On the distribution of residue classes of quadratic forms and integer-detecting sequences in number fields (Q2714369)

An increasing sequence \((m_n)_{n\geq 0}\) of positive integers is said integer-detecting if any rational number \(r\), satisfying \(\|m_n r\|<\varepsilon\) for all \(n\geq 0\), is an integer (the number \(\varepsilon\) is called the height). This notion was introduced in [\textit{J. W. Sander} and \textit{J. Steinig}, Indag. Math., New Ser. 9, 305--315 (1998; Zbl 0920.11043)] in order to study the residue classes to given moduli which are represented by a quadratic form.NEWLINENEWLINEThe aim of this paper is to extend this study to norms in quadratic number fields. Some sufficient criteria are first deduced from the rational case using growth and divisiblility properties, for a subset of the ring of integers \(\mathcal O_K\) of the number field \(K\) to be integer norm-detecting. Then, an algorithmic computation of maximal heights with respect to the discriminant of the field \(K\) in quadratic number fields is given, based on the Pólya-Vinogradov inequality. The paper ends with a series of criteria for a set \(A\) to be integer norm-detecting, based on the fact that the set \(A\) contains sufficiently enough residue classes with respect to every modulus. In particular, a set \(A\subset \mathcal O_{\mathbb Q(\sqrt - D)}\) is integer norm-detecting if \(A\) contains all algebraic integers lying in some arbitrarily small angle with vertex at the origin in the complex plane.