Questions tagged [additive-combinatorics]
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.
756 questions
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On sums of a prime and a central binomial coefficient
Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that
$$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$
...
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What proportion of vectors in ${\mathbb{F}_2^n}$ have more than $\frac{n+\sqrt{n}}{2}$ ones
Ben Green's "Finite field models in additive combinatorics" (proof of theorem 9.4) states that for sufficiently large $n$, the set $A$ of vectors in ${\mathbb{F}_2^n}$ with more than $\frac{...
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On even numbers of the form $p+p'+2^k$
Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that
$$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$
...
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Whether $2n>10$ can be written as $p+p'+2^a+2^b$ with $p$ and $p'$ consecutive primes?
In a paper published in 1971, R. Crocker proved that there are infinitely many positive odd numbers not of the form $p+2^a+2^b$ with $p$ prime and $a,b\in\mathbb Z^+=\{1,2,3,\ldots\}$. The proof makes ...
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Can $w^2+bx^2+cy^2+dz^2$ be universal over a sparse subset of $\mathbb N$?
Let $b,c,d\in\mathbb N=\{0,1,2,\ldots\}$ with $1\le b\le c\le d$. For a subset $S$ of $\mathbb N=\{0,1,2,\ldots\}$, if
$$\{w^2+bx^2+cy^2+dz^2:\ w,x,y,z\in S\}=\mathbb N$$
then we say that $w^2+bx^2+cy^...
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Lower bounds for difference sets of finite integer sets
Let $S$ be a finite subset of integer. Let $\{p \leq X\}$ be the set of primes bounded by $X$. Is it true that the set $S-S$ has a subset $A$ of positive density such that
$p \mid a$ for all prime $p \...
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A question related to matrix inverse diagonal zero property
$\DeclareMathOperator\supp{supp}$Let a symmetric nonsingular matrix $A \in \mathbb{R}^{2n \times 2n} $ have the following block form
$$ A = \begin{bmatrix}
X & D \\
D^{\top} ...
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The number of distinct possible values of $\pm a_1 \pm a_2 \pm \cdots \pm a_n$ for sufficiently large $n$
Conjecture. Assume that $(a_i)_{i = 1}^{\infty}$ is a sequence of positive integers such that $a_{n+1} \leq 1+\sum_{i = 1}^n a_i$ for sufficiently large $n$. If $A_n$ denotes the number of distinct ...
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Finding a sum that is always divisible by $1+2+3+\cdots+n$ [closed]
There are $n$ consecutive integers $m,m+1, \ldots, m+n-1$. Prove that you can choose some nonempty subset of these numbers whose sum is divisible by $1+2+\dots+n$.
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Ternary Goldbach-type problems
I am looking for problems comparable to the ternary Goldbach problem, which says that every positive odd integer may be written as the sum of three primes. For instance, something of the shape
Is ...
user574181
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Can $D-D$ be a set of $2$-topological recurrence if $D$ is lacunary?
Background.
For $k \in \mathbb{N}=\{1,2,3,\dots\}$, a set $R \subseteq \mathbb{N}$ is a set of $k$-topological recurrence if for every minimal topological dynamical system $(X,T)$ and every nonempty ...
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A universal additive complement
Let's say that two subsets of a group are additive complements of each other if their sumset is the whole group. Suppose that the group is finite of prime order. For a fixed $\alpha\in(0,1)$, what is ...
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Equality of multisets
I have tested this statement with several examples and it seems to hold true in all cases. Is there an elegant way to prove it, assuming it is indeed correct? A proof that avoids case-by-case analysis ...
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On the set $\{\sum_{k=1}^n p_k:\ n = 1,2,3,\ldots\}$
For any positive integer $n$, let $S(n)$ be the sum of the first $n$ primes. Then
$$S(1) = 2,\ S(2)=2+3=5,\ S(3)=2+3+5 =10,\ S(4) = 2+ 3+5+7 =17.$$
By the Prime Number Theorem,
$$S(n)\sim \frac{n^2}2\...
- 18k
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Weighted sums of four primes
Sums of primes have been studied by number theorists for many years. Goldbach's conjecture is the most famous unsolved problem in this direction.
Here I'd like to consider weighted sums of primes. For ...
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