Diophantine approximation with one prime and three squares of primes

Abstract

Let λ1,…,λ4 be non-zero real numbers, not all of the same sign, with λ1/λ2 irrational, and ϖ be a real number. We prove that for any ε>0, the inequality|λ1p1+λ2p22+λ3p32+λ4p42+ϖ|≤(max⁡{p1,p22,p32,p42})−114+ε has infinitely many solutions in prime variables p1,…,p4. If we further assume that λ1/λ3 is also irrational and λ1/λ2, λ1/λ3 are both algebraic, then we may replace the exponent by −340+ε.