Diophantine approximation over primes with different powers

Abstract

Assume non-zero real numbers κ1,…,κ6 are not all negative. Assume also κ1κ2 is algebraic and not rational, the sequence G is well-spaced and ϑ>0. In this work, we showed that the quantity of γ∈G satisfying 1≤γ≤X and making the Diophantine inequality|κ1p12+κ2p22+κ3p33+κ4p43+κ5p54+κ6p64−γ|<γ−ϑunsolvable in primes p1,…,p6 is not more than O(X67+2ϑ+ϵ) for any ϵ>0.