zbMATH — the first resource for mathematics

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

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 \(1\)