Diophantine Approximation by Cubes of Primes and an Almost Prime

Abstract

Let λ1, . . . , λs be non-zero with λ1/λ2 irrational and let S be the set of values attained by the form λ1x 3 1 + · · ·+ λsxs when x1 has at most 6 prime divisors and the remaining variables are prime. In the case s = 4, we establish that most real numbers are “close” to an element of S. We then prove that if s = 8, S is dense on the real line.