  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 $N \equiv 3 \pmod{6}$ 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.
