Let $\mathcal {P}$ be the set of primes and $\pi (x)$ the number of primes not exceeding x. Let $P^+(n)$ be the largest prime factor of n, with the convention $P^+(1)=1$, and $ T_c(x)=\#\{p\le x:p\in \mathcal {P},P^+(p-1)\ge p^c\}. $ Motivated by a conjecture of Chen and Chen [‘On the largest prime factor of shifted primes’, Acta Math. Sin. (Engl. Ser.) 33 (2017), 377–382], we show that for any c with $8/9\le c<1$,

$$ \begin{align*} \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\le 8(1/c-1), \end{align*} $$

which clearly means that

$$ \begin{align*} \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\rightarrow 0 \quad \text{as } c\rightarrow 1. \end{align*} $$

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.

Let ${{b}_{1}}$, ${{b}_{2}}$ be any integers such that $\gcd \left( {{b}_{1}},{{b}_{2}} \right)=1$ and ${{c}_{1}}\left| {{b}_{1}} \right|\,<\,\left| {{b}_{2}} \right|\,\le \,{{c}_{2}}\left| {{b}_{1}} \right|$ , where ${{c}_{1}}$, ${{c}_{2}}$ are any given positive constants. Let $n$ be any integer satisfying $\gcd \left( n,\,{{b}_{i}} \right)\,=\,1$ , $i\,=\,1,\,2$. Let ${{P}_{k}}$ denote any integer with no more than $k$ prime factors, counted according to multiplicity. In this paper, for almost all ${{b}_{2}}$ , we prove (i) a sharp lower bound for $n$ such that the equation ${{b}_{1}}p\,+\,{{b}_{2}}m\,=\,n$ is solvable in prime $p$ and almost prime $m\,=\,{{P}_{k}}$ , $k\,\ge \,3$ whenever both ${{b}_{i}}$ are positive, and (ii) a sharp upper bound for the least solutions $p$, $m$ of the above equation whenever ${{b}_{i}}$ are not of the same sign, where $p$ is a prime and $m\,=\,{{P}_{k}}$ , $k\,\ge \,3$.

In this paper the authors prove the following result.

Let $\alpha$ be an irrational number. Then for any $\varepsilon > 0$, there are infinitely many prime numbers p such that $\| \alpha p \| < p^{-16/49 +\varepsilon}$.

The exponent $\frac{16}{49}$ improves on $\frac{9}{28}$, which was obtained recently by the second author [Sci. China Ser. A 43 (2000) 703-721]. The result is very close to the exponent $\frac{1}{3}$, which can be obtained under the Generalized Riemann Hypothesis.

Previous approaches to this problem have all used the same basic estimates for the trigonometric sums that arise. However, the present proof uses new bounds, which depend on the Kloosterman sum and also on a counting problem in the geometry of numbers. In addition new techniques for the sieve method are applied. The most significant feature of the new approach is that, unlike previous methods, the exponential sum estimates remain non-trivial for the exponent $\frac{1}{3}$. This gives one hope for an unconditional result as good as that available under the Generalized Riemann Hypothesis.

2000 Mathematical Subject Classification: 11N36, 11J71.