Polynomials at iterated spectra near zero

Central   sets in NN are sets known to have substantial combinatorial structure. Given x∈Rx∈R, let w(x)=x−⌊x+12⌋. Kroneckerʼs Theorem (Kronecker, 1884) [18] says that if 1,α1,α2,…,αv1,α1,α2,…,αv are linearly independent over QQ and U   is an open subset of (−12,12)v, then {x∈N:(w(α1x),…,w(αvx))∈U}{x∈N:(w(α1x),…,w(αvx))∈U} is nonempty and Weyl (1916) [21] showed that this set has positive density. In a previous paper we showed that if 0¯ is in the closure of U  , then this set is central. More generally, let P1,P2,…,PvP1,P2,…,Pv be real polynomials with zero constant term. We showed that{x∈N:(w(P1(x)),…,w(Pv(x)))∈U}{x∈N:(w(P1(x)),…,w(Pv(x)))∈U} is nonempty for every open U   with 0¯∈cℓU if and only if it is central for every such U and we obtained a simple necessary and sufficient condition for these to occur.In this paper we show that the same conclusion applies to compositions of polynomials with functions of the form n↦⌊αn+γ⌋n↦⌊αn+γ⌋ where α   is a positive real and 0<γ<10<γ<1. (The ranges of such functions are called nonhomogeneous spectra   and by extension we refer to the functions as spectra.) We characterize precisely when we can compose with a single function of the form n↦⌊αn⌋n↦⌊αn⌋ or n↦⌊αn+1⌋n↦⌊αn+1⌋. With the stronger assumption that U   is a neighborhood of 0¯, we show when we can allow the composition with two such spectra and investigate some related questions.