Completeness of sparse, almost integer, and finite local complexity sequences of translates in Lp(R)

A real sequence is called ‐generating if there exists a function whose translates span the space . While the ‐generating sets were completely characterized for and , the case remains not well understood. In this case, both the size and the arithmetic structure of the set play an important role. In this paper, (i) we show that a ‐generating set of positive real numbers can be very sparse, namely, the ratios may tend to 1 arbitrarily slowly; (ii) we prove that every “almost integer” sequence , that is, satisfying , , is ‐generating; and (iii) we construct ‐generating sets such that the successive differences attain only two different positive values. The constructions are, in a sense, sharp: it is well known that cannot be Hadamard lacunary and cannot be contained in any arithmetic progression.