Quadratic estimates for the number of integer points in convex bodies
Let \(\Omega\) be a convex domain in \(\mathbb R^d\) with smooth boundary with positive Gaussian curvature and let \(\mathcal{D}(r,x)\) be the discrepancy between volume and number of points with integer coordinates in the dilated and translated domain \(r\Omega-x\). This discrepancy is a periodic function in \(x\) and so can be expanded into a multiple Fourier series on the torus \(\mathbb T^d=\mathbb R^d/\mathbb Z^d\). There are standard approximation techniques in fourier theory to handle such expansions which are only conditionally convergent. Using an approximation kernel and Poisson summation allows the estimation of \[ \left\{\int_{\mathbb T^d}|\mathcal{D}(r,x)|^2\,d\mu(x)\right\}^{1/2} \] where \(\mu(x)\) is a probability measure on the torus with Fourier coefficients \(\hat\mu(k)\) and leads to the upper bound \[ c\epsilon r^{d-1}+cr^{(d-1)/2}\left\{\sum_{k\in\mathbb Z^d} (1+\epsilon|k|)^{-\alpha}(1+|k|)^{-1}|\hat\mu(k)|\right\}^{1/2} \] where \(\alpha\) is a large positive number and \(\epsilon>0\) is chosen to minimise the sum. The estimate generalises classical results of Kendall for \(\int_{\mathbb T^d}|\mathcal{D}(r,x)|^2\,dx\) and Hlawka for \(\sup_x|\mathcal{D}(r,x)|\) and contains several other interesting special cases. A similar approach is also used to estimate the discrepancy with respect to dilations.
scientific article; zbMATH DE number 5121517
scientific article