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In this paper we prove two results concerning Vinogradov’s three primes theorem with primes that can be called almost twin primes. First, for any $m$, every sufficiently large odd integer $N$ can be written as a sum of three primes $p_{1},p_{2}$ and $p_{3}$ such that, for each $i\in \{1,2,3\}$, the interval $[p_{i},p_{i}+H]$ contains at least $m$ primes, for some $H=H(m)$. Second, every sufficiently large integer $N\equiv 3~(\text{mod}~6)$ can be written as a sum of three primes $p_{1},p_{2}$ and $p_{3}$ such that, for each $i\in \{1,2,3\}$, $p_{i}+2$ has at most two prime factors.

We prove character sum estimates for additive Bohr subsets modulo a prime. These estimates are analogous to the classical character sum bounds of Pólya–Vinogradov and Burgess. These estimates are applied to obtain results on recurrence $\bmod \,p$ by special elements.

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Vinogradov’s three primes theorem with almost twin primes

Character Sums Over Bohr Sets

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2 results

Vinogradov’s three primes theorem with almost twin primes

Character Sums Over Bohr Sets