Representation of natural numbers by sums of four squares of integers having a special form

Abstract

This paper obtains an asymptotic formula for the number of solutions to the equation \( l_1^2 + { }l_2^2 + l_3^2 + l_4^2 = N \) in integers l 1, l 2, l 3, l 4 such that a < {ηl j } < b, where η is a quadratic irrational number, 0 ≤ a < b ≤ 1, j = 1, 2, 3, 4.