Self-avoiding generating sequences for fourier lattice designs

Good sets of sampling points, or designs, for multidimensional Fourier regression are based on integer lattices whose positive integer generators have special self-avoiding properties. In this paper we discuss the smallest sample size to achieve the properties (the generalized Nyquist rate) and show that it depends on statements about the existence of integer vectors that do not satisfy a special set of linear equations. It follows that some solutions are derived from problems of intrinsic interest related to Sidon sets, the Thue-Morse sequence, and constructions based on integers with prohibited digits also related to the Cantor set.