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.
4 questions from the last 7 days
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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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