Ergodic theory on SL ( n ), diophantine approximations and anomalies in the lattice point problem

1998 ◽  
Vol 132 (1) ◽  
pp. 1-72 ◽  
Author(s):  
M. M. Skriganov
Keyword(s):  
Ergodic Theory ◽  
Lattice Point ◽  
Point Problem ◽  
Lattice Point Problem ◽  
Diophantine Approximations

On logarithmically small errors in the lattice point problem

2000 ◽  
Vol 20 (5) ◽  
pp. 1469-1476 ◽  
Author(s):  
M. M. SKRIGANOV ◽  
A. N. STARKOV
Keyword(s):  
Asymptotic Formula ◽  
Lattice Point ◽  
Orthogonal Matrix ◽  
Point Problem ◽  
Hidden Symmetries ◽  
Integer Points ◽  
Lattice Point Problem ◽  
Diophantine Approximations ◽  
Almost All ◽  
Compact Polyhedron

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 $\varepsilon>0$.

Steinhaus' lattice point problem for polyhedra

2017 ◽  
Vol 146 (1) ◽  
pp. 123-128 ◽  
Author(s):  
Hiroshi Maehara
Keyword(s):  
Lattice Point ◽  
Point Problem ◽  
Lattice Point Problem

The Generalized Lattice-Point Problem

1973 ◽  
Vol 21 (1) ◽  
pp. 141-155 ◽  
Author(s):  
Fred Glover ◽  
D. Klingman
Keyword(s):  
Lattice Point ◽  
Point Problem ◽  
Lattice Point Problem
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