JournalsrmiVol. 36, No. 6pp. 1597–1626

Discrepancy for convex bodies with isolated flat points

  • Luca Brandolini

    Università degli Studi di Bergamo, Dalmine, Italy
  • Leonardo Colzani

    Università di Milano-Bicocca, Italy
  • Bianca Gariboldi

    Università degli Studi di Bergamo, Dalmine, Italy
  • Giacomo Gigante

    Università degli Studi di Bergamo, Dalmine, Italy
  • Giancarlo 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 LpL^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 pp, 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