Weighted sieves with switching

Weighted sieves are used to detect numbers with at most S prime factors with S ∈ N as small as possible. When one studies problems with two variables in somewhat symmetric roles (such as Chen primes, that is primes p such that $p+2$ has at most two prime factors), one can utilize the switching principle. Here we discuss how different sieve weights work in such a situation, concentrating in particular in detecting a prime along with a product of at most three primes. As applications, we improve on the works of Yang and Harman concerning Diophantine approximation with a prime and an almost prime, and prove that, in general, one can find a pair (p, P₃) when both the original and the switched problem have level of distribution at least $0.267$.