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On logarithmically small errors in the lattice point problem
In the present paper we give an improvement of a previous result of the paper [M. M. Skriganov. Ergodic theory on $SL(n)$, diophantine approximations and anomalies in the lattice point problem. Inv. Math. 132(1), (1998), 1–72, Theorem 2.2] on logarithmically small errors in the lattice point problem...
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Published in: | Ergodic theory and dynamical systems 2000-10, Vol.20 (5), p.1469-1476 |
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Main Authors: | , |
Format: | Article |
Language: | English |
Citations: | Items that cite this one |
Online Access: | Get full text |
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Summary: | In the present paper we give an improvement of a previous result of the paper [M. M. Skriganov. Ergodic theory on $SL(n)$, diophantine approximations and anomalies in the lattice point problem. Inv. Math. 132(1), (1998), 1–72, Theorem 2.2] on logarithmically small errors in the lattice point problem for polyhedra. This improvement is based on an analysis of hidden symmetries of the problem generated by the Weyl group for $SL(n,\mathbb{B})$. Let $UP$ denote a rotation of a given compact polyhedron $P\subset\mathbb{B}^n$ by an orthogonal matrix $U\in SO(n)$, $tUP$ a dilation of $UP$ by a parameter $t>0$ and $N(tUP)$ the number of integer points $\gamma\in\mathbb{Z}^n$ which fall into the polyhedron $tUP$. We show that for almost all rotations $U$ (in the sense of the Haar measure on $SO(n)$) the following asymptotic formula
\[
N(t\UP)=t^n{\rm vol} P+ O((\log t)^{n-1+\varepsilon}),\quad t\to\infty,
\]
holds with arbitrarily small $\varepsilon>0$. |
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ISSN: | 0143-3857 1469-4417 |
DOI: | 10.1017/S0143385700000791 |