Mixed $L^{p}(L^{2})$ norms of the lattice point discrepancy
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- by Leonardo Colzani, Bianca Gariboldi and Giacomo Gigante PDF
- Trans. Amer. Math. Soc. 371 (2019), 7669-7706 Request permission
Abstract:
We estimate some mixed $L^{p}\left ( L^{2}\right )$ norms of the discrepancy between the volume and the number of integer points in $r\Omega -x$, a dilation by a factor $r$ and a translation by a vector $x$ of a convex body $\Omega$ in $\mathbb {R}^{d}$ with smooth boundary with nonvanishing Gaussian curvature, \[ \left \{ {\displaystyle \int _{\mathbb {T}^{d}}}\left ( \dfrac {1}{H}{\displaystyle \int _{R}^{R+H}}\left \vert \sum _{k\in \mathbb {Z}^{d}}\chi _{r\Omega -x}(k)-r^{d}\left \vert \Omega \right \vert \right \vert ^{2}dr\right ) ^{p/2}dx\right \} ^{1/p}. \] We obtain estimates for fixed values of $H$ and $R\to \infty$, and also asymptotic estimates when $H\to \infty$.References
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Additional Information
- Leonardo Colzani
- Affiliation: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via R. Cozzi 55, 20125 Milano, Italy
- MR Author ID: 50785
- Email: leonardo.colzani@unimib.it
- Bianca Gariboldi
- Affiliation: Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione, Università degli Studi di Bergamo, Viale Marconi 5, 24044 Dalmine (BG), Italy
- Email: biancamaria.gariboldi@unibg.it
- Giacomo Gigante
- Affiliation: Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione, Università degli Studi di Bergamo, Viale Marconi 5, 24044 Dalmine (BG), Italy
- MR Author ID: 666574
- Email: giacomo.gigante@unibg.it
- Received by editor(s): June 8, 2017
- Received by editor(s) in revised form: January 22, 2018
- Published electronically: March 7, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 7669-7706
- MSC (2010): Primary 11H06, 42B05, 52C07
- DOI: https://doi.org/10.1090/tran/7624
- MathSciNet review: 3955532