Representation of natural numbers by sums of four squares of integers having a special form (Q549516)

For fixed \(a\) and \(b\), let \(A=\{l\mid 0\leq a<\{\eta\}<b\leq 1\},\;J(N)=\#\{N=\sum\limits_{i=1}^{4}l_{i}^{2},\;l_{i}\in A\},\;I(N)=\#\{N=\sum\limits_{i=1}^{4}l_{i}^{2},\;l_{i}\in \mathbb{Z}\}\). In this paper, the authors obtain: \[ J(N)=(b-a)^{4}I(N)+O(N^{0.9+\varepsilon}). \] The proof uses the Vinogradov's little-glass functions, Hardy-Littlewood method and the Weil estimate for Kloosterman sum.