Orthogonal decomposition of the space of algebraic numbers and Lehmerʼs problem

Building on work of Dubickas and Smyth regarding the metric Mahler measure and the authors regarding extremal norms associated to the Mahler measure, the authors introduce a new set of norms associated to the Mahler measure of algebraic numbers which allow for an equivalent reformulation of problems like the Lehmer problem and the Schinzel-Zassenhaus conjecture on a single spectrum. We present several new geometric results regarding the space of algebraic numbers modulo torsion using the Lp Weil height introduced by Allcock and Vaaler, including a canonical decomposition of an algebraic number into an orthogonal series with respect to the L2 height.