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On lattices in algebraic number fields. (English. Russian original) Zbl 0693.41029
Sov. Math., Dokl. 39, No. 3, 538-540 (1989); translation from Dokl. Akad. Nauk SSSR 306, No. 3, 553-555 (1989).
For a totally real number field of degree s let $$L_ F$$ be the lattice in $${\mathbb{R}}^ s$$ generated by F in the usual way [cf. P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, 2nd ed. (1987; Zbl 0611.10017)]. For a compact body B with measure v(B) and a (non- degenerate) lattice L in $$R^ s$$ with determinant d(L) let $$r=r(B,L)$$ denote the remainder $$r=(d(L))^{-1}v(B)-\#(B\cap L).$$ Then one has $(*)\quad...r(tQ,L_ F)\quad \ll \quad (\log (t))^ s$ as $$t\to \infty$$ uniformly for parallelepipeds Q with edges parallel to the axes. Similarly one has $$r(tP,{\mathbb{Z}}^ 2)\ll (\log (t))^ 2$$ for compact polygons P in the plane with finitely many sides, all of whose normals are proportional to vectors $$(1,a_ j)$$, the $$a_ j$$ being quadratic irrationalities. As regards the distribution of the points of $$t^{- 1}L_ F$$ in the s-dimensional unit cube $$K^ s$$, the following estimate for the discrepancy $$D_ t=D(K^ s\cap t^{-1}L_ F)$$ can be derived from $(*)\quad D_ t\quad \ll \quad t^{-s}(\log (t)^ s\quad \ll \quad N_{t-1}(\log (N_ t)^ s\quad as\quad t\to \infty;$ here $$N_ t=\#(K^ s\cap t^{-1}L_ F)$$. Combined with the Koksma-Hlawka inequality, this can be applied to obtain the following estimate for the error term in numerical integration: For a function f with bounded variation on $$K^ s$$, one has $\int_{K^ s}f d{\mathfrak x}-(N_ t)^{-1}\sum \{f({\mathfrak y})| \quad {\mathfrak y}\in K^ s\cap t^{- 1}L_ F\}\quad \ll \quad V(f)(N_ t)^{-1}(\log (N_ t))^ s.$ Finally, the behaviour of $$\delta =\delta (f,t^{-1}L_ F)=\int_{R^ s}f d{\mathfrak x}-d(L_ F)\sum \{f({\mathfrak y})|$$ $${\mathfrak y}\in t^{- 1}L_ F\}$$ is studied for the class S(a,s) of functions satisfying the following conditions:
(i) $$| f({\mathfrak x})| <c((1+| x_ 1|)...(1+| x_ s|))^{-b}$$ with some $$b=b(f)>1;$$
(ii) $$| F({\mathfrak x})| <c((1+| x_ 1|)...(1+| x_ s|))^{-a}$$ with a fixed $$a>1.$$
(Here F denotes the Fourier transform of f.) For $$f\in S(a,s)$$, one has $\delta \quad \ll \quad (a-1)^{-1}\| f\| t^{-as}(\log (t)^{s-1},$ where $$\| f\| =\sup \{| F({\mathfrak x})| ((1+| x_ 1|)...(1+| x_ s|))^ a\}$$.
Reviewer: G.Ramharter

##### MSC:
 41A55 Approximate quadratures 11H06 Lattices and convex bodies (number-theoretic aspects) 11K38 Irregularities of distribution, discrepancy 11P21 Lattice points in specified regions 11R27 Units and factorization 11K06 General theory of distribution modulo $$1$$