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A Gauss-Kuzmin theorem for the Rosen fractions

Gabriela I. Sebe (2002)

Journal de théorie des nombres de Bordeaux

Using the natural extensions for the Rosen maps, we give an infinite-order-chain representation of the sequence of the incomplete quotients of the Rosen fractions. Together with the ergodic behaviour of a certain homogeneous random system with complete connections, this allows us to solve a variant of Gauss-Kuzmin problem for the above fraction expansion.

A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

Kathryn E. Hare, Shuntaro Yamagishi (2014)

Acta Arithmetica

Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define r N ( m ) ( ω ) to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that r N ( m ) ( ω ) < K for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

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