# Tagged Questions

Questions on the subject additive combinatorics, also known as arithmetic combinatorics, such as questions on: additive bases, sum sets, inverse sum set theorems, sets with small doubling, Sidon sets, Szemerédi's theorem and its ramifications, Gowers uniformity norms, etc. Often combined with the ...

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### When are these sums consecutive integers? [closed]

It is possible to construct $\frac{n(n-1)}{2}$ sums which each contain two distinct summands chosen from a set $n$ numbers. For which $n\geq 3$ do there exist a set of $n$ (distinct) integers such ...
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### Small set such that $\{1 , \ldots , n\} \cdot A = \mathbb{Z} / p \mathbb{Z}$

Let $p$ be a large prime and $n < p$. What is the smallest size of a set $A \subset \mathbb{Z} / p \mathbb{Z}$ such that $A \cdot \{1 , \ldots , n\} = \mathbb{Z} / p \mathbb{Z}$? Here $\cdot$ ...
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### Is budget-additive function a modular set function?

We know that budget-additive function $$f(S) = \min\{B,\sum_{i \in S}w_i\}$$ where $w_i$ is positive constant and $B \ge 0$ is called additive budget. Is it also a modular set function?
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### Combinatorics problem about sum of natural numbers

Following combinatorics problem is claimed to be an open problem in "The Princeton Companion to Mathematics" (pp. 6) Let $a_1,a_2,a_3,...$ be a sequence of positive integers, and suppose that each ...
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### Find a subset such that its sum is divisible by $n$

It is said that the following proposition is true. $\forall S \subset \mathbb{Z}, |S| = 2n-1.\ \exists A \subset S, |A| = n$ which satisfies $$n \ | \ \sum_{a \in A}a.$$ Could someone gives a ...
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For any set $B\subseteq \mathbb{N}$ one can associate the formal series $$f_B(z) = \sum_{b\in B}z^b$$ and obtain $$f_B(z)^k = \sum_{n\geqslant 0} r_{B,k}(n)z^n,$$ where $r_{B,k}(n) = |\{(x_1,\cdots,... 2answers 332 views ### Is the sumset or the sumset of the square set always large? Let A be a finite subset of$\mathbb{N}$,$\mathbb{R}$, or a sufficiently small subset of$\mathbb{F}_{p}$. Do we have a lower bound of the form$|A|^{1+\delta}$on the following quantity: $$\max (|\... 1answer 314 views ### Additivity of upper densities with respect to arithmetic progressions of integers Let \mathsf{d}^\star be the asymptotic upper density, defined on the power set of positive integers \mathbf{N}^+, so that$$ \mathsf{d}^\star\colon \mathcal{P}(\mathbf{N}^+) \to\mathbf{R}\colon X\... 0answers 211 views ### Area defined with$\pm$closedness Denote$B_n\subset\Bbb R^n$to be unit ball at origin. Denote$S\subset B_n$to region of type$\mathsf I$if it satisfies $$s\in S\iff\forall t\in S, s+t\in S\mbox{ or }s-t\in S$$ I am convinced$\...
The question in brief:   When does a subset $S$ of the complex $n$th roots of unity have the property that $$\prod_{\alpha\, \in \,S} (z-\alpha)$$ gives a polynomial in $\mathbb R[z]$ with ...
In an abelian group, the additive energy between two sets is $$E(A,B)=|\{(a_1,a_2,b_1,b_2) \in A\times A\times B\times B:a_1+b_1=a_2+b_2\}|$$ which is ranges from $|A||B|$ to $(|A||B|)^{3/2}$. What I ...