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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 {(n + ) : n 2 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

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