On the Images of ℕ-Rational Sequences Counting Multiplicities

Abstract

A mapping ϕ:ℤ → ℤ is called piecewise affine if there exist integers a≥ 1 and uj≥ 1, vj for 0≤ j<a such that ϕ(an+j)=ujn+vj whenever n∈ ℤ and 0≤ j<a. We prove that if s=(s(n))n≥ 0 and t=(t(n))n≥ 0 are ℕ-rational sequences such that s takes each value exactly as many times as t, then there exists a piecewise affine mapping ϕ:ℤ → ℤ such that s(n)=t(ϕ(n)) for almost all n≥ 0. As an application we solve the HD0L language equivalence problem in some cases.