Hurwitz Constant - Proof of Lemma 6.1.1. in Esmonde and Murty's Problems in Algebraic Number Theory

Question

I am studying from Esmonde & Murty's book, Problems in Algebraic Number Theory. I'm having trouble in the last step in the proof of the first Lemma in Chapter 6. Here $K$ is a number field of degree $n$ over $\mathbb{Q}$, and $\mathcal{O}$ is its ring of integers.

Lemma 6.1.1. There is a constant $H_K$ such that given $\alpha \in K$, $\exists\, \beta \in \mathcal{O}$, and a non-zero integer $t$, with $|t| \leq H_K$, such that $$ |N(t\alpha - \beta)| < 1. $$

The proof given in the book runs this way:

Let $\{ \omega_1,\dots,\omega_n \}$ be an integral basis for $\mathcal{O}$. $\alpha \in K$ so there is a non-zero integer $c$ such that $c \alpha \in \mathcal{O}$. So we can write $$ \alpha = \sum_{i=1}^n c_i \omega_i $$ where $c_i \in \mathbb{Q}$. Let $L \in \mathbb{N}$ be a positive integer, which we will specify later. Divide the interval $[0,1]$ into $L$ equal parts. Then, $[0,1]^n$ will be divided into $L^n$ subcubes. Define $\varphi : \alpha \mathbb{Z} \to [0,1]^n$ by $$ \varphi(t \alpha) = (\{ t c_1\},\dots,\{ tc_n \}) $$ where $t$ is an integer and $\{ \cdot \}$ denotes the fractional part. Let $t$ run from $0$ to $L^n$. By the pigeonhole principle, there exist $t_1, t_2$ in this range such that $t_1 \alpha$ and $t_2 \alpha$ are mapped to the same subcube. So, $| \{ t_1 c_i \} - \{ t_2 c_i \} | \leq 1 / L$ for $i = 1,\dots,n$. Let $$ \beta = \sum_{i=1}^n ([t_1 c_i] - [t_2 c_i]) \omega_i $$ where $[\cdot]$ is the greatest integer function. Let $t = t_1 - t_2$. Then, \begin{align*} |N(t\alpha - \beta)| &= \left| N\left( \sum_{i=1}^n (\{ t_1 c_i \} - \{ t_2 c_i \}) \omega_i \right) \right|\\ &= \left| \prod_{j = 1}^n \sigma_j \left( \sum_{i=1}^n (\{ t_1 c_i \} - \{ t_2 c_i \}) \omega_i \right) \right|\\ &\leq \prod_{j=1}^n \sum_{i=1}^n | (\{ t_1 c_i \} - \{ t_2 c_i \}) \sigma_j(\omega_i) |\\ &\leq \frac{1}{L^n} \prod_{j=1}^n \sum_{i=1}^n | \sigma_j(\omega_i) | \end{align*} where the $\sigma_j$ are the $n$ embeddings of $K$ in $\mathbb{C}$. Let $H_K = \prod_{j=1}^n \sum_{i=1}^n | \sigma_j(\omega_i) |$. If $L^n > H_K$ then $|N(t \alpha - \beta)| < 1$. Also, $|t| \leq L^n$, so if we choose $L = H_K^{1/n}$ we are done.

It is not clear to me how we are done by choosing $L = H_K^{1/n}$. I need $L^n > H_K$ to get $N(t\alpha - \beta) < 1$, but $L = H_K^{1/n} \Rightarrow L^n = H_K$, so these two conditions on $L$ are incompatible. Also, $L$ is a natural number so I also need to show that $H_K^{1/n}$ is a natural number. What steps am I missing in this proof?

EDIT : The book says that the constant $H_K$ is called the Hurwitz constant, hence my title.

Answer 1

The proof itself needs fixing. It should read:

If we take $L^n > \prod_{j=1}^n(\sum_{i=1}^n |\omega_i^{(j)}|)$, then $$|N(t\alpha - \beta)| < 1.$$ Furthermore, since $0 \leq t_1, t_2 \leq L^n$, we have $|t| \leq L^n$. Thus, if we choose $H_K = L^n$, we are done.