Mixed $L^p(L^2)$ norms of the lattice point discrepancy
| Autor: | Giacomo Gigante, Leonardo Colzani, Bianca Gariboldi |
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| Přispěvatelé: | Colzani, L, Gariboldi, B, Gigante, G |
| Rok vydání: | 2017 |
| Předmět: |
General Mathematics
Mathematics::Analysis of PDEs Boundary (topology) 11H06 42B05 52C07 01 natural sciences Combinatorics Dilation (metric space) symbols.namesake Mathematics - Analysis of PDEs Integer Integer lattice Settore MAT/05 - Analisi Matematica Convex body Gaussian curvature FOS: Mathematics Lattice points Discrepancy Number Theory (math.NT) 0101 mathematics MAT/05 - ANALISI MATEMATICA Discrepancy Mathematics Mathematics - Number Theory Applied Mathematics 010102 general mathematics symbols Analysis of PDEs (math.AP) |
| DOI: | 10.48550/arxiv.1706.04419 |
| Popis: | 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 dilated by a factor $r$ and translated by a vector $x$ of a convex body $\Omega$ in $\mathbb{R}^{d}$, $ \left\{ {\int_{\mathbb{T}^{d}}}\left( \frac{1}{H} {\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 $H\to\infty$. |
| Databáze: | OpenAIRE |
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