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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 top-level tags nt.number-theory or co.combinatorics. Some additional tags are available for further specialization, including the tags sumsets and sidon-sets.

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### Bins-and-primes (prime divisors of $\prod(a_i-a_j)$, II)

This is a refined version of a question I have recently posted. For a prime $p$, let $\varphi_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the ...
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### Prime divisors of $\prod(a_i-a_j)$

For a prime $p$, let $\varphi_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the group of order $p$. Given an integer $n\ge 3$, what is the smallest ...
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### Decomposition of a natural number as sum of positive integers

Let $n \in \mathbb{N}$ be a positive natural number and denote by $f(n)$ the number of decompositions of $n$ of the form $n = a+b+c+d$ where $a,b,c,d > 0$ are also positive natural numbers such ...
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### Multidimensional van der Waerden, bounds for squares

Given $r$, let $f(r)$ be the smallest $N$ such that for any $r$-coloring $C:\{1,\dots,N\}^2 \to \{1,\dots,r\}$, there exists $x,y,d\neq 0$ such that $C((x,y)) = C((x+d,y))= C((x,y+d))=C((x+d,y+d))$. I ...
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### On Behrend's construction

Fix $\alpha>0$. Does there exist $\epsilon = \epsilon(\alpha)>0$ such that if $S\subset [N]:=\{1,\dots,N\}$ has $\ge \alpha N$ elements, then for any function $f:S\to [0,1]$, there exist some ...
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### Reference request for a result in additive combinatorics

Let $p$ be a prime number and $[p-1]=\{1, 2, \ldots, p-1\}$. The following proposition is proved: (but I cannot find out where) Proposition: The non-empty subset sums of $[p-1]$ are equally ...
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### Bounds for Szemerédi’s theorem for GAP’s

Let a $(k,D)$-AP refer to sets of the form $\{n_0+l_1n_1+\dotsb +l_Dn_D: l_1,\dotsc,l_D \in [k]\}$ with cardinality $k^D$ (i.e. a proper $D$-dimensional GAP with width $k$). Let $r(N,k,D)$ be the ...
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### Diameter of Cayley graphs whose generators intersect all large cosets

Let $G = G_n= \Bbb{F}_2^n$. We say $A\subset G$ is $t$-blocking, if it has non-empty intersection with all cosets $C\subset G$ with codimension at most $t$. Given a group $G$ and set $A\subset G$, we ...
Fix $\epsilon>0$. For all large $N$, does there exist $A\subset [N]:=\{1,\dots,N\}$ such that both $A+A$ and $A^c:=[N]\setminus A$ lack arithmetic progressions of length $N^\epsilon$? I am aware ...