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

Abstract

I P ∗ IP^* sets and central sets are subsets of N \mathbb 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 sets of the form { [ n α + γ ] : n ∈ N } \{[n\alpha +\gamma ]\colon n\in \mathbb N\} . Iterated spectra are similarly defined with n 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 \alpha >0 and 0 > γ > 1 0>\gamma >1 , then { [ n α + γ ] : n ∈ N } \{[n\alpha +\gamma ]\colon n\in \mathbb N\} is an I P ∗ IP^* set and consequently contains an infinite sequence together with all finite sums and products of terms from that sequence without repetition.