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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.

256 questions with no upvoted or accepted answers
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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 ...
Louis Deaett's user avatar
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17 votes
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722 views

Consider the sequence $$a_n:=\min_{\substack{A\subseteq\mathbb{Z}\\|A|=n}}|(A+A)\cup (A\cdot A)|.$$ This is A263996 in OEIS. (Actually, they restrict to $\mathbb{N}$, but it makes little difference to ...
Dustin G. Mixon's user avatar
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14 votes
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Let $A$ and $B$ ($A\subset B$) be subsets of a finite abelian group $G$. (For the sake of argument, you can take $G$ to be $\mathbb{Z}/p\mathbb{Z}$ for large $p$, say.) Write $1_S$ for the ...
H A Helfgott's user avatar
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12 votes
0 answers
233 views

The Stone–Čech compactification $\beta \mathbb N$ of the positive integers has extensive applications in combinatorial number theory. A feature of $\beta \mathbb N$ that makes these applications ...
Jakub Konieczny's user avatar
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11 votes
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This is just a summer-time curiosity arisen after a recent question by Dominic van der Zypen. For a finite subset $S$ of $\mathbb{Z}\times\mathbb{Z}$ and a function $f$ on $\mathbb{Z}\times\mathbb{...
Pietro Majer's user avatar
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11 votes
0 answers
624 views

A tuple $(A_1,\dots,A_n)$ of subsets of a finite group $G$ is called a factorization of $G$ if $G=A_1\cdots A_n$ and $|A_1|\cdots|A_n|=G$. In Cryptology factorizations of groups are known as ...
Taras Banakh's user avatar
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11 votes
0 answers
638 views

A linear equation $c_1x_1 + \cdots + c_sx_s = 0$ is partition regular if for every partition of the natural numbers into colour classes $A_1, \ldots, A_r$, there is a solution to the equation in which ...
Ben Barber's user avatar
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11 votes
0 answers
881 views

The following question is motivated by pure curiosity; it is not a part of any research project and I do not have any applications. The question comes as an interpolation between two notoriously ...
Seva's user avatar
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11 votes
0 answers
903 views

To construct gaps between primes which are marginally larger than average, Erdős and Rankin covered an interval $[1,y]$ with arithmetic progressions with prime differences. A nice short exposition is ...
Douglas Zare's user avatar
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10 votes
0 answers
194 views

Define a set of numbers with small radicals (A341645 in OEIS) by $$A_2=\{n\in\mathbb{N} \;|\; \text{rad}(n)^2\le n\}$$ The asymptotic density of $A_2\cap \{1,\dots N\}$ is $\sqrt{N}\times e^{2(1+o(1))\...
Yaakov Baruch's user avatar
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10 votes
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The old formula for the Frobenius number of a numerical semigroup generated by two elements can be stated as follows: assume $\gcd\{a+1,b+1\}=1$, then the Frobenius number of $S= \left<a+1,b+1\...
Hailong Dao's user avatar
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9 votes
0 answers
321 views

Let $d=10^{10}$, let $F=\{-1000,\dots,1000\}^d\subseteq\mathbb{Z}^d$. Let $L$ be an extremely big number, e.g. $L>10^{100}$. Is there a subset $A$ of $\{1,\dots,L\}^d$ such that $\frac{|A|}{L^d}>...
Saúl RM's user avatar
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9 votes
0 answers
318 views

Let $b \ge 1$ and $A\subseteq [0,b]$ be a set of integers (all intervals will be of integers). Write $hA := \underbrace{A + \ldots + A}_{h\text{ summands}} = \{ \sum_{i=1}^h a_i ~|~a_i \in A,\, \...
Alufat's user avatar
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9 votes
0 answers
332 views

I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
monkeymaths's user avatar
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9 votes
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450 views

Suppose $\mathscr{A} \subset \mathbb{N}$ enjoys for all large enough cutoffs $X$ the following properties: $|\mathscr{A} \cap [1,X]| > \sqrt{X}/10$; and the discriminant $\prod_{\alpha \neq \beta}|\...
Vesselin Dimitrov's user avatar
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