Questions tagged [additive-combinatorics]

Additive combinatorics is about giving combinatorial estimates of addition and subtraction operations on Abelian groups or other algebraic objects. Key words: sum set estimates, inverse theorems, graph theory techniques, crossing numbers, algebraic methods, Szemerédi’s theorem.

Additive combinatorics is about giving combinatorial estimates of addition and subtraction operations on Abelian groups or other algebraic objects. Key words: sum set estimates, inverse theorems, graph theory techniques, crossing numbers, algebraic methods, Szemerédi’s theorem.

337 questions
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What is $\tau(A_n)$?

Suppose G is a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$. What is $\tau(A_n)$? Similar problems for some different classes of groups are already…
42
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A Nice Problem In Additive Number Theory

$\color{red}{\mathrm{Problem:}}$ $n\geq3$ is a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a multiple of $n$. Prove there are at least $n$…
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Number of vectors so that no two subset sums are equal

Consider all $10$-tuple vectors each element of which is either $1$ or $0$. It is very easy to select a set $v_1,\dots,v_{10}= S$ of $10$ such vectors so that no two distinct subsets of vectors $S_1 \subset S$ and $S_2 \subset S$ have the same sum.…
graffe
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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…
Bartek
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21
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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…
21
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A problem in additive number theory.

Original Problem: Counterexample given below by user francis-jamet. Let $A\subset \mathbb Z_n$ for some $n\in \mathbb{N}$. If $A-A=\mathbb Z_n$, then $0\in A+A+A$ New Problem: Is the following statement true? If not, please give a…
Terry Zhou
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Is every prime the average of two other primes?

$\forall {p_1\in\mathbb{P}, p_1>3},\ \exists {p_2\in\mathbb{P},\ p_3\in\mathbb{P}};\ (p_1 \neq p_2) \land (p_1\neq p_3) \land (p_1 = \frac{p_2+p_3}{2})$ Now I'm not a 100% sure about this, but I vaguely remember proving this once, but I cannot…
19
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2 answers

Sumset that covers $\mathbb{Z}/p\mathbb{Z}$.

Let $p$ be a prime. Let $S$ be a set of residues modulo $p$. Define $$S^2 = \{a \cdot b \mid a \in S, b \in S\}.$$ Question: How small can we make $|S|$ such that $\{0, 1, \cdots, p-2, p-1\} \in S^2$ ? It seems that the optimal bound should be…
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An old question about sumsets and difference sets

Let $A$ be a finite set. Define the symbols $+$ and $-$ as follows: $$A+A=\{a+b:a,b\in A\};$$ $$A-A=\{a-b:a,b\in A\}.$$ Prove or disprove $|A+A|\leq|A-A|$, where $|A|$ denotes the cardinality of $A$. This is a seemingly correct conjecture. But…
Colescu
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If a finite set tiles the integers, must it be an arithmetic progression?

Let $P$ be a finite subset of ${\mathbb Z}$ containing at least three elements. I say that $P$ tiles $\mathbb Z$ if $\mathbb Z$ can be written as a disjoint union of translates of $P$. Trivially if $P$ is an arithmetic progression of the form…
Ewan Delanoy
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A Combinatorial problem involving $\mathbf{Z}/2^k\mathbf{Z}$

Let $k$ be a positive integer. Let $A=\{a_1,\dots, a_{2^k}\}$ be a subset of $\mathbf{Z}/2^{k+1}\mathbf{Z}$ whose image in $\mathbf{Z}/2^k\mathbf{Z}$ is the whole $\mathbf{Z}/2^k\mathbf{Z}$. Let $B=\{b_1,\dots, b_{2^k}\}$ be a subset with the same…
14
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4 answers

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…
Eugene
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Is the Green-Tao theorem a consequence of the Euler's theorem?

The Erdős-Turán conjecture states that If $A\subset\mathbb{N}$ is such that $$ \sum_{n\in A} \frac{1}{n} = \infty,$$ then $A$ contains arithmetic progressions of any given length. I'm interest when $A$ is the set of all prime number . In this…
13
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3 answers

Special subdivision of numbers from 1 to 99

I've been lately working on a problem I still can't solve. The problem is: Can we divide numbers from 1 to 99 into 33 groups of three numbers, such that in every group one number is the sum of the two remaining elements? Thank you in advance for any…
mounaim
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