Quasiregular heterostructures: An overview of the current situation

Mathematical research on quasiregular sequences has provided a sound foundation for the analysis of their formal properties. However, physical research on heterostructures following these sequences appears to largely ignore this formal basis. The topology of the spectra is seldom studied in a sufficiently complete way and claims of self-similarity usually lack precision and rigour. In a critical overview of the current situation, limitations are pointed out and directions for further progress are suggested. The discussion focuses on the possible prospects, so far comparatively scarce, for device design. Theoretical research, which mostly dominates the scene, appears to have scarcely stimulated experimental work.