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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### A combinatorial property of uncountable groups

Let $A,B$ be two uncountable sets in a group $G$ such that for any elements $x,y\in G$ the intersection $xA\cap yB$ is finite. Let $\Phi:G\to 2^G$ be a function assigning to each element $x\in G$ some ...

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### On decomposition of finite Abelian groups

It is easy to see that for any finite Abelian group $G$ and any numbers $a,b$ with $|G|=ab$ there exist a subgroup $A\subset G$ and a subset $B\subset G$ such that $|A|=a$, $|B|=b$ and $G=A+B$, where $...

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179 views

### Sidon Sets and Diophantine Equation

Suppose $X$ is a subset of $\{1, \cdots, n\}$ such that the equation $ax_i+bx_j=cx_k+dx_{\ell}$ where $a+b=c+d,$ $a,b,c,d \in \mathbb{N}$ and $x_i, x_j, x_k, x_{\ell} \in X,$ has only trivial solution....

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### A Freiman-type of question for sets with small doubling costant

I start by saying that I have posted a similar question a few years back, but now I have refined the question a bit more. I have stumbled on this working in finite group theory. The question reminds ...

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257 views

### Escaping from a centralizer

Let $G = Sym(n)$, $n$ even. Let $H<G$ be the stabilizer of the partition $\{\{1,2\},\{3,4\},\dotsc,\{n-1,n\}\}$, or, what is the same, the centralizer of $(1\;2) \dotsc (n-1\; n)$.
By Stirling's ...

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81 views

### Polynomials passing through points with tangential conditions

In corollary here http://math.mit.edu/~lguth/Exposition/erdossurvey.pdf on polynomial methods it is said "(Parameter counting) If $S\subset\mathbb F^n$ is a finite set, then there is a non-zero ...

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47 views

### Equivalent condition for Poincare polynomial

I have found a statement in the introduction of the paper 'Sets of Recurrence and Generalized Polynomials' by Bergelson & Haland, which is
Result: Given a polynomial $p \in \mathbb{R}[x]$ such ...

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78 views

### Infinite difference length of integer subsets

Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition:
We say $A$ has infinite difference length, if
(a) For every integer $n$ there exist a ...

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312 views

### The soccer splitting problem in arbitrary commutative ring

There's a folklore problem:
Let $x_1, \cdots, x_{23} \in \mathbb{Z}$ be the weights of $23$ soccer players. Now Master Yoda want's to form two soccer teams with $11$ players each. Turns out for ...

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### The number of co-circular four tuples

Let $A,B ⊂ \mathbb{R}$ such that $|A| = |B| = n$. What is the best-known upper bound on the number of four-tuples in $A \times B$ where the four points are co-circular, they lie on the same circle?

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222 views

### Additive basis of order 2

Can we find $\alpha>1$ such that $u=(\lfloor n^\alpha\rfloor)_{n\geqslant0}$ is an additive basis of order $2$ (i.e. $\forall x\in\mathbb{N}, \exists(n,m)\in\mathbb{N}^2, x=u_n+u_m$) ?
Remark : ...

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### Density version of the Erdos-Graham conjecture

In 2003 E. S. Croot [Ann. of Math. 157(2)(2003), 545-556] proved the Erdos-Graham Conjecture which states that if $\{2,3,\ldots\}$ is partitioned into ﬁnitely many subsets then one of the subsets ...

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247 views

### A permutation problem

Here I ask a question on permutations of $n$ distinct real numbers.
QUESTION: Let $a_1,a_2,\ldots,a_n\ (n>1)$ be (pairwise) distinct real numbers. Is there a permutation $b_1,\ldots,b_n$ of $a_1,\...

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### Does every $n\times n\times n$ Latin cube contain a Latin transversal?

In 1967 H. J. Ryser conjectured that every Latin square of odd order has a Latin transversal. Similar to Latin squares, we may consider Latin cubes.
QUESTION: Let $n$ be any positive integer. Does ...

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55 views

### Maximum-size sum sets

Let $A, B \subseteq G$ for abelian group $G$. It is obvious that $|A+B| \leq |A| \cdot |B|$. Is there an easy characterization of sets $A, B$ meeting this bound? I am especially interested in the case ...

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### A permutation problem for finite subsets of an abelian group

Here I ask the following question in additive combinatorics.
QUESTION: Let $A$ be any finite subset of an additive abelian group $G$ with $|A|=n>3$. Can we write $A$ as $\{a_1,\ldots,a_n\}$ so ...

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193 views

### Is every integer $n>1$ the sum of two squares and two central binomial coefficients?

Those integers $\binom{2n}n\ (n=0,1,2,\ldots)$ are called central binomial coefficients. By Stirling's formula,
$$\binom{2n}n\sim \frac{4^n}{\sqrt{n\pi}}\ \ \ \ (n\to+\infty).$$
Of course, the ...

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159 views

### Balancing points with lines

$\newcommand{\F}{\mathbb F}$
Suppose that $p$ is a prime, and $k<p/2$ a positive integer.
Consider a system of $k$ distinct directions in the affine plane $\F_p^2$, and the system of $k$ pencils ...

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### Is every integer $n>1$ the sum of two triangular numbers and two powers of $5$?

Recall that the triangular numbers are those integers
$$T_n=n(n+1)/2\ \ \ (n=0,1,2,\ldots).$$
In 1796 Gauss proved that each $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three triangular numbers, ...

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232 views

### Asymptotic for number of partitions of $n$ into $k$ squares, uniform in $n,k \to +\infty$

Let $p^{(s)}(n)$ be the number of ways of writing the positive integer $n$ as a sum of perfect $s$-powers, where the order does not matter. For example, $p^{(2)}(9) = 4$ since
$$9 = 1^2 + 1^2 + 1^2 + ...

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### Decomposing a subset of $\mathbf Z$ into a sumset of irreducibles

We say that a subset $A$ of $\mathbf Z$ is irreducible if $|A| \ge 2$ and there do not exist $X, Y \subseteq \mathbf Z$ with $|X|, |Y| \ge 2$ such that $A = X + Y$.
If $X \subseteq \mathbf Z$, we ...

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### $m$-thick sets with small $n$-fold sumsets in finite cyclic groups

Problem. Is it true that for every positive integers $n,m$ there exists a subset $A_{n,m}$ of a finite cyclic group $G$ having the following two properties:
$(\Sigma_n)$ the $n$-fold sum $A_{n,m}^{+...

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### Is there an “analytical” version of Tao's uncertainty principle?

Let $p$ be a prime. For $f: \mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{C}$ let its Fourier transform be:
$$\hat f(n) = \frac{1}{\sqrt{p}}\sum_{x \in \mathbb{Z}/p \mathbb{Z}} f(x)\, e\left(\frac{-xn}{...

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### Prove A Skipping Prime Conjecture For Rio?

I am writing a paper to accompany a Short Communication I plan to give in Rio this August. The paper regards work on jumping primes, a project on which Jose Brox has been working with me. I was going ...

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164 views

### Behrend's Construction

Consider the following statement due to F.Behrend (1946)
Theorem: Let $N$ be a large integer. Then there exists a subset $A\subset [1,N]$ with $\frac{|A|}{N}\geq \exp(-4\sqrt{\ln N})$ which does not ...

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### Existence of Arithmetic Progression from density inequality

Let $A\subset \{0,1,\dots, N-1\}$ such that $$|A\cap [0,N/3)|\geq \left(\delta+\dfrac{\delta}{8}\right)\cdot \dfrac{N}{3},$$ where $\delta\in (0,1]$. Prove that exists arithmetic progression $P$ with $...

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285 views

### Sets $A$ stable under $(x,f(x))\mapsto x+f(x)$

Let $A$ be a finite set of real numbers or integers. We know how to characterize, broadly speaking, sets $A$ such that $A+A$ is not much larger than $A$ (Freiman's theorem). I have a question that ...

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### Square weighted integer partition

As a part of my research I have arrived at the following generating function
$\prod_{k=1}^{\infty}\left(\frac{1}{1-k^{2}x^{k}}\right)$
This is almost similar to the generating function of the ...

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### Inequality in the proof of Roth's theorem on progressions

I was working out through the proof of Roth's theorem on arithmetic progression. I ran into the following inequality which is not clear to me: Let $A\subset \mathbb{Z}_N$ and $M_A:=A\cap [N/3,2N/3)$ ...

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111 views

### Bounding the size of certain sumsets in the plane

Let $A$ be a finite set in $\mathbb{R}^2$ of $k^2$ elements and consider a set $B=\{x_1,x_2,x_3,x_4\}$ such that the points in $B$ are in general position (no three points on a line).
Question 1: Is ...

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### Lemma towards Roth's theorem

I am trying to work out the proof of the following lemma which is used in the proof Roth's theorem.
However, I guess that this lemma has some flaws in the proof and I would be very grateful is ...

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199 views

### Separating points of shifts of a finite set in the plane

Let $A\subset \mathbb{R^2}$ be a finite set such that $|A|=k^2$. Let $x_i\in \mathbb{R^2}$, $i=1,2,3,4$, be four points in the plane in general position (no three lie on any line).
Let us form the ...

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### Density of intersection with shifted sets

Given a subset $S$ of the positive integers $\mathbf{N}$, let $\mathrm{d}^\star(S)$ be its upper asymptotic density, that is,
$$
\mathrm{d}^\star(S)=\limsup_{n\to \infty}\frac{|S \cap [1,n]|}{n}.
$$
...

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### Smallest $k$ such that every vector is a linear combination of at most $k$ generators

Let $V$ be a finite dimensional vector space over a finite field $\mathbb{F}_q$ (the most interesting case to me is $q=O(1)$). Let $A$ be an arbitrary subset of $V$ that spans $V$ over $\mathbb{F}_q$. ...

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236 views

### Partition set into equal-sum subsets

What is known about how to divide $\{1,4,9,16,...,81^2\}$ into $3$ equal-sum subsets with $27$ elements per subset?

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112 views

### Properties of Zero Line-Sum Matrices

By a Zero Line-Sum (ZLS) matrix I mean matrices with the property, that each row sum and each column sum equals zero:
$$A\in\mathbb{R}^{m\times n}:\ \sum_{i=1}^{n}a_{ij}=\sum_{j=1}^{m}a_{ij}=0$$
...

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### large arithmetic progression modulo p (II)

Is it possible to construct a $B$ $\subseteq$ $Z_p(=Z/pZ)$ of cardinal $cp^{\frac{1}{3}}$, for some constant $c$, such that there exists an arithmetic progression of size $c_1p^{\frac{2}{3}}$, for ...

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### A formula for Frobenius number of certain numerical semigroups

The old formula for the Frobenius number of a numerical semigroup generated by two elements can be stated as follows: assume $\gcd\{a+1,b+1\}=1$, then the Frobenius number of $S= \left<a+1,b+1\...

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### Large arithmetic progression modulo $p$

Let $B$ be a subset of $Z_p(=\mathbb{Z}/p\mathbb{Z})$ of cardinal $Cp^{\frac{1}{3}}$, for some constant $C$. How to construct an arithmetic progression of length $C_1p^{\frac{2}{3}}$ where $C_1$ is ...

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378 views

### Near-linear mappings from $\mathbb F_p$ to $\mathbb R$

$\newcommand{\F}{{\mathbb F}}$
$\newcommand{\R}{{\mathbb R}}$
$\renewcommand{\phi}{\varphi}$
Let $p\ge 5$ be a prime.
If the functions $\phi_1,\phi_2,\phi_3\colon\F_p\to\R$ satisfy $\phi_1(x)+\...

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134 views

### Disjoint union of affine subspaces contains a larger affine subspace

I'd like to say that a large structured subset of the $n$-dimensional Boolean cube $\{0,1\}^n$ contains a non-trivial affine subspace. To be more specific, I want to prove/disprove that for some ...

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548 views

### Increasing sequences in polynomial progressions modulo p

In a random permutation on $n$ elements one expects the largest increasing and decreasing sequences to have size $(2+o(1))\sqrt{n}$. Is it known if this same property holds in sequences given by ...

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457 views

### Which of these sums appear most often?

Let $N=\{1,2,3,\ldots, n\}$.
We sum all the elements of every nonempty subset of $N$.
Which sum(s) appears most often? (Let's call this sum a champion).
Using a simple pigeonhole argument a champion ...

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87 views

### Smallest size of integral vector with certain inner product

Let $v =(r,s,t) \in \mathbb{N}^3$ be a vector such that $\gcd(r,s,t)=1$. We know that there are vectors $x= (x_1,x_2,x_3) \in \mathbb{Z}^3 $ such that $v.x =1$. For each $v$, let $O(v)$ be the ...

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319 views

### Binary weight of shifted integers

Suppose $n_2$ denotes the binary representation of the integer number $n$. Let $X_2(n)=[1_22_2\ldots n_2]$,$n\geq3$, be a binary vector which is obtained by concatenating of binary representation of ...

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41 views

### Questions on “The condition number of a randomly perturbed matrix”

This question is about the two vectors $w'$ and $y$ that are necessary for the argument in section $7$ (page 6) of this paper by Terence Tao and Van Vu,
https://arxiv.org/abs/math/0703307 (that ...

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126 views

### A question about finite free convolution

For any square matrix $Y$ let $\chi_x(Y) = det(xI -Y)$ denote its characteristic polynomial.
Say $A$ and $B$ are two $n-$dimensional symmetric matrices with constant row sums $a$ and $b$. Lets ...

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142 views

### Large submatrices of circulant matrices

What properties of circulant matrices are inherited by their principal submatrices? To be more specific:
Given a square (zero-one) matrix $M$ of order $n$ with all elements on its main diagonal ...

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303 views

### A question about the paper “The Condition Number of a Randomly Perturbed Matrix”

My question pertains to this paper by Terence Tao and Van Vu, https://arxiv.org/abs/math/0703307
Both my questions pertain to the argument presented in this paper in its section 6 (page 5). We are ...

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26 views

### Generating larger atoms from smaller ones in a simple $\text{C}_0$-monoid

Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$ (which I'll write multiplicatively), $H$ a submonoid of $\mathscr F(P)$, and $\mathcal A(H)$ the set of atoms of $H$ (...