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Distribution of integral lattice points in an ellipsoid with a Diophantine center. (English) Zbl 1327.11066
In this paper, the authors present some asymptotic values of the normalised deviations of the number of lattice points inside a rational ellipsoid $$E_t^M( \alpha)\in\mathbb{R}^n$$, with centre $$\alpha$$. Let $E_t^M(\alpha)=\left \{x\in \mathbb{R}^n:Q_M(x-\alpha)\leq t^2\right \},\qquad t\in\mathbb{R}_{>0},$ where $$Q_M$$ is the quadratic form with corresponding positive definitive symmetric $$n\times n$$ matrix $$M$$, and where the central vector $$\alpha$$ is said to be of “Diophantine type $$\kappa$$” if there exists a constant $$c_0>0$$, such that $\left |\alpha-\frac{m}{q}\right |>\frac{c_0}{q^\kappa}\quad \text{for all}\,\,\, m\in\mathbb{Z}^n,\,\,\, \text{and}\,\,\, q\in \mathbb{N}.$ Then denoting by $$N_M(t)=\#\left \{\mathbb{Z}^n\cap E_t^M(\alpha)\right \}$$, the number of lattice points inside the ellipsoid $$E_t^M(\alpha)$$, and by $$|E^M_t|$$ the $$n$$-dimensional volume of the ellipsoid $$E^M_t$$ (which is independent of the choice of $$\alpha$$), the authors consider the asymptotics of the normalised deviation $$F_M(t)$$ defined by $F_M(t)=\frac{N_M(t)-\left |E^M_1\right |t^n}{t^{(n-1)/2}},\qquad \text{as}\,\,\, t\rightarrow \infty,$ as well as the normalised deviation $$S_M(t,\eta)$$ of the number of lattice points inside the shell between the elliptic spheres of radii $$t$$ and $$t+\eta$$ given by $S_M(t,\eta)=\frac{N_M(t+\eta)-N_M(t)-\left |E^M_1\right |\left ((t+\eta)^n-t^n\right )}{\sqrt{\eta}\,\,t^{(n-1)/2}},\qquad \text{as}\,\,\, t\rightarrow \infty\,\,\,\text{and as}\,\,\,\eta \rightarrow 0.$ For $$\eta\geq 0$$, and provided that there exists some $$L>0$$ such that $$T^{-L}<\eta\ll1$$, it is shown (Theorem 1.2) that $\mathop{\text{lim}}_{T\rightarrow \infty}\langle F_M\rangle_T=0,\qquad \text{and}\qquad \mathop{\text{lim}}_{T\rightarrow \infty}\langle S_M(\cdot\,,\eta)\rangle_T=0.$ With the constraints $$n\geq 2$$; $$\alpha\in \mathbb{R}^n$$ a vector of Diophantine type $$\kappa <(n-1)/(n-2)$$; $$({\alpha},1)\in \mathbb{R}^{n+1}$$ a vector whose components are linearly independent over $$\mathbb{Q}$$; $$M=\text{diag}(a_1,\ldots a_n)$$ a diagonal matrix with entries in $$\mathbb{N}$$, and $$\eta\gg T^{-\gamma}$$ for some $$\gamma\in (0,1)$$, it is also shown (Theorem 1.5) that $\mathop{\text{lim}}_{T\rightarrow \infty}\langle |S_M(\cdot\,,\eta)|^2\rangle_T=n|E_1^M|,\qquad \text{as}\,\,\, \mu\rightarrow 0.$ In translating the ellipsoid from the origin to a Diophantine vector, exponential sums appear in the expansion of the counting function $$N_M(t)$$. The mean square limits of these exponential sums are considered, yielding further results and extending a previous result of J. Marklof on Euclidean balls to some ellipsoids [Acta Arith. 117, No. 4, 353–370 (2005; Zbl 1075.11023)].
##### MSC:
 11P21 Lattice points in specified regions 11H06 Lattices and convex bodies (number-theoretic aspects) 11L07 Estimates on exponential sums 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 42B05 Fourier series and coefficients in several variables
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