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Given a subsequence of a uniformly distributed sequence, relations between the asymptotic densities of sets of its indices and the Lebesgue measure of the set of all its limit points are studied.

On normal numbers and powers of algebraic numbers.

Kaneko, Hajime (2010)

Integers

On Riemann Sums and Lebesgue Integrals.

R. Nair (1995)

Monatshefte für Mathematik

On the convergence to 0 of mₙξmod 1

Bassam Fayad, Jean-Paul Thouvenot (2014)

Acta Arithmetica

We show that for any irrational number α and a sequence ${m_{l}}_{l \in ℕ}$ of integers such that $l i m_{l \to \infty} | | | m_{l} α | | | = 0$ , there exists a continuous measure μ on the circle such that $l i m_{l \to \infty} \int_{} | | | m_{l} θ | | | d μ (θ) = 0$ . This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any α ∈ ℝ - ℚ, there exists a sequence ${m_{l}}_{l \in ℕ}$ of integers such that $| | | m_{l} α | | | \to 0$ and such that $m_{l} θ [1]$ is dense on the circle if and only if θ ∉ ℚα + ℚ.

On the discrepancy of (nα)

Johannes Schoißengeier (1984)

Acta Arithmetica

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.

On the distributation of p ... modulo one.

R.C. Baker, G. Kolesnik (1985)

Journal für die reine und angewandte Mathematik

On the distribution function of certain sequences (mod 1)

A. Ostrowski (1980)

Acta Arithmetica

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