Assuming Martin's axiom we show that if X is a dyadic space of weight at most continuum then every Radon measure on X admits a uniformly distributed sequence. This answers a problem posed by Mercourakis [10]. Our proof is based on an auxiliary result concerning finitely additive measures on ω and asymptotic density.

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 $ℝ/\mathbb{Z}\times ℝ?$

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.

We construct infinite-dimensional chains that are L¹ good for almost sure convergence, which settles a question raised in this journal [N]. We give some conditions for a coprime generated chain to be bad for L² or $L^{\infty }$, using the entropy method. It follows that such a chain with positive lower density is bad for $L^{\infty }$. There also exist such bad chains with zero density.