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  $d = \gcd(a - 1, b - 1)$ where $a, b \in
  \mathcal{S}$ are multiplicatively independent. We
  assume: $d > \max(a, b)^\eps$ for some $\eps > 0$.
  Our goal is to prove $d < C(\eps)$.
  \[
    \prod_{v \in S = \{\infty, 2, 3\}} \prod_{j =
    1}^{n^ - 1} |\Lambda_j^{(v)}(e / d)|_v
    \le da^{n - 1} b^{n - 1} d^{-n^2}
    \tag{$*$}
  .\]
  $e_1, \ldots, e_{n^2}$ is an enumeration of $a^k
  b^l$, $k, l = 0, \ldots, n - 1$.
  \[
    (*)
    \le \max(a, b)^{2n - 2} \cdot \max(a,
    b)^{-\eps(n^2 - 1)}
  .\]
  Let's take $n > 3\eps^{-1}$, $(*) < \max(a,
  b)^{-n}$.
  \[
    H(\Lambda_1^{(0)}(e / d), \ldots, \Lambda_{n -
    1}^{(0)}(e / d)) \le a^{n - 1} b^{n - 1}
  .\]
  $(*) < H(\cdots)^{-\half}$.
  Subspace theorem applies hence there exists a
  linear relation between $e_1, \ldots, e_{n^2} \in
  \mathcal{S}$ (distinct by multiplicative
  independence of $a, b$).

  Proposition implies
  \[
    |e_i - e_j|_\infty |e_i - e_j|_2 |e_i - e_j| < C
    = C(\eps)
  \]
  for some $i \neq j$. Then $e_i \neq e_j$ so $e_i -
  e_j \neq 0$. However, $d \mid e_i - e_j$.
  \[
    d \le |e_i - e_j|_\infty |e_i - e_j|_2 |e_i -
    e_j|_3 < C
  .\]
\end{proof}

\begin{theorem}[Feldman]
  Let $\alpha \in \ol{\Qbb}$ of degree $d \ge 3$.
  Then there exists effective $C = C(\alpha) > 0$
  and $\eps = \eps(\alpha) > 0$ such that for all
  $\frac{p}{q} \in \Qbb$,
  \[
    \left| \alpha - \frac{p}{q} \right| >
    \frac{C}{q^{d - \eps}}
  .\]
\end{theorem}

\begin{remark*}
  This is enough to solve $P(x, y) = m$, where $P$
  is a degree $d$ homogeneous polynomial without
  repeated factors. Thue equation.
\end{remark*}

\begin{proposition*}
  Let $K$ be a number field. Then there exists $r
  \in \Zbb_{\ge 0}$ and $u_1, \ldots, u_r \in
  \O_K^\times$ and a constant $C = C(K)$ such that
  $\forall \alpha \in \O_K$, there exists
  $\tilde{\alpha} \in \O_K$ and $b_1, \ldots, b_r
  \in \Zbb$ such that
  \begin{align*}
    H(\tilde{\alpha})
    &\le C \cdot |N_{K /
    \Qbb}(\alpha)|^{\frac{1}{[K : \Qbb]}} \\
    |b_1|, \ldots, |b_r|
    &\le C\log H(\alpha) \\
    \alpha
    &= \tilde{\alpha} u_1^{b_1} \cdots u_r^{b_r}
  \end{align*}
\end{proposition*}

Define $\Phi : K^\times \to \Rbb^{M_{K, \infty}} :
(\Phi(\alpha))_v = d_v \cdot \log|\alpha|_v$
(logarithmic embedding). Note that here $K^\times$
is the group under multiplication, while
$\Rbb^{M_{K, \infty}}$ is the additive group.

\begin{align*}
  |N_{K / \Qbb}(\alpha)|
  &= \exp \left( \sum_{v \in M_{K, \infty}}
  (\Phi(\alpha))_v \right) \\
  H(\alpha)^{[K : \Qbb]}
  &= \exp \left( \sum_{v \in M_{K, \infty}}
  \max(0, (\Phi(\alpha))_v) \right)
\end{align*}
For $\alpha \in \O_K$, $\Sigma(\Phi(\alpha))_v \ge
0$. Then:
\[
  \exp(\|\Phi(\alpha)\|_1 / 2)
  \le H(\alpha)^{[K : \Qbb]}
  \le \exp(\|\Phi(\alpha)\|_1)
.\]
For $\alpha \in \O_K^\times$, $N_{K /
\Qbb}(\alpha) = 1$. So
\[
  \Phi(\alpha)
  \in W
  = \{x \in \Rbb^{M_{K, \infty}} : \sum x_v = 0\}
.\]
Kronecker's theorem: $\Phi^{-1}(0) = \Ker\Phi$ are
the roots of unity.

Dirichlet's unit theorem: $\Phi(\O_K^\times)$ is a
lattice in $W$ that is a $\Zbb$-module of rank
$\dim W = r$ which spans $W$.

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/ef90967a273d4727.png}
\end{center}

Let $u_1, \ldots, u_r$ be a fundamental system of
units, that is $\Phi(u_1), \ldots, \Phi(u_r)$ is a
basis for the lattice $\Phi(\O_K^\times)$. Fix
some $\alpha \in \O_K$. Pick some $x \in \Rbb_{\ge
0}^{M_{K, \infty}}$ such that
\[
  \sum x_v = \ub{\log |N_{K / \Qbb}(\alpha)|}_{\ge 0}
.\]
$x \in \Phi(\alpha) + W$.

Then there exist $y_1, \ldots, y_r \in \Rbb$ such
that
\[
  x = \Phi(\alpha) + y_1 \Phi(u_1) + \cdots + y_r
  \Phi(u_r)
.\]
There exists $C = C(K)$ such that $|y_j| \le C
\cdot \|\Phi(\alpha)\|_1$.

Let $b_j \in \Zbb$ with $|y_j - b_j| \le 1$ and
$|b_j| \le |y_j|$. This gives $|b_j| \le C \cdot
\|\Phi(\alpha)\|_1 \le C' \log H(\alpha)$.

Take: $\tilde{\alpha} = \alpha u_1^{b_1} \cdots
u_r^{b_r}$.
\[
  \Phi(\tilde{\alpha})
  = \Phi(\alpha) + b_1 \Phi(u_1) + \cdots + b_r
  \Phi(u_r)
  = x + \ub{(b_1 - y_1) \Phi(u_1) + \cdots}_{(*)}
.\]
$(*)$ is in a fixed, compact region of $W$.

$\|\Phi(\tilde{\alpha})\|_1 \le C + \|x\|_1$.
\[
  H(\tilde{\alpha})^{[K : \Qbb]}
  \le \exp(\|\Phi(\tilde{\alpha})\|_1)
  \le \exp(C) \cdot N_{K / \Qbb}(\alpha)
.\]
