Articles of additive combinatorics

Some co-finite subsets of rational numbers

Is there any $A\subseteq \mathbb{Q}$ such that $A-A$ is a non-trivial co-finite subset? Note that $A-A=\{a_1-a_2: a_1,a_2\in A\}$, also see A type of integer numbers set.

Questions on Erdős–Ginzburg–Ziv theorem for primes and understanding related lemmas and their applications.

While trying to prove the prime case of Erdős–Ginzburg–Ziv theorem: Theorem: For every prime number $p$, in any set of $2p-1$ integers, the sum $p$ of them divisible by $p$. I came across with the following lemma: Lemma 1: If $p$ is a prime number and $A,B$ are subsets of $\mathbb{Z}_p$ with $\varnothing\neq A\neq \mathbb{Z}_p […]

Number writable as sum of cubes in $9$ “consecutive” ways

Let’s say that a given $n\in\mathbb{N}$ is writable as sum of cubes in $k$ consecutive ways if it can be written as sum of $j,j+1,\ldots, j+(k-1)$ nonzero cubes, for some $j\geqslant 1$. For example, $26$ is writable as sum of squares in $4$ consecutive ways, for \begin{align} 26 &= 1+25 &&\text{($2$ squares)}\\ 26 &= 1+9+16 […]

Constructing arithmetic progressions

It is known that in the sequence of primes there exists arithmetic progressions of primes of arbitrary length. This was proved by Ben Green and Terence Tao in 2006. However the proof given is a nonconstructive one. I know the following theorem from Burton gives some criteria on how large the common difference must be. […]

Representation of integers by Fibonacci numbers

Denote by $F$ the set of all Fibonacci numbers. It is our conjecture that: (a) For every integer $n$ there exist a number $k=2^q$ (for some positive integer $q$) and numbers $a_1,\cdots,a_k\in F$ such that $$ n=a_1+\cdots+a_{\frac{k}{2}}-(a_{\frac{k}{2}+1}+\cdots+a_k). $$ Now, denote by $k(n)$ the least $k$ obtained from (a). (b) The set of all $k(n)$, where […]

About translating subsets of $\Bbb R^2.$

I’m looking for a pair of sets $A,B$ of points in $\Bbb R^2$ such that $A$ is a union of translated (only translations are allowed) copies of $B;$ $B$ is a union of translated copies of $A;$ $A$ is not a a single translated copy of $B$ (and the other way around, which follows). The […]

A type of integer numbers set

Is there any $A\subseteq \mathbb{Z}$ such that $A-A$ is a non-trivial co-finite subset? Note that $A-A=\{a_1-a_2: a_1,a_2\in A\}$.

Cardinality of sum-set

Let $X\subset R$ and $Y \subset R$ where X and Y have finite cardinalities. Let also, $a,b \in R$. How to show that $|aX+bY|=|X||Y|$ almost everywhere (measure of $(a,b) \subset R^2$ such that $|aX+bY|<|X||Y|$ is 0) ? Where we define $|aX+bY|=\{ax+by:x \in X, y \in Y \}$. More, specifically let $X=\{x=d_x*z: z \in \mathcal{Z}, d_x […]

Sum of multiples of $2$ coprime numbers cover positive integers

This question already has an answer here: The Frobenius Coin Problem 4 answers

Sum of two squares modulo p

I have heard somewhere that for all primes $p$, for all $k$, there exist $x, y$ s.t. $x^2 + y^2\equiv k \pmod{p}$? I recall that the proof is very elementary, but I can’t remember such a proof. How would one prove this? One way is to use Cauchy-Davenport, but I don’t think that this is […]