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 study the concentration of the distribution of an additive function f when the sequence of prime values of f decays fast and has good spacing properties. In particular, we prove a conjecture by Erdős and Kátai on the concentration of $f\left(n\right)={\sum }_{p|n}{\left(logp\right)}^{-c}$ when c > 1.

We show that for any irrational number α and a sequence ${m_l}_{l\in \mathbb{N}}$ of integers such that $li{m}_{l\to \infty }\left|\right||m_l\alpha \left|\right||=0$, there exists a continuous measure μ on the circle such that $li{m}_{l\to \infty }{\int }_{}\left|\right||m_l\theta \left|\right||d\mu \left(\theta \right)=0$. 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 ${m_l}_{l\in \mathbb{N}}$ of integers such that $\left|\right||m_l\alpha \left|\right||\to 0$ and such that $m_l\theta \left[1\right]$ is dense on the circle if and only if θ ∉ ℚα + ℚ.

In an earlier paper Gyarmati introduced the notion of f-correlation for families of binary pseudorandom sequences as a measure of randomness in the family. In this paper we generalize the f-correlation to families of pseudorandom sequences of k symbols and study its properties.

Given an integer base $q\ge 2$ and a completely $q$-additive arithmetic function $f$ taking integer values, we deduce an asymptotic expression for the counting function$$N_f\left(x\right)=\#\left\{0\le n\<x\left|f\right(n)\mid n\right\}$$under a mild restriction on the values of $f$. When $f=s_q$, the base $q$ sum of digits function, the integers counted by $N_f$ are the so-called base $q$ Niven numbers, and our result provides a generalization of the asymptotic known in that case.