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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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