers
626 views

Inverse Length 3 Arithmetic Progression Problem for sets with positive upper density

It is a famous theorem of Roth, which Szemerédi famously generalized, that if a set of natural numbers has positive upper density then it contains arithmetic progressions of length $k$. The famous ...
11
votes
2answers
699 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 < ...
4
votes
1answer
290 views

Is the generalized Erdős–Heilbronn problem true for finite cyclic groups?

The generalized Erdős–Heilbronn (GEH) theorem, which is proved by da Silva and Hamidoune in 1994, states that: Theorem. If p is a prime and $X$ is a subset of $\mathbb{Z}_p$, then $|\hat{k}X| \geq ...
2
votes
2answers
1k views

Almost periodic functions in Tao's ergodic proof of Szemerédi's theorem

Do we say that a function $f$ is uniformly almost periodic in the aforementioned proof if $f$ is bounded (in the sense that $||f||_{L^\infty}\leq 1$) and that there exists a natural number $d>0$ ...
29
votes
4answers
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 ...
15
votes
3answers
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 ...
6
votes
1answer
336 views

Upper bound for size of subsets of a finite group that contains a sum-full set

Problem I'm looking for an upper bound for the number $k(G)$ of a finite group $G$, defined as follow: Let $\mathcal{F}_k$ be the family of subsets of $G$ with size $k$, and we define $k(G)$ be ...
11
votes
7answers
830 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 ...
3
votes
1answer
1k views

Queries about the Skolem-Mahler-Lech theorem (integer zeros of exponential polynomials)

The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. ...
7
votes
1answer
585 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 ...
4
votes
1answer
507 views

Cauchy-Davenport strengthening?

Is the following statement, refining classical Cauchy-Davenport Theorem (that states that for sets $A$, $B$ of residues modulo prime $p$, $|A+B|\geq |A|+|B|-1$ provided that RHS does not exceed $p$) ...
6
votes
1answer
813 views

Additive combinatorics and large Fourier coefficients

Elon Lindenstrauss explains in his talk at the MSRI in Fall 2008 (the relevant comment is at minute 41 of the video) that the set of large Fourier coefficients of a probability measure $\mu$ on the ...
4
votes
2answers
495 views

Sum of sets modulo a square

I would be glad to see a reference to the following easy lemma in additive combinatorics: if $A_1$ and $A_2$ are two sets of remainders modulo $n^2$, each has cardinality $n > 1$ and all elements ...
20
votes
1answer
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 ...
7
votes
1answer
433 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 ...
2
votes
3answers
218 views

Given N points on a number line and m total distances between those points, are there efficent ways to optimize for particular values in m?

Given a set of distances S, choose N unique points P on a number line such that the distances between the N points occur in S as much as possible. That is, maximize the occurence in S of the distances ...
13
votes
1answer
601 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 ...
5
votes
2answers
376 views

k-pseudorandom measures

In reading the paper of Green and Tao on arithmetic progressions within the primes, I became very interested in the notion of a k-pseudorandom measure discussed in that paper. A measure here is a ...