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Lattices in algebraic number fields and uniform distribution mod 1. (English. Russian original) Zbl 0714.11045
Leningr. Math. J. 1, No. 2, 535-558 (1990); translation from Algebra Anal. 1, No. 2, 207-228 (1989).
The author considers a totally real algebraic number field F of degree s and the lattice $$\Gamma_ M\subset {\mathbb{R}}^ s$$ induced by a full module $$M\subset F$$. Let $$B=B(x,t)=$$ $$[x_ 1,x_ 1+t_ 1]\times [x_ 2,x_ 2+t_ 2]\times...\times [x_ s,x_ s+t_ s]$$ and $$r(x,\Gamma_ M)=\#(B\cap \Gamma_ M)-t_ 1t_ 2...t_ s/d(\Gamma_ M),$$ where $$d(\Gamma_ M)$$ is the discriminant of $$\Gamma_ M$$. Then the $$L^{\infty}$$- and the $$L^ q$$-norm, $$q\geq 2$$, of $$r(x,\Gamma_ M)$$ with respect to x are estimated by $$O(\log^ s(2+t_ 1t_ 2...t_ s)).$$ Analogous estimates are derived for the discrepancy of $$B(0,K^ s)\cap T'\Gamma_ M,$$ where $$K^ s$$ is the unit torus in $${\mathbb{R}}^ s$$, $$T'=(1/t_ 1,...,1/t_ s),$$ and for the remainder term in quadrature formulas. For $$s=2$$ the estimates $$O(\log^{s-1}(2+t_ 1t_ 2...t_ s))$$ are achieved. - The author claims that his choice of $$\Gamma_ M$$ is well-suited for numerical integration.
Reviewer: P.Schatte

##### MSC:
 11K38 Irregularities of distribution, discrepancy 11H06 Lattices and convex bodies (number-theoretic aspects) 65D32 Numerical quadrature and cubature formulas 11R80 Totally real fields 11P21 Lattice points in specified regions