Contributions to Metric Number Theory

Paul Rowe 

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 $\mathbb{R}^{m}$.