A paucity problem associated with a shifted integer analogue of the divisor function (Q2079504)

Let \(\theta\) be a complex number, \(k\in\mathbb{N}\) and, for \(\nu\in\mathbb{Z}[\theta]\), \[ \tau_k(\nu,x,\theta)=\mathop{\sum\limits_{1\leqslant d_1\leqslant x} \ldots \sum\limits_{1\leqslant d_k\leqslant x}}\limits_{(d_1+\theta)+\ldots+(d_k+\theta)=\nu}1. \] The authors compare the sum \[ \sum\limits_{\nu\in\mathbb{Z}[\theta]}\Big(\tau_k(\nu,x,\theta)\big)^2 \] with the quantity \(T_k(x)\), where \(T_k(x)\) is the number of \(k\)-tuples \((z_1,\ldots,z_k)\) and \((y_1,\ldots,y_k)\) in which \(1\leqslant z_i,y_i\leqslant x\) and \((z_1,\ldots,z_k)\) is a permutation of \((y_1,\ldots,y_k)\). The quantities are compared differently in cases when \(\theta\) is either transcendental or algebraic.