All questions with tag [math: additive-combinatorics]

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Combinatorics question in the style of Van der Waerden's theorem

I would really appreciate some help with the following problem. It resembles Van der Waerden a lot but I don't know how to proceed. I was told an averaging argument might do the trick but I can't see it. Let $N, r$ be positive integers. Then, there exists a subset $X$ of $\left\{1,2,\ldots,N\right\}$ which contains arithmetic progression of leng...
2022-07-25 13:21:19
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Example of sets with additive property

The following is a question from Tao and Vu's Additive combinatorics, which I'm stuck at and I could really use some help. First, a definition: for two sets $X$ and $Y$ define the additive energy $W(X,Y)=|\left\{(a,a',b,b') \in X \times X \times Y \times Y: x+y = x' + y' \right\}|$. Now, the exercise just asks for...
2022-07-17 14:18:06
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Using Bohr sets to locate arithmetic progressions

I've just started to read about additive combinatorics and I'd like to know how I can use Bohr sets to make a statement about arithmetic progressions in a given subset $A$ of an Abelian group $Z$ (the ambient group). For example: Green-Tao states that if $A$ is the set of all primes then we can always find an arithmetic progression of length $k$...
2022-07-16 13:46:23
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Arithmetic progressions

What are the biggest recognized lower bounds for $B_k$, the topmost amount of the reciprocals of the participants of parts of the favorable integers which have no math developments of size $k$? for $k=3,4,5,6...$ $B_k\leq$ sup (S has no math developments of size k ) And is the bound confirmed to be limited for any kind of k?ยข Can there exist a...
2022-07-11 02:57:47
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Given a list of $2^n$ nonzero vectors in $GF(2^n)$, do some $2^{n-1}$ of them sum to 0?

Let $G=(\mathbb{Z/2Z})^n$ written additively, $n>1$. (you can think of it as $\mathbb{F}_{2^n}$ but I didn't find that useful... yet) Let $v_i$ be nonzero elements of $G$ for $i \in \{1 \dots 2^n \}$ so that $\sum_{i=1}^{2^n} v_i=0$. Is it possible to find a subset $J \subset \{1 \dots 2^n \}$, $|J|=2^{n-1}$ so that $$\sum_{j \in J} v_j=0...
2022-07-08 17:32:28
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Farey fractions in arithmetic progression

Let $\mathcal{F}_{Q;r,q}=\{\gamma=\frac{m}{n} | 0\leq m \leq n \leq Q, \gcd(m,n)=1, n \equiv r \mod q, \gcd(r,q)=1\}$. Generally, without problem on math development, after that $\# \mathcal{F}_{Q}$ (cardinality) $=\varphi(1)+\varphi(2)+\varphi(3)+\cdots+\varphi(Q)$ $=\frac{3Q^2}{\pi^2}+O(Q\log Q)$ as $Q\rightarrow \infty$, today just how to get...
2022-06-27 15:19:11
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Uses of Chevalley-Warning

In the current IMC 2011, the last trouble of the 1st day (no. 5, the hardest of that day) was as adheres to: We have $4n-1$ vectors in $F_2^{2n-1}$: $\{v_i\}_{i=1}^{4n-1}$. The trouble asks: Confirm the presence of a part $A \subseteq [4n-1], |A| = 2n$ such that $\sum_{i \in A} v_i = \vec{0}$. Exists a remedy that makes use of the Chevalley-Wa...
2022-06-08 22:05:34