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On the study of lattice points in multidimensional ellipsoids. (Zur Gitterpunktlehre von mehrdimensionalen Ellipsoiden.) (German) Zbl 0127.27502
Let \(Q(u) = Q(u_1,\ldots, u_r)\) be a positive definite quadratic form. The form \(Q\) is said to be rational if there exists \(a > 0\) such that \(Q(u) = \alpha Q_1(u)\), where \(Q_1\) has integer coefficients ; otherwise \(Q\) is said to be irrational. Let \(x > 0\) and let \(A(x)\) denote the number of points of the integer lattice in the ellipsoid \(Q(u)\leq x\). Write \(P(x)= A(x)-V(x)\), where \(V(x)\) denotes the volume of the ellipsoid. It is known that for every form \(Q\), \(P(x) = \Omega(x^{(r-1)/4})\) and that for rational \(Q\), with \(r> 4\), \(P(x)=O(x^{r/2-1})\). [See L. K. Hua, Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie. Leipzig: B. G. Teubner Verlag (1959; Zbl 0083.03601)].
In a series of papers the author and A. Walfisz considered a class of irrational forms
\[ Q(u)=\alpha_1 (u_{1,1}^2+\ldots+u_{1,\tau_1}^2)+\ldots+\alpha_r (u_{r,1}^2+\ldots+u_{r,\tau_r}^2) \] \((\alpha_j>0,\;r_j>0,\;r_1+\ldots+r_r=r>4,\;\tau>2)\) and showed that for such forms \(P(x) =O(x^{r/2-1})\). Moreover, for almost all systems (in the sense of Lebesgue measure) of positive numbers \((\alpha_1, \ldots, \alpha_r)\) and for every \(\varepsilon> 0\)
\[ P(x) = O(x^{r/2-\lambda +\varepsilon})\quad (r> 4,\;\tau > 2)\quad\text{where }\lambda=\sum_{j=1}^r\min (1, \tfrac 14 r_j).\tag{1} \]
The author now proves the following result. Let \(A\) denote the set of all number pairs \((\alpha_1, \alpha_2)\), \(\alpha_1 > 0\), \(\alpha_2 > 0\), such that for every \(\varepsilon > 0\) there is a constant \(c(\varepsilon)\) for which
\[ | q((\alpha_1/\alpha_2)-p| > c(\varepsilon)/q^{1+\varepsilon}\quad\text{ for all }p, q\in \mathbb Z, q > 0. \] It is well known that almost all pairs \((\alpha_1, \alpha_2)\) of positive numbers are in \(A\). Let \(r_1+\ldots+r_r = r > 4\), \(\tau> 2\), \(r_j > 0\) and suppose that \((\alpha_1, \alpha_2)\in A\). Then for almost all systems \((\alpha_3, \ldots, \alpha_r)\) of positive numbers and for every \(\varepsilon > 0\) the estimate (1) holds.

MSC:
11P21 Lattice points in specified regions
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