A note on the exceptional set for Diophantine approximation with mixed powers of primes

Suppose that lambda(1),lambda(2), lambda(3), lambda(4 )are nonzero real numbers, not all negative, and lambda(1)/lambda(2) is irrational and algebraic. Let nu be a well-spaced sequence, delta > 0. It is proved that for any epsilon > 0, the number of u is an element of nu with u <= N for which vertical bar lambda(1)p(1)(2) + lambda(2)p(2)(3) + lambda(3)p(3)(4) + lambda(4)p(4)(5) - v vertical bar < v(-delta) has no solution in primes p(1), p(2), p(3), pa does not exceed O (N-347/360 + 2(delta + epsilon)). -Fe This result constitutes an improvement upon that of Ge and Zhao.