# Tagged Questions

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.

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### Additive combinatorics and $|S \cap (S+t)| \geq K |S|$ over $\mathbb{F}_2^n$

I don't know much about additive combinatorics, and I am wondering if there are results concerning the size of $|S \cap (S+t)|$ where $S \subset \mathbb{F}_2^n$ and $t \in \mathbb{F}_2^n$. Especially, ...
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### Understanding Ziv's proof of zero sum problem .

I was going through the proof of zero sum problem in one dimension as provided by Abraham Ziv. The problem statement is to prove that given a set of $2n+1$ integers, we can find at least $n$ integers ...
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The definitions: Let $n \in \mathbb{Z}^+$. A positive integer $L$ is called $n$ -magnificent if in every partition of $\{ 1, ... , L \}$ into $n$ non-empty parts, at least one part contains an ...
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### Additive combinatorics, moments, generating functions, Cauchy's formula and asymptotics

Let $A, B$ be finite sets of positive integers, let $$f(z) = \left(\sum_{a \in A} z^a\right)\left(\sum_{b \in B} z^b \right) = \sum_{m \geq 0} r_{A, B}(m) z^m,$$ where $r_{A, B}(m)$ is the number of ...
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### Number of ways, modulo a prime $p$, to write $n$ as a sum $n = x_1^k + x_2^k + \cdots + x_s^k$

Removing the restriction on $p$, this is known as Waring's problem. The circle method has been used successfully to tackle this. Using analysis, nice estimates can be given. I wonder what analytic ...
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### Extremal set theory problem concerning translations of a set of integers

Let $A$ be a subset of $B = \{1, 2,\ldots,n\}$. Suppose that $F$ is a family of subsets of $B$, each of which is a translation of $A$ and no two of which intersect more than once. What is the maximum ...
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### Sumset of a subset of a group

I am interested in the following which I believe is known: Let $S$ be a subset of a finite group $G$ containing more than half of $G$'s elements. Then $S+S = G$. I have been looking but can not ...
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### Prove: A 9 element subset of ${1,2,…,99}$ must have two distinct subsets with the same sum.

APMO 2014 Problem 4: Prove: A 9 element subset of ${1,2,...,99}$ must have two distinct subsets with the same sum. I am having a lot of trouble with this problem. The official solution: ...
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Given $n,m\in\Bbb N$. We want to find two disjoint sets $A$ and $B$ such that $$|A|=|B|=n$$ $$\min\{a\in A,b\in B\}>m$$ $$|A+B|=2n$$ where $A+B=\{a+b:a\in A, b\in B\}$. What is the minimum ...
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### Understanding an example of a prime-like non-finite additive basis

In this answer on MO, the user Gene S. Kopp gives an example of a relatively "big" set $A\subset \mathbb{N}$ with relatively "small" gaps that fails to be an asymptotic finite basis. I'm having a ...
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### Probability addition rule over 100 percent?

In Texas holdem, one is dealt a Decent Hand (Any pocket pair or any two broadway cards) ~15 percent of the time. If there are three people left in the hand, I can use the probability addition rule to ...
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### Is there any general method for solving $(a_1+a_2+..a_n)^2=a_1^3+a_2^3+…+a_n^3$ in positive integers $a_1,a_2,…a_n$?

We know the identity $(1+2+...+n)^2=1^3+2^3+...+n^3$ . So I was thinking , for given $n\in \mathbb N$ , is there any general method for solving $(a_1+a_2+..a_n)^2=a_1^3+a_2^3+...+a_n^3$ in positive ...
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### Rusza triangle inequality and approximate groups.

Feel free to scroll down to the "Question" section if you're familiar with the notation of Tao and Vu's Additive Combinatorics, which I believe is standard notation for the field. Notation Let $Z$ ...
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### 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 ...
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### Simple $\{-1,0,1\}$ equation set

I'm trying to find the shortest path, getting from $x=0$ to $x=k$ in a certain problem, where I can slowly accelerate and decelerate. It comes down to finding the smallest $n$ and set of values ...
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### Combinatorics problem on the size of A+B

Let $A$, $B$ be finite subsets of $\mathbb{Z}$ with $|A|=n$, $|B|=m$. Denote $A+B=\{a+b:a \in A, b \in B\}$. It's fairly easy to show that $|A+B| \geq n+m-1$. My question is: If $|A+B|=n+m-1$, ...
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### Does there exist an integer $s$ such that every integer $> 1$ can be written as a sum of at most $s$ primes?

Does there exist an integer $s$ such that every integer $> 1$ can be written as a sum of at most $s$ primes ?
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### Is it true that every sufficiently large positive integer can be written as a sum of a square free number and a perfect square ?

Is it true that $\exists k \in \mathbb Z^+$ such that every integer $n >k$ can be written as a sum of a square free number and a perfect square ?
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### Asymptotic expression for $3$ term arithmetic progression in the primes

I have found an asymptotic for the following sum using the circle method: \begin{align} R(n)=\sum_{\substack{p_1,p_2,p_3 \le n \\p_1+p_2=2p_3 }} \log (p_1) \log (p_2) \log ...
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### Inequality of sizes of finite sets

For a set $A\subset \mathbb N$, define $$A+c\cdot A=\{a+c\cdot a'|a,a'\in A\}$$ Is it true that for every such finite set $A$ $$\frac{|A+2A|}{|A|}\leq\left(\frac{|A+A|}{|A|}\right)^3$$?
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### Approximate groups

I am reading the article "The structure of approximate groups" by Breuillard, Green and Tao. At some point they state the following theorem 1.6: Let A be a K-approximate subgroup of a group G. Then ...
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### How to prove such inequality?

I have some problems with proof of the inequality: A - is a set. I tried to use Plünnecke-Ruzsa inequality but didn't get any result. It will be great if someone help a little wuth this problem
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### Ex. 1.3.10 in Tao's Additive Combinatorics

For $k \geq 2$ and $B \subset \mathbb N$ set $r_{k,B}(n) = |\{ (x_1,\dots,x_k) \in B^k: \sum x_i = n\}|$. The problem at hand is to show that it can not happen that for some $m \geq 1$ that ...
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### additive number theory: sums and products of subsets of integers

Suppose that $A$, $B$ are finite subsets of the integers. Consider the subset $E$ of $A+B$ consisting of all elements $s$ of $A+B$ that can be written uniquely as $s=a+b$, where $a\in A$ and $b\in B$, ...
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### Find the constant c > 0?

Find a constant c > 0 such that for every finite set of integers B not containing 0, there is a subset A of B such that A is sum-free and |A| ≥ c|B|, where |A| means the number of elements of A.
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### Choosing sets yielding large sumsets

I am working on a problem which basically boils down to wanting to choose sets $A_1, \ldots, A_m$ over a group $G=(\mathbb{Z}_C,+)$, where $C$ and $m$ are constants, such that the sumsets defined with ...
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### How many ways are there of choosing $k$ distinct items from a set of $n$?

Specifically, say I have the integers $1,2,3,\dots,n$ (a set of $n$ integers). I want to select numbers one after another (not at the same time) until I have $k$ distinct numbers. How many ways are ...
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### Tilde Notation in Additive Combinatorics

A, B are some finite subsets of a abelian group. |A + B| ~ |A| The problem is I couldn't precisely understand what does ~ mean in that case. For example, it could be found in notes of Tao's ...