Iterated Spectra of Numbers—elementary, Dynamical, and Algebraic Approaches

IP* sets and central sets are subsets of N which arise out of applications of topological dynamics to number theory and are known to have rich combinatorial structure. Spectra of numbers are often studied ([3], [15], [16], [30], [31], [32]) sets of the form {[nα+ γ] : n ∈ N}. Iterated spectra are similarly defined with n coming from another spectrum. Using elementary, dynamical, and algebraic approaches we show that iterated spectra have significantly richer combinatorial structure than was previously known. For example we show that if α > 0 and 0 < γ < 1, then {[nα + γ] : n ∈ N} is an IP* set and consequently contains an infinite sequence together with all finite sums and products of terms from that sequence without repetition.