Number Systems and Repartition Original Research Article

Given a number system image = ((0, 1), Tj, (Jn, j)n set membership, variant Ej)j ≥ 0, we define a measurable mapping Φimage: (0, 1)image → (0, 1) such that λ∞(Φ−1image(A)) = λ(A), A set membership, variant image(0, 1). A measurable section (tn(·))n ≥ 0 is defined for Φimage; tn(·) has uniform distribution for any n ≥ 0. Conditions relative to λ-a.e. repartition properties of (tn(·))n ≥ 0 are studied. Applications to (α, γ)-expansions, Cantor products, and continued fractions are given.