A problem on square roots of integers (Q1112086)

In their paper ``On the divisor and circle problems'' [J. Number Theory 29, No.1, 60-93 (1988; Zbl 0644.10031)], \textit{H. Iwaniec} and \textit{C. J. Mozzochi} required an upper bound for the number B(\(\delta\),K,L) of solutions in integers for \[ \ell_ 1+\ell_ 2=\ell_ 3+\ell_ 4,\quad \ell_ 1k_ 1+\ell_ 2k_ 2=\ell_ 3k_ 3+\ell_ 4k_ 4, \] \[ \ell_ 1\sqrt{k_ 1}+\ell_ 2\sqrt{k_ 2}=\ell_ 3\sqrt{k_ 3}+\ell_ 4\sqrt{k_ 4}+O(\delta L\sqrt{K}),\quad K\leq k_ i<2K,\quad L\leq \ell_ i<2L\quad (i=1,...,4), \] where \(\delta\), K, L are fixed, \(0\leq \delta \leq 1\) and K and L are positive integers. In Section 14 of the same paper they gave an elementary proof of the bound \[ B(\delta,K,L)=O_{\epsilon}((KL^ 3+K^ 2L^ 2+\delta K^ 3L^ 2)(KL)^{\epsilon})\quad (\epsilon >0). \] The author has borrowed some of their ideas and added a few of his own to show that \[ B(\delta,K,L)=O(KL^ 3+K^ 2L^ 2(\log 2K)+\delta K^ 3L^ 2(\log 2K)). \] For an application see \textit{M. N. Huxley}'s forthcoming paper ``Exponential sums and lattice points'' [Proc. Lond. Math. Soc., III. Ser. (to appear); see the preview in Zbl 0659.10057)].