Related papers: Equidistribution Mod $1$ And Normal Numbers

A real number is called simply normal to base $b$ if every digit $0,1,\ldots ,b-1$ should appear in its $b$-adic expansion with the same frequency $1/b$. A real number is called normal to base $b$ if it is simply normal to every base $b,…

Given a real number $0.a_1a_2 a_3\dots$ that is normal to base $b$, we examine increasing sequences $n_i$ so that the number $0.a_{n_1}a_{n_2}a_{n_3}\dots$ are normal to base $b$. Classically it is known that if the $n_i$ form an arithmetic…

In this paper we study the property of normality of a number in base 2. A simple rule that associates a vector to a number is presented and the property of normality is stated for the vector associated to the number. The problem of testing…

Defined by Borel, a real number is normal to an integer base $b$, greater than or equal to $2$, if in its base-$b$ expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider…

Despite the fact that almost all real numbers are absolutely normal---that is, the digits in their expansions to any base occur in all possible configurations with the expected frequency---not one specific example of an absolutely normal…

Let $\alpha,\beta$ be real numbers and $b\geq2$ be an integer. Allouche and Shallit showed that the sequence $\{\lfloor\alpha n+\beta\rfloor\}_{n\geq0}$ is $b$-regular if and only if $\alpha$ is rational. In this paper, using a…

In this paper, we show that the concatenation of the Fibonacci sequence is \textit{normal} in base $10$, meaning every string of a given length, $k$, occurs as frequently as every other string of length $k$ (there are as many $1$'s as $2$'s…

Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0, 1, ..., s-1 occurs with the same frequency 1/s. Let X be the set of positive integers that are not perfect…

Wall's theorem on arithmetic progressions says that if $0.a_1a_2a_3\dots$ is normal, then for any $k,\ell\in \mathbb{N}$, $0.a_ka_{k+\ell}a_{k+2\ell}\dots$ is also normal. We examine a converse statement and show that if…

A number is normal in base $b$ if, in its base $b$ expansion, all blocks of digits of equal length have the same asymptotic frequency. The rate at which a number approaches normality is quantified by the classical notion of discrepancy,…

It is well known that all numbers that are normal of order $k$ in base $b$ are also normal of all orders less than $k$. Another basic fact is that every real number is normal in base $b$ if and only if it is simply normal in base $b^k$ for…

In this paper, we compute the asymptotic average of the decimals of some real numbers. With the help of this computation, we prove that if a real number cannot be represented as a finite decimal and the asymptotic average of its decimals is…

Let $(p_n)$ denote the sequence of prime numbers, with $2=p_1<p_2<\ldots$. We demonstrate the existence of an irrational number $\alpha$ having the property that the sequence $(\alpha p_n)$ is not well-distributed modulo $1$.

Let $x$ be a periodic continued fraction with the initial block $0$ and the repeating block $c_1,\ldots,c_n$. So $x$ is a quadratic irrational of the form $x=a+\sqrt b$, where $a$, $b$ are rational numbers, $b>0$, $b$ not a square. The…

It is well known that if $0.a_1a_2a_3\dots$ is the base-$b$ expansion of a number normal to base-$b$, then the numbers $0.a_ka_{m+k}a_{2m+k}\dots$ for $m\ge 2$, $k\ge 1$ are all normal to base-$b$ as well. In contrast, given a continued…

We construct the base $2$ expansion of an absolutely normal real number $x$ so that, for every integer $b$ greater than or equal to $2$, the discrepancy modulo $1$ of the sequence $(b^0 x, b^1 x, b^2 x , \ldots)$ is essentially the same as…

We consider numbers formed by concatenating some of the base b digits from additive functions f(n) that closely resemble the prime counting function \Omega(n). If we concatenate the last \lceil y \frac{\log \log \log n}{\log b} \rceil…

In the present paper the authors construct normal numbers in base $q$ by concatenating $q$-adic expansions of prime powers $\lfloor\alpha p^\theta\rfloor$ with $\alpha>0$ and $\theta>1$.

We show that the sequence $(\alpha n)_{n\in \mathcal{B}}$ is uniformly distributed modulo 1, for every irrational $\alpha$, provided $\mathcal{B}$ belongs to a certain family of integer sequences, which includes the prime, almost prime,…

The equidistribution of the double sign patterns of the Liouville function $\lambda$ is proved unconditionally. As application, it is shown that the computable real number $$\sum_{n \geq 1} \frac{1+\lambda(n)}{2^{n}}$$ is a simply normal…