The values of binary linear forms at prime arguments

Suppose that λ 1 and λ 2 are positive real numbers such that λ 1 / λ 2 is irrational and algebraic. Let V be a well-spaced sequence and δ > 0 . Denote by E ( V , X , δ ) the number of v ∈ V with v ≤ X such that the inequality | λ 1 p 1 + λ 2 p 2 − v | v − δ has no solution in primes p 1 , p 2 . We prove that for all X ≥ 1 , E ( V , X , δ ) ≪ X f ( δ ) + e for any e > 0 with f ( δ ) = max ⁡ ( 5 / 9 + 2 δ , 2 / 3 + 4 δ / 3 ) , which improves the earlier result.