Quadratic algebraic numbers with finite b‐adic expansion on the unit circle and their distribution

Abstract

AbstractWe consider elements x + y$ \sqrt {-m} $ in the imaginary quadratic number field ℚ($ \sqrt {-m} $) such that the norm x2 + my2 = 1 and both x and y have a finite b–adic expansion for an arbitrary but fixed integer base b. For m = 2, 3, 7 and 11 a full description of this set is given. Ordered by the number of digits in the b–adic expansion of the coordinates, the corresponding sequence of points on the unit circle, if infinite, is uniformly distributed. This continues work of P. Schatte who treated the case m = 1. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)