The fundamental lemma of Brun's sieve in a new setting (Q1069982)

Through his recent investigations [Acta Arith. 43, 425-440 (1984; Zbl 0544.12013)] the author was led to various generalizations of results from ''Sieve methods'' [\textit{H. Halberstam} and \textit{H.-E. Richert} (1974; Zbl 0298.10026)]. The general outset is as follows: Let \({\mathcal A}\) be a finite set and \(\{\) \({\mathcal T}_ p\}^ a \)family of sets indexed by primes from a certain set \({\mathcal P}\). Assume \((\Omega_ 1)\), \((\Omega_ 2(k))\) (cf. loc. cit.) and for the remainders \[ | | {\mathcal A}\cap \cap_{p\in {\mathcal P}, p| d}{\mathcal T}_ p | -\frac{\omega (d)}{d}X | \leq A_ 3 X^{1-(1/k)} d^{k-1} \omega (d). \] Then several results about S(\({\mathcal A},{\mathcal P},z)=| {\mathcal A}\setminus \cup_{p\in {\mathcal P}, p<z}{\mathcal T}_ p |\) including applications are given.