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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
##### Keywords:
lattice points in multidimensional ellipsoids
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