A remark on the values of binary linear forms at prime arguments

Let λ₁, λ₂ be positive real numbers such that ({frac{{lambda_1}}{{lambda_2}}}) is irrational and algebraic. For any (C, c) well-spaced sequence ({mathcal {V} = {{v_i}}_{i = 1}^infty}) and δ > 0 let ({E( {mathcal {V},X,delta})}) denote the number of elements ({v in mathcal {V}, v le X}) for which the inequality

$| {lambda_1 p_1 + lambda_2 p_2 - v} | < X^{- delta}$

is not solvable in primes p ₁, p ₂. In this paper it is proved that

$E( {mathcal {V},X,delta}) ll X^{frac{4}{5} + delta + varepsilon}$

for any ({varepsilon > 0}). This result constitutes an improvement upon that of Brüdern, Cook, and Perelli for the range ({frac{2}{{15}} < delta < frac{1}{5}}).