Revista Matemática Iberoamericana


Full-Text PDF (368 KB) | List online-first RMI articles | RMI summary
Published online first: 2020-02-14
DOI: 10.4171/rmi/1177

Discrepancy for convex bodies with isolated flat points

Luca Brandolini[1], Leonardo Colzani[2], Bianca Gariboldi[3], Giacomo Gigante[4] and Giancarlo Travaglini[5]

(1) Università degli Studi di Bergamo, Dalmine, Italy
(2) Università di Milano-Bicocca, Italy
(3) Università degli Studi di Bergamo, Dalmine, Italy
(4) Università degli Studi di Bergamo, Dalmine, Italy
(5) Università di Milano-Bicocca, Italy

We consider the discrepancy of the integer lattice with respect to the collection of all translated copies of a dilated convex body having a finite number of flat, possibly non-smooth, points in its boundary. We estimate the $L^p$ norm of the discrepancy with respect to the translation variable, as the dilation parameter goes to infinity. If there is a single flat point with normal in a rational direction we obtain, for certain values of $p$, an asymptotic expansion for this norm. Anomalies may appear when two flat points have opposite normals. Our proofs depend on careful estimates for the Fourier transform of the characteristic function of the convex body.

Keywords: Discrepancy, integer points, Fourier analysis

Brandolini Luca, Colzani Leonardo, Gariboldi Bianca, Gigante Giacomo, Travaglini Giancarlo: Discrepancy for convex bodies with isolated flat points. Rev. Mat. Iberoam. Electronically published on February 14, 2020. doi: 10.4171/rmi/1177 (to appear in print)