TY - JOUR

T1 - Iterated spectra of numbers - Elementary, dynamical, and algebraic approaches

AU - Bergelson, Vitaly

AU - Hindman, Neil

AU - Kra, Bryna

PY - 1996

Y1 - 1996

N2 - 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 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 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.

AB - 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 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 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.

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U2 - 10.1090/s0002-9947-96-01533-4

DO - 10.1090/s0002-9947-96-01533-4

M3 - Article

AN - SCOPUS:21344464858

SN - 0002-9947

VL - 348

SP - 893

EP - 912

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 3

ER -