Lattices in algebraic number fields and uniform distribution mod 1. (English. Russian original) Zbl 0714.11045

Skriganov, M. M.

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

Keywords:

totally real algebraic number field; lattice; discrepancy; quadrature formulas; numerical integration