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

In this paper we extend Champernowne’s construction of normal numbers in base $b$ to the ${\mathbb{Z}}^d$ case and obtain an explicit construction of the generic point of the ${\mathbb{Z}}^d$ shift transformation of the set ${\{0,1,...,b-1\}}^{{\mathbb{Z}}^d}$. We prove that the intersection of the considered lattice configuration with an arbitrary line is a normal sequence in base $b$ .

Let $q,q_1,\cdots ,q_s\ge 2$ be integers, and let ${\alpha }_1,{\alpha }_2,\cdots$ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence$${(\left\{{\alpha }_m{q}^n\right\},\cdots ,\left\{{\alpha }_{m+s-1}{q}^n\right\})}_{m,n=1}^{MN}$$coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences ${\left(xn\right)}_{n=1}^{MN}$ in $s$-dimensional unit cube $(s,M,N=1,2,\cdots )$. We also find a lower bound of the discrepancy (up to a logarithmic factor) of the sequence ${(\left\{{\alpha }_1{q}_1^n\right\},\cdots ,\left\{{\alpha }_s{q}_s^n\right\})}_{n=1}^N$ (Korobov’s problem).

It is proved that a real-valued function $f\left(x\right)=exp\left(\pi i{\chi }_I\left(x\right)\right)$, where I is an interval contained in [0,1), is not of the form $f\left(x\right)=\overline{q\left(2x\right)}q\left(x\right)$ with |q(x)|=1 a.e. if I has dyadic endpoints. A relation of this result to the uniform distribution mod 2 is also shown.

Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1’s in the binary expansions of real algebraic numbers. A central result is that if a real $y$ has algebraic degree $D\>1$, then the number $\#\left(\right|y|,N)$ of 1-bits in the expansion of $\left|y\right|$ through bit position $N$ satisfies$$\#\left(\right|y|,N)\>C{N}^{1/D}$$for a positive number $C$ (depending on $y$) and sufficiently large $N$. This in itself establishes the transcendency of a class of reals ${\sum }_{n\ge 0}1/2^{f\left(n\right)}$ where the integer-valued...

We construct a Markov normal sequence with a discrepancy of $O\left(N^{-1/2}{log}^{2}N\right)$. The estimation of the discrepancy was previously known to be $O\left(e^{-c{(logN)}^{1/2}}\right)$.

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