Representation of natural numbers by sum of four squares of almost-prive having a special form

Abstract

In this paper we consider the equation $x_1^2+x_2^2+x_3^2+x_4^2=N$, where $N$ is a sufficiently large integer and prove that if $\eta$ is quadratic irrational number and $0<\lambda <\frac{1}{10}$, then it has a solution in almost-prime numbers $x_1,\dots ,x_4$, such that $\{\eta x_i\}<N^{-\lambda }$ for $i=1,\dots ,4$.