Polynomials at iterated spectra near zero

Central sets in N are sets known to have substantial combinatorial structure. Given x ∈ R , let w ( x ) = x − ⌊ x + 1 2 ⌋ . Kroneckerʼs Theorem (Kronecker, 1884) [18] says that if 1 , α 1 , α 2 , … , α v are linearly independent over Q and U is an open subset of ( − 1 2 , 1 2 ) v , then { x ∈ N : ( w ( α 1 x ) , … , w ( α v x ) ) ∈ 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 P 1 , P 2 , … , P v be real polynomials with zero constant term. We showed that { x ∈ N : ( w ( P 1 ( x ) ) , … , w ( P v ( x ) ) ) ∈ U } is nonempty for every open U with 0 ¯ ∈ c ℓ U if and only if it is central for every such U and we obtained a simple necessary and sufficient condition for these to occur. s paper we show that the same conclusion applies to compositions of polynomials with functions of the form n ↦ ⌊ α n + γ ⌋ where α is a positive real and 0 < 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 ↦ ⌊ α n ⌋ or n ↦ ⌊ α n + 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.