# 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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### When does the equality hold in Dias da Silva - Hamidoune Theorem?

Let $p$ be prime number and let $A$ be a $k$-elements subset of $\mathbb{Z}/p\mathbb{Z}$. Dias da Silva - Hamidoune Theorem states that $|h^{\hat{}}A| \geq \min(p, hk -h^2 + 1)$, where $h$ is an ...
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### Convolution of indicator functions with values in a finite field

This is something I haven't seen online yet, indicator functions with values in a finite field. Probably for a good reason, but I would like to know why, and if there are still things that can be said....
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### Contradiction in Frobenius coin problem

The Frobenius coin problem guarantees that if $(a,b)=1$, then $$ax+by$$ does not represent exactly $(a-1)(b-1)/2$ numbers all below $ab-a-b$ if $x,y\geq0$ holds. I am confused by following argument. ...
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### Erdős and Szemerédi sums and producs

Erdős and Szemerédi proved that: $$\max(|A+A|,|A \cdot A|) \gg |A|^{1+\epsilon}$$ It might be that the work of Erdős and Szemerédi does not help me here at all, I did not have to deal with math much ...
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### Is $1072$ the sum of $51$ cubes?

Inspired by this superb question Number writable as sum of cubes in $9$ "consecutive" ways I wonder, whether $1072$, being the sum of $2,3,4,...,50$ cubes, is also a sum of $51$ cubes. Of ...
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### 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 ...
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### Invariance of a set under permutations

Given a positive integer $n$, let $f$ be a function $$\{1,\ldots,n\}^2 \to \{1,\ldots,n\}.$$ Then, it is possible that there exists a permutation $\{\sigma_1,\ldots,\sigma_n\}$ of $\{1,\ldots,n\}$ ...
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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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### To determine number of arrangements of 4 letters in a word so that the transitions remains conserved

A 10 letter word is composed of $A,\ B,\ C,\ D$. The problem is to find the number of arrangements of these alphabets which could lead to fixed number of transitions between each pair of alphabets. ...
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I'm working in an exercise and I need some results about additive theory of numbers, I encountered this problem: Given an element $x\in \mathbb{Z}/n\mathbb{Z}$ and two subsets $A$ and $B$ of $\... 1answer 83 views ### Asymptotic expression for$3term arithmetic progression in the primes I have found an asymptotic for the following sum using the circle method: \begin{align} R(n)=\sum_{\substack{p_1,p_2,p_3 \le n \\p_1+p_2=2p_3 }} \log (p_1) \log (p_2) \log (p_3)=\mathfrak{S}\frac{n^2}... 1answer 48 views ### Inequality of sizes of finite sets For a setA\subset \mathbb N$, define $$A+c\cdot A=\{a+c\cdot a'|a,a'\in A\}$$ Is it true that for every such finite set$A$$$\frac{|A+2A|}{|A|}\leq\left(\frac{|A+A|}{|A|}\right)^3$$? 1answer 48 views ### 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 ... 1answer 75 views ### 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 2answers 117 views ### 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$r_{k,B}(...
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 $b\in B$, ...