Diophantine approximation by cubes of primes and an almost prime. II

Abstract

Let $\lambda_1,\dots,\lambda_4$ be non-zero with $\lambda_1/\lambda_2$ irrational and negative, and let $\mathcal S$ be the set of values attained by the form $ \lambda_1x_1^3 + \dots + \lambda_4x^3_4 $ when $x_1$ has at most 3 prime divisors and the remaining variables are prime. We prove that most real numbers are close to an element of $\mathcal S$.