On the representation of large integers as sums of four almost equal squares of primes (Q1039628)

Let \(E(N,U)\) be the number of all positive integers \(n\equiv 4\pmod{24}\) satisfying \(N\leq n\leq N+\sqrt{N}U\), which can not be written as \[ \begin{cases} n=p_1^2+p_2^2+p_3^2+p_4^2;\\ |p_j-\sqrt{\frac{N}{4}}|\leq U,&j=1,2,3,4, \end{cases} \] with \(p_1,p_2,p_3,p_4\) prime numbers. The authors prove: Theorem 1. For \(U=N^{\frac{21}{50}+\varepsilon}\) we have \[ E(N,U)\ll(\sqrt{N}U)^{1-\varepsilon}. \] Theorem 2. For \(U=N^{\frac{9}{20}+\varepsilon}\) we have \[ E(N,U)\ll U^{1-\varepsilon}. \] The proof is based on the new estimates for exponential sums over primes in short intervals and a technique to get the asymptotic formula on the enlarged major arcs in the circle method.