EUDML

Let $T : ℝ / ℤ \to ℝ / ℤ$ be a von Neumann-Kakutani $q$ - adic adding machine transformation and let $ϕ \in C^{1} ([0, 1])$ . Put $ϕ_{n} (x) : = ϕ (x) + ϕ (T x) + ... + ϕ (T^{n - 1} x), x \in ℝ / ℤ, n \in ℕ .$ We study three questions: 1. When will $(ϕ_{n} (x))_{n \geq 1}$ be bounded? 2. What can be said about limit points of $(ϕ_{n} (x))_{n \geq 1} ?$ 3. When will the skew product $(x, y) \mapsto (T x, y + ϕ (x))$ be ergodic on $ℝ / ℤ \times ℝ ?$

On the discrepancy of sequences associated with the sum-of-digits function

Gerhard Larcher; N. Kopecek; R. F. Tichy; G. Turnwald — 1987

Annales de l'institut Fourier

If $w = (q_{k})_{k \in N}$ denotes the sequence of best approximation denominators to a real $α$ , and $s_{α} (n)$ denotes the sum of digits of $n$ in the digit representation of $n$ to base $w$ , then for all $x$ irrational, the sequence $(s_{α} (n) \cdot x)_{n \in N}$ is uniformly distributed modulo one. Discrepancy estimates for the discrepancy of this sequence are given, which turn out to be best possible if $α$ has bounded continued fraction coefficients.

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Quantitative rearrangement theorems

Optimale Koeffizienten bezüglich zusammengesetzter Zahlen.

A Best Lower Bound for Good Lattice Points.

On the Distribution of Sequences Connected with Good Lattice Points.

On the Distribution of Sequences connected with Digit-Representation.

Metric results on the approximation of zero by linear combinations of independent and of dependent rationals.

Corrigendum to the paper "On the distribution of s-dimensional Kronecker sequences" Acta Arith. 51 (1988), 335-347

Nets obtained from rational functions over finite fields

On the distribution of s-dimensional Kronecker-sequences

A bound for the discrepancy of digital nets and its application to the analysis of certain pseudo-random number generators

Kronecker-type sequences and nonarchimedean diophantine approximations

Some number-theoretical properties of generalized sum-of-digit functions

On existence and discrepancy of certain digital Niederreiter-Halton sequences

On functions with bounded remainder

On the discrepancy of sequences associated with the sum-of-digits function