Contributions to Metric Number Theory

Abstract

The aim of this work is to investigate some arithmetical properties of real numbers, for example by considering sequences of the type $([b^{n}\alpha ])$ , $n=1,2,\ldots $ where $b\in \mathbb{N},\alpha \in \mathbb{R}$, the terms of the sequences being in arithmetical progression, square-free, sums of two squares or primes. The results are most commonly proved for almost all $\alpha\in \mathbb{R}$ or $(\alpha_{1},\ldots ,\alpha _{m})\in \mathbb{R}^{m}$ (in the sense of Lebesgue measure). In the first chapter normal numbers are studied. The concept of a normal number is generalised by defining normal points in higher dimensions, and through the link between normal numbers and uniform distribution, it is proved that almost all points on the curve $(\alpha ,\alpha^{2}, \ldots ,\alpha ^{m}) \in \mathbb{R}^{m}$ are normal. The second chapter includes a construction that yields normal numbers. This follows on from a result by Davenport and Erd\H{o}s which shows that $0.f(1)f(2)f(3)\cdots $ is normal for any polynomial $f(x)$ which takes only positive integer values at $x=1,2,\ldots $. The result proved here replaces $f(x)$ by $[g(x)]$ where $g(x)=a_{1}x^{\alpha_{1}}+a_{2}x^{\alpha_{2}}+\cdots + a_{k}x^{\alpha_{k}}$ for the $\alpha _{i},a_{i}$ any positive real numbers. The third chapter considers square-free numbers and gives for almost all $\alpha $, an asymptotic formula for the number of solutions in $n$ to $[10^{n}\alpha^{a_{1}}]$, $[10^{n}\alpha^{a_{2}}]$, $ \ldots , [10^{n}\alpha^{a_{k}}]$ simultaneously square-free for $n\leq N$, where each $a_{i}\in \mathbb{N}$. The fourth chapter considers sums of two squares and gives for almost all $(\alpha, \beta)\in \mathbb{R}^{2}$ an asymptotic formula for the number of solutions to $[10^{n}\alpha]$ and $ [10^{n}\beta]$ simultaneously sums of two squares for $n\leq N$. The final chapter investigates the set of $(\alpha_{1},\ldots , \alpha _{m})\in \mathbb{R}^{m}$ such that $[10^{n}\alpha_{1}],[10^{n}\alpha_{2}],\ldots , [10^{n}\alpha_{m}]$ are simultaneously prime infinitely often. This set is shown to have Hausdorff dimension $m$ and to be dense in $\mathbb{R}^{m}$.