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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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 ...
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Proof that $G(3)\le 7$

Let $G(k)$ be the minimal $n$ s.t. every sufficiently large integer is the sum of $n$ nonnegative $k$th powers. Does anybody know where I can find Vaughan's proof that $G(3)\le 7$? I can't find a ...
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I need a set that enables me to identify specific containing elements by any summation of any of its subsets (see example to understand)

My question is more practically understood by example. I need a set A that behaves like the one below: Set A: {1,3,5} Set B (all subsets of A): {1}, {3}, {5}, {1,3}, {1,5}, {3,5}, {1,3,5} Set C ...
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How can the smallest set of integers be generated such that the sums cover a given set?

I have a set of positive integers S. I want to generate a set of positive integers T such that every member of S is the sum of some combination of members from T. I am looking for the smallest ...
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Cardinality of finite sets: if $B=\{a-b \mid a,b \in A\}$ and $C=\{a+b \mid a,b \in A\}$, then $|C|^2\ge |A||B|$

Let A be a finite set of real numbers. Suppose $B=\{a-b \mid a,b \in A\}$ and $C=\{a+b \mid a,b \in A\}$. Prove that $|C|^2\ge |A||B|$. I tried to solve this in this way: We claim that the function ...
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A sequence of subsets of $\Bbb Z$

Is there a sequence $(A_n)$ of nonempty subsets of $\Bbb Z$ such that for each $n$, $$\{a-b\mid a,b\in A_{n+1}\}\subsetneqq A_n$$ ?
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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 ...
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Are there any “nontrivial” sets with small difference sets?

I'm trying to find finite sets $S$ of natural numbers with "small" difference sets. One option is just taking an arithmetic progression $S = \{0, , \ldots, n-1\}$. Then $|S - S| = 2 |S| - 1$, which ...
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“Spanning” the difference set of $S$

Suppose that $S$ is a finite set of natural numbers, and $\{(x_i, y_i)\}$ is a set of tuples of numbers in $S$ with $$ \{x_i - y_i\} = S - S := \{a - b \mid a, b \in S\} $$ that is, $\{(x_i, y_i)\}$ ...
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Which basis orders [for the natural numbers] have been proven?

The set $A$ of nonnegative integers is called an additive basis of order $h$ if every nonnegative integer can be written as the sum of $h$ not necessarily distinct elements of $A$. For example, the ...
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What is Vandermonde's formula with multisets?

I need Vandermonde's formula in multi-set form. I modified the original formula but I get a mess with too many letters everywhere, is there a nice representation? Here's the original: $$ ...
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Szemeredi Trotter and additive combinatorics on A+AA

I am trying to get a lower bound on $|A+AA|$ where $A$ is a set, and $A+AA=\{a+bc: a,b,c \in A\}$ using Szemeredi Trotter. I would think we need to form lines of the form $y=ax+b$ where $a,b \in A$, ...
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Generating finite additive groups using sumsets

Let $G$ be a finite additive group and let $A,B \subset G$. The sumset $A+B$ is defined as $A+B = \{a+b \mid a \in A, b \in B\}$, where addition is in $G$. We use $kA$ for short of $A+A+\ldots+A$ with ...
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Looking for references

I am looking for reference on the following problem. Let $S= \{ ax_1+bx_2 \mid x_1 \in X_1 , x_2 \in X_2, \}$ where $X_1,X_2\subseteq \mathbb{R}^n$ and $a,b \in \mathbb{R}$. Note that $a$ and $b$ ...
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Multiplicative subgroup of a finite field with prescribed trace.

Any suggestions/methods/estimates for the following problem would be very appreciated. $l,p$ are primes with $p \equiv 1 \!\! \pmod l$. $r$ is a positive integer with $r \equiv 1 \!\! \pmod p$ and $q ...
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Inequalities that show if a distribution decays slowly

Often, one is often interested in theorems/inequalities of the following kind: Let $X$ be a random variable then the probability that $X$ is close to typically $\mu$ (or larger than some constant) is ...
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28 views

Cardinality of sum-set of two arithmetic progressions

Suppose that sum-set $A+B$ between sets $A$ and $B$ is defined as $A+B=\{a+b|a \in A, b \in B \}$. We further assume that $A=\{d_az|z \in \mathbb{Z} \}$ and $B=\{d_bz|z \in \mathbb{Z} \}$ where $d_a$ ...
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Finding an imperfect finite difference set for large N

I want to find a set of integers $N$ for which there always exists a pair of numbers $(a, b)$ both $\in N$ such that $a-b = x$ for all $0<x<2^{32}$. Obviously one possible set N is all the ...
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Sumset magnification ratio strictly smaller for subset.

Do there exist sets $X \subset A \subset \mathbb{Z}$ such that $$\frac{|A+X|}{|X|} < \frac{|A+A|}{|A|} $$? I would also be happy if one can replace $\mathbb{Z}$ with any other abelian group.
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$|(A+x) \cap A| \leq 1$, for all $x \in \mathbb{Z}_n$, $x \neq 0$.

Let $n$ be a positive integer. If $k$ is an integer such that $2^{k+1} \leq n$, then $A=\{1,2,2^2,...,2^{k-1},2^k\}$ is a subset in $\mathbb{Z}_n$ such that $|(A+x) \cap A| \leq 1$, for all $x \in ...
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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 ...
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Number-Theoretic Coin Puzzle

There are three piles of coins. You are allowed to move coins from one pile to another, but only if the number of coins in the destination pile is doubled. For example, if the first pile has 6 coins ...
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Proving that any sufficiently large sequence of vectors has zero-summed subsequence

Given $k,d$ I want to prove that any sufficiently large sequence of vectors over ${\pm 1, 0}$ whose sum is in $[-k,k]^d$ has a nonempty proper subsequence whose sum is $\bar 0$. Using probabilisitic ...
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Intersection of subsets of additive groups

Given two subsets $A$ and $B$ of a finite additive group $Z$, how can one show that there exists an element $x$ in $Z$ such that $$1 - \frac{|A \cap (B + x)|}{|Z|} \le \left(1 - ...
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44 views

Bounds on cardinality of sum-sets. [duplicate]

Let $X\subset \mathbb{R}$ and $Y \subset \mathbb{R}$ where X and Y have finite cardinalities. Let also, $a,b \in \mathbb{R}/0$. What can we say about cardinality $|aX+bY|=????$ For example we can ...
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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 ...
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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 ...
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23 views

Testing density with a countable family

Let $d$ denote the lower density on $\mathbb{N}$, $a>0, $ $\mathbb{N}_{a}:=\left\{ B\subset\mathbb{N}:{d}(B)\geq1-a\right\} $ and $A\subset\mathbb{N}$. If $A\cap B\neq\emptyset$ for every ...
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Reasoning about Schnirelmann Density: Proving that $d(C) \ge d(A) + d(B)$

I am taking this argument from Gelfond & Linnik's Elementary Methods in the Analytic Theory of Numbers. They state if for every $n \ge 1$, there exists $m \in [1,n]$ where $C(n) - C(n-m) \ge ...
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Linear algebra and combinatorics. For a family with even size sets and even intersections prove that $|F| \le 2^{n/2}$

Let $F \subset P(n)$ be a family such that for all i and j $ |f_i \cap f_j|$ and $|f_i|$ are even Prove that $|F| \le 2^{n/2}$ Now I think we go by contradiction and say if $|F| \ge 2^{n/2}+1$ ...
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4 squares almost in an arithmetic progression

It is well known that the exists no arithmetic progression of squares of length $4$. But consider the following arithmetic progression of length $5$: $49,169,289,409,529$. All terms apart from the ...
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Rate of convergence of $n$-fold convolution over $\mathbb{Z}_p$.

Suppose that $\mu$ is a probabilist measure on $\mathbb{Z}_p$ such that $d_{TV}(\mu,\mu_U)\leqslant \delta < 1$. What are the best upper bound on $$ d_{TV}(\underbrace{\mu*\mu*\ldots*\mu}_{n ...
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Bounds on an additive combinatorics problem (just looking for references)

I'm looking for known results of a problem, but i don't know the right terms to look for. What is the minimal number $a$, s.t. any subset $A\subseteq \mathbb{Z}_3^n$ $\ \ \ |A|\ge a$, contains 3 ...
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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 ...
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1answer
96 views

Cauchy-Davenport theorem and its extension

According to Cauchy-Davenport Theorem, if $A,B$ are subsets of a prime field ($F_p$) then we have the following bound on the number of elements within the sumset $A + B = \left\{ {\left. {a + b} ...
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Maximal size of an almost-disjoint linearly independent family in $K^{\mathbb{N}}$

Let $K$ be a field, say infinite, and denote by $L$ the $K$-vector space $K^{\mathbb{N}}$. What is the maximal cardinality of a $K$-linearly independent subset $X$ of $L$ such that any two distinct ...
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73 views

Lower bound for quotient sets

Exercise 2.8.6 from Tao and Vu's Additive Combinatorics asks: Given a subset $A\subset\mathbb{F}_p$ with size $|A|>p^\frac{1}{k}$ $(k\geq 2)$, show $\frac{A-A}{A-A}$ has size at least ...
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Show there’s at most $n\choose \left \lfloor\frac{n}{2} \right\rfloor$ subsets $A\subset[n]$ such that $\displaystyle\sum\limits_{i\in{A}} a_i=\alpha$

Let $a_1, a_2, a_3, ... , a_n$ and $\alpha$ be n+1 non-zero real numbers. Prove that there are at most $n\choose \left\lfloor\frac{n}{2}\right\rfloor$ subsets $A\subset[n]$ such that ...
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Cubic sum of Gauss integers

It's known that any Gauss integer can be represented as the sum of three Gauss integer squares. (See my another problem.) Let $A$ be a set, define $nA=\{x\mid x=a_1+a_2+\cdots a_k,a_i\in A,1\leq ...
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Quadratic sum of Gauss integers

Let $A$ be a set, define $nA=\{x\mid x=a_1+a_2+\cdots a_k,a_i\in A,1\leq k\leq n\}$. Denote $G=\{z\mid z=(a+bI)^2,a,b\in \mathbb Z,I=\sqrt{-1}\},K=\{z\mid z=a+2bI,a,b\in \mathbb Z,I=\sqrt{-1}\}$ ...
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478 views

The number of solutions for $x+y+z=n$ [duplicate]

How do I approach this problem? I know the formula but do not how it had come. Could you please explain to me the procedure, with examples if possible. stars and bars theorem
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2-colorings of arithmetic progression

Consider the following classical result: $\forall r \in \mathbb N : \exists N\in \mathbb N$ such that every 2-colored arithmetic progression of length N contains a monochromatic arithmetic ...
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152 views

Learning roadmap for additive combinatorics

I have read Calculus by Michael Spivak. Now I want to learn additive combinatorics though I have no experience with combinatorics or probability theory. To my understanding, there is a book on the ...
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Learning roadmap for additive combinatorics

I have read Calculus by Michael Spivak. Now I want to learn additive combinatorics though I have no experience with combinatorics or probability theory. To my understanding, there is a book on the ...
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1answer
63 views

Sum of two finite sets in torsion-free abelian groups

Suppose $G$ is a torsion-free abelian group (written additively) and $A$ and $B$ are two nonempty finite subsets (not subgroups) of $G$. Is it true that there is an element of $G$ which may be ...
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154 views

Combination problem

There are N advertisement boards of which M consecutive boards should have at least K advertisements. How to find number of ways in which this is possible keeping cost minimum. Eg: N=6,M=3,K=2 which ...
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92 views

Property of finite basis free abelian group

I am trying to understand the following (from Munkres) Thm: If G, a free abelian group, has basis $\{a_1,...,a_n\}$, then $n$ is uniquely determined by G. What does this mean ? What exactly is being ...
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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 ...
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2answers
98 views

Combinatorics Catastrophe

How will you solve $$\sum_{i=1}^{n}{2i \choose i}\;?$$ I tried to use Coefficient Method but couldn't get it! Also I searched for Christmas Stocking Theorem but to no use ...
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147 views

Converse of the Erdős Conjecture on Arithmetic Progressions

Clearly, the question of whether large sets always contain arbitrarily long arithmetic progressions is an open question. So my question is not about this conjecture. Instead, it is about the ...