Discrepancy for convex bodies with isolated flat points
Luca Brandolini
Università degli Studi di Bergamo, Dalmine, ItalyLeonardo Colzani
Università di Milano-Bicocca, ItalyBianca Gariboldi
Università degli Studi di Bergamo, Dalmine, ItalyGiacomo Gigante
Università degli Studi di Bergamo, Dalmine, ItalyGiancarlo Travaglini
Università di Milano-Bicocca, Italy
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Abstract
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 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 , 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.
Cite this article
Luca Brandolini, Leonardo Colzani, Bianca Gariboldi, Giacomo Gigante, Giancarlo Travaglini, Discrepancy for convex bodies with isolated flat points. Rev. Mat. Iberoam. 36 (2020), no. 6, pp. 1597–1626
DOI 10.4171/RMI/1177