This is the second volume in a series of books exploring the mathematics of aperiodic order. While the first volume was meant as a general introduction to the field, we now start to develop the theory in more depth. To do justice to the rapidly expanding field, we decided to work with various authors or teams of authors, which means that this book is somewhere intermediate between a monograph and a review selection. Future volumes will also be structured in this way.

Clearly, almost periodicity is a central concept of crystallography, as it reflects and captures the coherent repetition of local motifs or patterns. The foremost tool to analyse such structures is provided by Fourier analysis of measures, which thus forms a substantial part of this volume. Other important aspects are usually analysed by group theoretic or general algebraic methods. In this respect, due to the availability of comprehensive reviews and several books, we decided to not include a chapter on space groups and their generalisation to quasicrystals.

The main text begins with a chapter on inflation tilings, contributed by Dirk Frettlöh. It augments the discussion of the first volume by presenting a panorama of less familiar constructions and recent developments. This is followed by a contribution to the inverse problem of discrete tomography, where special emphasis lies on the comparison between notions from classical (periodic) crystallography and their extensions to quasicrystals. A similar interplay is prevalent in the ensuing chapter on enumeration problems for lattices versus embedded ℤ-modules, which highlights the power of numbertheoretic methods in the theory of aperiodic order.

The substantial part on almost periodicity and its facets begins with a thorough exposition of the general theory of almost periodic measures on locally compact Abelian groups, contributed by Robert V. Moody and Nicolae Strungaru. This comprehensive summary emerged from the need to understand the spectral structure of aperiodic systems. Perhaps the most important connection exists with the structure of Meyer sets and their description via cut and project schemes, which is developed in the ensuing chapter by Nicolae Strungaru.