On differences of semicubical powers (Q1424407)

In this paper the author establishes several upper bounds for the function \(I_8(X,Y,\Delta)\) which, for given \(\Delta ,X,Y>0\), is defined to equal the number of integer points \({\mathbf x}\in{\mathbb Z}^8\) such that \(x_i,x_{i+4}\in(X,2X]\) and \(| x_i-x_{i+4}| \in(Y,2Y]\), for \(i=1,\ldots ,4\), and \(\left| \sum_{i=1}^4(x_i^{\beta}-x_{i+4}^{\beta}) \right| \leq\delta(\beta)X^{\beta}\), for \(\beta =1,3/2,2\), where \(\delta(1)=\delta(2)=0\) and \(\delta(3/2)=\Delta\). As is briefly reported in the paper, the bounds established for \(I_8(X,Y,\Delta)\) are of interest in connection with an application of the Bombieri-Iwaniec method that yields strong mean-square bounds for a Dirichlet \(L\)-function over short intervals, in case when certain conditions are met [see the author's paper, Acta Arith. 111, 307--403 (2004; Zbl 1049.11096)]. The proofs in the paper are largely elementary, although technically complicated and are based on several lemmas. They involve consideration of a second function \(I_8^*(X,Y,\Delta)\) which is an upper bound for \(I_8(X,Y,\Delta)\). A central feature is the iterative use of results that bound \(I_8^*(X,Y,\Delta)\) in terms of \(I_8^*(X,Y',\Delta)\) with some \(Y'\in(0,Y)\).