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The spatial distribution of algebraic integers on constant-norm hypersurfaces

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  26 February 2010

R. W. K. Odoni
R. W. K. Odoni
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, Scotland.
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Abstract

Let K be an algebraic number field, [ K: ] = KΣ. Most of what we shall discuss is trivial when K = , so that we assume that K ≥ 2 from now onwards. To describe our results, we consider the classical device [2] of Minkowski, whereby K is embedded (diagonally-) into the direct product MK of its completions at its (inequivalent) infinite places. Thus MK is -algebra isomorphic to , and is to be regarded as a topological -algebra, dimRMK = K, in which K is everywhere dense, while the ring Zx of integers of K embeds as a discrete -submodule of rank K. Following the ideas implicit in Hecke's fundamental papers [6] we may measure the “spatial distribution” of points of MK (modulo units of κ) by means of a canonical projection onto a certain torus . The principal application of our main results (Theorems I–III described below) is to the study of the spatial distribution of the which have a fixed norm n = NK/Q(α). In §2 we shall show that, with suitable interpretations, for “typical” n (for which NK/Q(α) = n is soluble), these α have “almost uniform” spatial distribution under the canonical projection onto TK. Analogous questions have been considered by several authors (see, e.g., [5, 9, 14]), but in all cases, they have considered weighted averages over such n of a type which make it impossible to make useful statements for “typical” n.

MSC classification

Secondary: 11R45: Density theorems
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 350 - 366
Copyright
Copyright © University College London 1992

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