Lattice points in three-dimensional convex bodies with points of Gaussian curvature zero at the boundary
Let \(K\) be a three-dimensional compact convex set with the origin as an inner point. Let \(A(x)\) denote the number of lattice points inside the blown-up body \(xK\) where \(x\) is a large positive parameter. If the boundary of \(K\) is smooth and has positive Gaussian curvature everywhere, then \(A(x)=\text{vol}(k)x^3+O(x^\theta)\) and the best known result is \(\theta=63/43+\varepsilon\) (which is a little less than 3/2). In his book ``Lattice points'' [Kluwer (1988; Zbl 0675.10031)], the author considered bodies whose surface contains points with Gaussian curvature 0. Here, he extends the results to the case where the surface contains a flat point. It is enough to study the pillar \(S=\{(z,t_1,t_2): 0\leq z\leq f(t_1,t_2)\), \(a_i<t_i\leq b_i\), \(i=1,2\}\) where \(f\) has continuous partial derivatives up to the \(k\)th order for some \(k>2\). For this body, \[ A_k(x)=\text{vol}(S)x^3-(A\psi(b_1x)-B\psi(a_1x))x^2+P_k(x) \] where \(\psi(u)=u-[u]-1/2\) and \[ P_k(x)=-\sum_{a_1x<n_1\leq b_1x}\sum_{a_2x<n_2\leq b_2x} \psi(xf(n_1/x,n_2/x))+O(x). \] The point \((a_1,a_2)\) is a flat point if the Gaussian curvature is 0 there and also possibly at the points \((a_1,t_2)\) and \((t_1,a_2)\) and this is assumed to be the only flat point. If the partial derivatives \(\partial^{\nu_1}f/\partial t_1^{\nu_1}(a_1,t_2)\) are 0 for \(2\leq\nu_1\leq k_1-1\) and bounded away from 0 for \(\nu_1=k_1\), the partial derivatives \(\partial^{\nu_2}f/\partial t_2^{\nu_2}(t_1,a_2)\) are 0 for \(2\leq\nu_1\leq k_2-1\) and bounded away from 0 for \(\nu_2=k_2\), and the Hessian \(H(f)\) behaves like \((t_1-a_1)^{k_1-2}(t_2-a_2)^{k_2-2}\) as \((t_1,t_2)\rightarrow (a_1,a_2)\), then \[ P_k(x)\ll x^{2-{1\over k_1}-{1\over k_2}}\log^2x+x^{5\over 3} \Bigl(x^{-{1\over k_1}}+x^{-{1\over k_2}}\Bigr) +x^{3\over 2}. \] Another similar result is obtained under slightly different hypotheses on \(f\). If, in addition, the flat point at \((a_1,a_2)\) has a rational tangent plane, a sharper estimate for \(P_k(x)\) is obtained with 3 terms. The first which is \(\ll x^{2-{1\over k_1}-{1\over k_2}}\) arises from the flat point itself, the second which is \(\ll x^{5\over 3}(x^{-{2\over 3k_1}}+x^{-{2\over 3k_2}})\) arises from the other points on the lines with Gaussian curvature 0, and the third is \(\ll x^{{3\over 2}+\varepsilon}\).
scientific article; zbMATH DE number 1887590