We develop an axiomatic formulation of the higher rank version of the classical Selberg sieve with an ``additive twist'' and provide asymptotic formulas for the same. As an application of this higher rank sieve, we obtain improvements of results of Heath-Brown and Ho-Tsang on almost prime $k$-tuples.
"A higher rank Selberg sieve with an additive twist and applications." Funct. Approx. Comment. Math. 57 (2) 151 - 184, December 2017. https://doi.org/10.7169/facm/1622