Related papers: Equidistribution Mod $1$ And Normal Numbers
Khabibullin's conjecture for integral inequalities has two numeric parameters $n$ and $\alpha$ in its statement, $n$ being a positive integer and $\alpha$ being a positive real number. This conjecture is already proved in the case where…
Copeland and Erd\H{o}s showed that the concatenation of primes when written in base $10$ yields a real number that is normal to base $10$. We generalize this result to Pisot number bases in which all integers have finite expansion.
In this paper we refine Ball-Rivoal's theorem by proving that for any odd integer $a$ sufficiently large in terms of $\epsilon>0$, there exist $[ \frac{(1-\epsilon)\log a}{1+\log 2}]$ odd integers $s$ between 3 and $a$, with distance at…
In this short note, we give a proof, conditional on the Generalized Riemann Hypothesis, that there exist numbers x which are normal with respect to the continued fraction expansion but not to any base b expansion. This partially answers a…
In this article we investigate different forms of multiplicative independence between the sequences $n$ and $\lfloor n \alpha \rfloor$ for irrational $\alpha$. Our main theorem shows that for a large class of arithmetic functions $a, b…
Mordechay Levin has constructed a number $\alpha$ which is normal in base 2, and such that the sequence $\left\{2^n \alpha\right\}_{n=0,1,2,\ldots}$ has very small discrepancy $D_N$. Indeed we have $N\cdot D_N = \mathcal{O} \left(\left(\log…
A classical problem in analytic number theory is to study the distribution of fractional part $\alpha p+\beta$ modulo 1, where $\alpha$ is irrational and $p$ runs over the set of primes. We consider the subsequence generated by the primes…
We prove some new theorems in additive number theory, using novel techniques from automata theory and formal languages. As an example of our method, we prove that every natural number > 25 is the sum of at most three natural numbers whose…
Let $\alpha$ be a real number such that $1< \alpha <2$ and let $x_0=x_0(\alpha)$ be a {\rm(}unique{\rm)} positive solution of the equation $$ x^{\alpha-1} -\frac{\pi}{e^2\sqrt{3}}x +1=0. $$ Then we prove that for each positive integer…
For two natural numbers $1<p_1<p_2$, with $\alpha = \frac{\log(p_1)}{\log(p_2)}$ irrational, we describe, in main Theorem $\Omega$ and in Note $1.5$, the factorization of two adjacent numbers in the multiplicatively closed subset $S =…
A seminal result due to Wall states that if $x$ is normal to a given base $b$ then so is $rx+s$ for any rational numbers $r,s$ with $r\neq 0$. We show that a stronger result is true for normality with respect to the continued fraction…
We show how Benford's Law (BL) for first, second, ..., digits, emerges from the distribution of digits of numbers of the type $a^{R}$, with $a$ any real positive number and $R$ a set of real numbers uniformly distributed in an interval $[…
Prime numbers play a key role in number theory and have applications beyond Mathematics. In particular, in the Theory of Codes and also in Cryptography, the properties of prime numbers are relevant, because, from them, it is possible to…
The rational base number system, introduced by Akiyama, Frougny, and Sakarovitch in 2008, is a generalization of the classical integer base number system. Within this framework two interesting families of infinite words emerge, called…
In order to prove irrationality of \sqrt{2} by using only decimal expansions (and not fractions), we develop in detail a model of real numbers based on infinite decimals and arithmetic operations with them.
We analyze the convergence order of an algorithm producing the digits of an absolutely normal number. Furthermore, we introduce a stronger concept of absolute normality by allowing Pisot numbers as bases, which leads to expansions with…
We use entropy rates and Schur concavity to prove that, for every integer k >= 2, every nonzero rational number q, and every real number alpha, the base-k expansions of alpha, q+alpha, and q*alpha all have the same finite-state dimension…
Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation…
For $\alpha>1$ we represent a real number in $(0,1]$ in the form \[ \sum_{i=1}^{\infty}(\alpha-1)^{i-1}\alpha^{-(d_{1}+\dots+d_{i})}\] with $d_{i}\in\mathbb{N}$. We discuss ergodic theoretical and dimension theoretical aspects of this…
We show that each number of the form, the square root of s for s not a perfect square, is simply normal to the base 2. The argument uses some elementary ideas from the calculus of finite differences.