1 Introduction
Throughout this paper, the (naïve) height
$H(\alpha )$
of an algebraic number
$\alpha $
is the (naïve) height
$H(P)$
of its minimal defining polynomial
$P(X)$
over
${\mathbb Z}$
, that is, the maximum of the absolute values of the coefficients of
$P(X)$
. The present note is a follow-up of [Reference Bajpai and Bugeaud1], where we investigated the effective approximation to complex algebraic numbers by algebraic numbers of degree at most
$4$
. Our previous results were restricted to totally complex numbers (that is, complex algebraic numbers having no real Galois conjugates). Here, we focus our attention on quadratic approximation to complex nonreal algebraic numbers having at least one real Galois conjugate.
Let
$\xi $
be a complex nonreal algebraic number of degree
$d \ge 2$
. By a Liouville-type argument (see e.g. [Reference Güting8] or below the statement of Theorem 1.1), there exists an effectively computable, positive number
$c_1(\xi )$
such that
$$ \begin{align} |\xi - \alpha|> c_1 (\xi) H(\alpha)^{- \frac{d}{2}}, \quad \text{for every quadratic complex number } \alpha \not= \xi. \end{align} $$
When
$d = 2, 3$
, this is best possible; see e.g. [Reference Bugeaud and Evertse5, Proposition 10.2]. For
$d \ge 4$
, it follows from [Reference Bugeaud and Evertse5, Corollary 2.4] (see also [Reference Bajpai and Bugeaud1, Lemma 4.7]) that (1.1) is best possible (up to the value of
$c_1(\xi )$
) if and only if
${\mathbb {Q}}(\xi )$
is a quartic CM-field (a number field K is a CM-field if it is a quadratic extension
$K/F$
such that the image of every complex embedding of F is contained in
${\mathbb R}$
, but there is no complex embedding of K whose image is contained in
${\mathbb R}$
). In the case where
${\mathbb {Q}}(\xi )$
is not a quartic CM-field, (1.1) can be considerably improved by means of the Schmidt Subspace Theorem. Namely, for every
${\varepsilon }> 0$
, [Reference Bugeaud and Evertse5, Corollary 2.4] asserts that there exists a positive number
$c_2 (\xi , {\varepsilon })$
such that
$$ \begin{align} |\xi - \alpha|> c_2 (\xi, {\varepsilon}) H(\alpha)^{- \frac{3}{2} - {\varepsilon}}, \quad \text{for every quadratic complex number } \alpha \not= \xi. \end{align} $$
The current techniques do not yield an explicit value for
$c_2 (\xi , {\varepsilon })$
. Hence, there is a large gap between the effective statement (1.1) and the ineffective statement (1.2). In the present note, we apply Baker’s theory of linear forms in logarithms to get an effective improvement of (1.1). We establish the
Theorem 1.1 Let
$\xi $
be a complex nonreal algebraic number of degree
$d \ge 4$
. If
$d=4$
, assume furthermore that
${\mathbb {Q}}(\xi )$
is not a CM-field. Then, there exist effectively computable positive real numbers
$\kappa (\xi )$
and
$c(\xi )$
such that
$$ \begin{align*}|\xi - \alpha|> c(\xi) H(\alpha)^{ - \frac{d}{2} + \kappa(\xi) }, \quad \mathrm{for\ every\ quadratic\ complex\ number}\ \alpha. \end{align*} $$
Under the additional assumption that
$\xi $
is totally complex, Theorem 1.1 has been proved in [Reference Bajpai and Bugeaud1]. Both proofs use the same method, the main difference being a simple observation that we overlooked in [Reference Bajpai and Bugeaud1]. Let us explain it below.
The question of effective approximation to algebraic numbers by algebraic numbers of bounded degree is deeply connected to the effective resolution of norm-form equations. Indeed, let
$\xi $
be an algebraic number of degree
$d \ge 3$
. Set
$K = {\mathbb {Q}} (\xi )$
. Put
$\delta = 1$
if
$\xi $
is real and
$\delta = 2$
otherwise. Let n be a positive integer with
$n \le d-2$
. Let
$\alpha $
be an algebraic number of degree n. Then, its minimal defining polynomial
$P_{\alpha } (X) := x_0 + x_1 X + \ldots + x_n X^n$
satisfies
$$ \begin{align*}|P_{\alpha} (\xi)| = |x_0 + x_1 \xi + \ldots + x_n \xi^n| \ll_{\xi, n} |\xi - \alpha| \, H(P_{\alpha}) \end{align*} $$
and
$$ \begin{align} 1 \le |\mathrm{Norm}_{K / {\mathbb {Q}}} (P_{\alpha} (\xi))| \ll_{\xi, n} |P_{\alpha} (\xi)|^{\delta} \cdot H(\alpha)^{d- \delta} \ll_{\xi, n} |\xi - \alpha|^\delta \, H(\alpha)^d, \end{align} $$
since
$|P_{\alpha } (\xi )| = |P_{\alpha } (\overline {\xi })|$
if
$\xi $
is nonreal with complex conjugate
$\overline {\xi }$
. Here and below, the constants implicit in
$\ll _{x, y, \ldots }$
and in
$\gg _{x, y, \ldots }$
are positive, effectively computable and depend at most on
$x, y, \ldots $
This gives immediately that
$$ \begin{align*}|\xi - \alpha| \gg_{\xi, n} H(\alpha)^{-d}, \quad \text{ if } \xi \text{ is real,} \end{align*} $$
and
$$ \begin{align*}|\xi - \alpha| \gg_{\xi, n} H(\alpha)^{-d/2}, \quad \text{if } \xi \text{ is complex nonreal}. \end{align*} $$
By (1.3), these lower bounds for
$|\xi - \alpha |$
obtained by a Liouville-type argument can be improved (at the level of the exponent of
$H(\alpha )$
) as soon as we can establish a lower bound for
$|\mathrm {Norm}_{K / {\mathbb {Q}}} (P_{\alpha } (\xi ))|$
of the form
$H(P_{\alpha })^c$
, valid for every integer polynomial
$P_\alpha (X)$
of height at least
$H_0$
, for some positive, effectively computable real numbers c and
$H_0$
, depending only on
$\xi $
and n. For
$n=1$
(and
$\xi $
real, otherwise there is nothing to do), this has been done by Feldman [Reference Fel’dman7] (see also [Reference Baker3]), who applied the theory of linear forms in logarithms, first developed by Alan Baker [Reference Baker2].
Recently, under suitable assumptions on
$\xi $
, we showed in [Reference Bajpai and Bugeaud1] that there exist positive, effectively computable, real numbers
$\delta (\xi )$
and
$H_0 (\xi )$
such that
$$ \begin{align} |\mathrm{Norm}_{K / {\mathbb {Q}}} (P(\xi))| \ge H(P)^{\delta (\xi)}, \end{align} $$
holds for every integer polynomial
$P(X)$
of height at least
$H_0 (\xi )$
and degree at most
$2$
(resp.,
$3$
,
$4$
). This implies that no roots of
$P(X)$
is very close to
$\xi $
and allowed us to obtain various effective results for approximation to
$\xi $
by algebraic numbers of degree at most
$2$
(resp.,
$3$
,
$4$
) of the form
$$ \begin{align*}|\xi - \alpha| \gg_{\xi} H(\alpha)^{-d/2 + \kappa(\xi)}, \end{align*} $$
for some positive, effectively computable, number
$\kappa (\xi )$
.
It is important to observe that if this method can be applied to
$\xi $
, then it applies as well to every Galois conjugate of
$\xi $
. This remark explains why we required in [Reference Bajpai and Bugeaud1] the algebraic number
$\xi $
to be totally complex: no effective result better than Liouville’s inequality is known for approximation to real algebraic numbers by quadratic numbers, and this question is likely to be very difficult. However, to get an effective improvement over Liouville’s inequality, we do not need to prove (1.4) for all integer polynomials
$P_{\alpha } (X)$
. Indeed, if there is a Galois conjugate
$\xi _0$
of
$\xi $
such that
$|P_{\alpha }(\xi _0)| < |P_{\alpha }(\xi )|$
, then the trivial upper bound
$|P_{\alpha }(\xi _0)| \ll _{\xi , n} H(P_{\alpha })$
used in (1.3) can be replaced by
$|P_{\alpha }(\xi _0)| < |P_{\alpha }(\xi )|$
, and we eventually obtain
$$ \begin{align*}1 \ll_{\xi, n} |\xi - \alpha|^3 \, H(\alpha)^d. \end{align*} $$
This provides an effective improvement on Liouville’s inequality. A version of this observation has already been used at the end of the proof of [Reference Bajpai and Bugeaud1, Theorem 2.3].
The next section gathers two estimates from the theory of linear forms in logarithms. Theorem 1.1 is established in Section 3.
Throughout the paper, we let h denote the logarithmic Weil height and we set
$h_* ( \cdot ) = \max \{h (\cdot ), 1\}$
and
$\log _* (\cdot ) = \max \{ \log (\cdot ), 1 \}$
.
2 Auxiliary results
We recall first a classical estimate for linear forms in complex logarithms of algebraic numbers.
Theorem 2.1 Let
$n \ge 1$
be an integer. Let
$\alpha _1, \ldots , \alpha _n, \alpha _{n+1}$
be nonzero algebraic numbers. Let D denote the degree of the algebraic number field generated by
$\alpha _1, \ldots , \alpha _{n+1}$
over
${\mathbb {Q}}$
. Let
$b_1, \ldots , b_n$
be nonzero integers and set
$$ \begin{align*}B = \max\{|b_1|, \ldots, |b_n|\}. \end{align*} $$
If
$\alpha _1^{b_1} \ldots \alpha _n^{b_n} \alpha _{n+1} \not = 1$
, then we have
$$ \begin{align*}\log |\alpha_1^{b_1} \ldots \alpha_n^{b_n} \alpha_{n+1} - 1|> - c(n, D) \, h_* (\alpha_1) \cdots h_*(\alpha_{n+1}) \, \log_* \Bigl( \frac{B}{h_* (\alpha_{n+1})} \Bigr). \end{align*} $$
Proof See [Reference Bugeaud4] or [Reference Waldschmidt10].
We also need the following auxiliary result on multiplicative dependence relations between algebraic numbers. A version of it was originally proved by Loxton and van der Poorten [Reference Loxton and van der Poorten9].
Theorem 2.2 Let
$m\ge 2$
be an integer and
$\alpha _1,\ldots ,\alpha _m$
be multiplicatively dependent nonzero algebraic numbers. Let
$\log \alpha _1, \ldots , \log \alpha _m$
be any determination of their logarithms. Let D be the degree of the number field generated by
$\alpha _1,\ldots ,\alpha _m$
over
${\mathbb {Q}}$
. For
$1\le j\le m$
, let
$A_j$
be a real number satisfying
$$ \begin{align*}\log A_j \ge \max\Bigl\{h(\alpha_j),{|\log \alpha_j| \over D}, 1\Bigr\}. \end{align*} $$
Then there exist rational integers
$n_1,\ldots ,n_m$
, not all of which are zero, such that
$$ \begin{align*}n_1 \log \alpha_1+ \cdots + n_m \log \alpha_m=0, \quad \alpha_1^{n_1} \cdots \alpha_m^{n_m} = 1, \end{align*} $$
and
$$ \begin{align*}|n_k|\le \bigl(11(m-1)D^3\bigr)^{m-1}{(\log A_1) \cdots(\log A_m)\over \log A_k}, \quad \text{for } 1\le k\le m. \end{align*} $$
Proof See [Reference Bugeaud4, Theorem 10.5].
3 Proof of Theorem 1.1
Set
$K = {\mathbb {Q}}(\xi )$
. Let
$\alpha $
be a quadratic complex (nonreal) algebraic number. Let
$P_\alpha (X) = x_0 + x_1 X + x_2 X^2$
denote its minimal defining polynomial over
${\mathbb Z}$
. Set
$$ \begin{align*}\mathrm{Norm}_{K/{\mathbb {Q}}} (x_0 + x_1\xi + x_2\xi^2 ) = m, \end{align*} $$
and
$$ \begin{align*}\mathbf{x} = x_0 + x_1\xi + x_2\xi^2 = P_\alpha (\xi). \end{align*} $$
Let
$\sigma _1, \ldots , \sigma _d$
denote the d embeddings of K into
${\mathbb C}$
numbered in such a way that
$$ \begin{align*}|\sigma_1( \mathbf{x} )| \ge |\sigma_2( \mathbf{x} )| \ge \cdots \ge |\sigma_{d-1}( \mathbf{x} )| \ge |\sigma_d( \mathbf{x} )|. \end{align*} $$
For later use, recall that
$$ \begin{align*}d^{-1} \log |\sigma_1( \mathbf{x} )| \le h(\mathbf{x}) \le \log |\sigma_1( \mathbf{x} )|. \end{align*} $$
and observe that
$$ \begin{align*}\sigma_1(\mathbf x) \cdot \sigma_2(\mathbf x) \cdots \sigma_d(\mathbf x) = m. \end{align*} $$
By [Reference Bajpai and Bugeaud1, Lemma 4.5] (which follows from the proof of [Reference Evertse and Győry6, Proposition 4.3.12]), there exists a unit u in K such that, putting
$\mu = \mathbf {x} / u$
, we have
$$ \begin{align*}C_1^{-1} |m|^{1/d} \le |\sigma (\mu) | \le C_1 |m|^{1/d}, \end{align*} $$
for every complex embedding
$\sigma $
of K, with a constant
$C_1 = C_1 (K) \ge 1$
.
The next lemma is a reformulation of [Reference Bajpai and Bugeaud1, Theorem 4.6].
Lemma 3.1 Keep the above notation. Assume there are effectively computable positive real numbers
$\kappa _1$
and
$h_0$
, depending only on
$\xi $
, such that
$$ \begin{align*}h(\mathbf{x}) \le \kappa_1 h_* (\mu), \end{align*} $$
holds when
$h(\mathbf {x})> h_0$
. Then, there exist effectively computable, positive
$c, \kappa $
, depending only on
$\xi $
, such that
$$ \begin{align*}|\xi - \alpha|> c \, H(\alpha)^{- \frac{d}{2} + \kappa }. \end{align*} $$
Our aim is to prove the existence of
$\kappa _1$
and
$h_0$
as in Lemma 3.1. Thus, without any loss of generality, we assume that
$$ \begin{align} h(\mathbf{x}) \ge 2 h_*(\mu), \end{align} $$
and also
$$ \begin{align*}|\sigma_d (\mathbf{x})| < 1. \end{align*} $$
Indeed, otherwise we get at once
$|\sigma _1 (\mathbf {x})| \le |m|$
and we obtain
$$ \begin{align*}d h_*(\mu) \gg \log_* |m| \ge \log |\sigma_1 (\mathbf{x})| \ge h(\mathbf{x}). \end{align*} $$
Here and below, the constants implicit in
$\gg $
and in
$\ll $
are positive, effectively computable and depend at most on
$\xi $
.
Since the totally complex case has been fully addressed in [Reference Bajpai and Bugeaud1], we assume that K is not totally complex. In particular, its unit group has rank
$2$
.
Case 1. Assume first that the embeddings can be numbered in such a way that
$\sigma _d (\mathbf {x}) = \mathbf {x} = P_\alpha (\xi )$
. It is a complex nonreal number. Without any loss of generality, we assume that
$\sigma _{d-1} (\mathbf {x})$
is its complex conjugate. The proof then goes as in [Reference Bajpai and Bugeaud1], except for Subsubcase
$(\star )$
below.
Subcase
$d = 4$
. It follows from [Reference Bajpai and Bugeaud1, Lemma 5.1] that there are nonzero
$a_0, a_1, a_2$
in the Galois closure of K such that
$$ \begin{align*}\sigma_1 (\mathbf{x}) + a_0 \sigma_2 (\mathbf{x}) + a_1 \sigma_3 (\mathbf{x}) + a_2 \sigma_4 (\mathbf{x}) = 0. \end{align*} $$
Upon dividing by
$\sigma _1 (\mathbf {x})$
and recalling that
$\mathbf {x} = \mu u$
, we get
$$ \begin{align*}|1 - \gamma v| \le \Bigl| \frac{a_1 \sigma_3 (\mathbf{x})}{\sigma_1 (\mathbf{x})} \Bigr| + \Bigl| \frac{a_2 \sigma_4 (\mathbf{x})}{\sigma_1 (\mathbf{x})} \Bigr|, \end{align*} $$
with
$$ \begin{align*}\gamma = - \frac{a_0 \sigma_{2} (\mu)}{\sigma_1 (\mu)}, \quad v = \frac{\sigma_{2} (u)}{\sigma_1 (u)}. \end{align*} $$
We distinguish two cases.
Subsubcase
$(\star )$
. We assume that
$1 - \gamma v = 0$
, that is,
$$ \begin{align*}\sigma_1 (\mathbf{x}) + a_0 \sigma_2 (\mathbf{x}) = 0. \end{align*} $$
Let
$\eta _1, \eta _2$
be a fundamental system of units of
${\mathbb {Q}}(\xi )$
. Then, there are a root of unity
$\zeta $
in
${\mathbb {Q}}(\xi )$
and integers
$b_1, b_2$
such that
$$ \begin{align*}\mathbf{x} = \mu u = \mu \zeta \eta_1^{b_1} \eta_2^{b_2}, \end{align*} $$
and we have
$$ \begin{align} \frac{\sigma_1 (\mu \zeta)}{- a_0 \sigma_2 (\mu \zeta)} \, \Bigl( \frac{\sigma_1 (\eta_1)}{\sigma_2 (\eta_1)} \Bigr)^{b_1} \, \Bigl( \frac{\sigma_1 (\eta_2)}{\sigma_2 (\eta_2)} \Bigr)^{b_2} = 1. \end{align} $$
By Theorem 2.2, this implies that there exist rational integers
$n_0, n_1, n_2$
, not all of which are zero, such that
$$ \begin{align*}\Bigl( \frac{\sigma_1 (\mu \zeta)}{- a_0 \sigma_2 (\mu \zeta)} \Bigr)^{n_0} \, \Bigl( \frac{\sigma_1 (\eta_1)}{\sigma_2 (\eta_1)} \Bigr)^{n_1} \, \Bigl( \frac{\sigma_1 (\eta_2)}{\sigma_2 (\eta_2)} \Bigr)^{n_2} = 1, \end{align*} $$
with
$$ \begin{align*}|n_0| \ll 1, \quad |n_1|, |n_2| \ll h_*(\mu). \end{align*} $$
If
$\frac {\sigma _1 (\eta _1)}{\sigma _2 (\eta _1)}$
and
$\frac {\sigma _1 (\eta _2)}{\sigma _2 (\eta _2)}$
are multiplicatively independent, then we derive from (3.2) that
$b_1 n_0 = n_1$
,
$b_2 n_0 = n_2$
, and
$$ \begin{align*}\max\{ |b_1|, |b_2| \} \ll h_*(\mu), \end{align*} $$
giving
$h(u) \ll h_* (\mu )$
and
$h(\mathbf {x}) \ll h_* (\mu )$
, as wanted.
Otherwise, there exist nonzero integers
$c_1$
,
$c_2$
, with
$|c_1|, |c_2| \ll 1$
such that
$$ \begin{align*}\Bigl( \frac{\sigma_1 (\eta_1)}{\sigma_2 (\eta_1)} \Bigr)^{c_1} \, \Bigl( \frac{\sigma_1 (\eta_2)}{\sigma_2 (\eta_2)} \Bigr)^{c_2} = 1. \end{align*} $$
Combined with (3.2), this gives
$$ \begin{align*}\Bigl( \frac{\sigma_1 (\mu \zeta)}{- a_0 \sigma_2 (\mu \zeta)} \Bigr)^{c_1} \, \Bigl( \frac{\sigma_1 (\eta_2)}{\sigma_2 (\eta_2)} \Bigr)^{-c_2 b_1 + c_1 b_2} = 1. \end{align*} $$
We deduce that
$$ \begin{align*}\max\{|b_1|, |b_2|\} \ll h \Bigl( \frac{\sigma_1 (\mu \zeta)}{- a_0 \sigma_2 (\mu \zeta)} \Bigr) \ll h_* (\mu), \end{align*} $$
and we conclude as above that
$h(\mathbf {x}) \ll h_* (\mu )$
.
Subsubcase
$(\star \star )$
. We assume that
$|1 - \gamma v|> 0$
. We treat this case below, simultaneously with the case
$d \ge 6$
.
Case
$d \ge 6$
. It follows from [Reference Bajpai and Bugeaud1, Lemma 5.1] that there are nonzero
$a_0, a_1, a_2, b_0, b_1, b_2$
in the Galois closure of K such that
$$ \begin{align*}\sigma_1 (\xi^j) + \sum_{i=0}^2 a_i \sigma_{d-i} (\xi^j) = 0 \quad \text{and} \quad \sigma_2 (\xi^j) + \sum_{i=0}^2 b_i \sigma_{d-i} (\xi^j) = 0, \quad \text{for } j = 0, 1, 2. \end{align*} $$
This implies
$$ \begin{align*}b_0 \sigma_1 (\xi^j) - a_0 \sigma_2 (\xi^j) + \sum_{i=1}^2 (a_i b_0 - a_0 b_i) \sigma_{d-i} (\xi^j)= 0, \quad \text{for } j = 0, 1, 2.\end{align*} $$
Consequently, we have
$a_1 b_0 \not = a_0 b_1$
. Thus,
$a_1 \sigma _{d-1} (\mathbf {x}) + a_0 \sigma _d(\mathbf {x})$
and
$b_1 \sigma _{d-1} (\mathbf {x}) + b_0 \sigma _d(\mathbf {x})$
cannot be both
$0$
. By permuting
$\sigma _1$
and
$\sigma _2$
if necessary, we assume that
$a_1 \sigma _{d-1} (\mathbf {x}) + a_0 \sigma _d(\mathbf {x})$
is nonzero. By linearity, we have
$$ \begin{align*}\sigma_1 (\mathbf{x}) + \sum_{i=0}^2 a_i \sigma_{d-i} (\mathbf{x}) = 0, \end{align*} $$
thus
$\sigma _1 (\mathbf {x}) + a_2 \sigma _{d-2} (\mathbf {x})$
is nonzero. Consequently, by dividing by
$\sigma _1 (\mathbf {x})$
and recalling that
$\mathbf {x} = \mu u$
, we get
$$ \begin{align*}0 < |1 - \gamma v| \le \sum_{i=0}^1 \Bigl| \frac{a_i \sigma_{d-i} (\mathbf{x})}{\sigma_1 (\mathbf{x})} \Bigr|, \quad \text{with} \quad \gamma = - \frac{a_2 \sigma_{d-2} (\mu)}{\sigma_1 (\mu)}, v = \frac{\sigma_{d-2} (u)}{\sigma_1 (u)}. \end{align*} $$
This corresponds exactly to the inequalities obtained in the case
$d=4$
. So, we treat both cases simultaneously.
Since
$|\sigma _{d-1} (\mathbf {x})| = |\sigma _d (\mathbf {x}) | < 1$
, this gives
$|1 - \gamma v| \ll |\sigma _1 (\mathbf {x}) |^{-1}$
, thus
$$ \begin{align*}\log |1 - \gamma v| \le - h(\mathbf{x}) / 2, \quad \text{if } h(\mathbf{x}) \text{ is large enough}. \end{align*} $$
As, by Dirichlet’s unit theorem, the unit v can be expressed as a product of a root of unity in K times integral powers of elements of a fundamental system of units in K, it follows from Theorem 2.1 that
$$ \begin{align} \frac{h(\mathbf{x})}{2} \le - \log |1 - \gamma v| \ll h_* (\gamma) \log_* \frac{h_*(v)}{h_* (\gamma)}. \end{align} $$
Furthermore, by (3.1), we get
$$ \begin{align*}h(v) \ll h(u) \ll (h(\mathbf{x}) + h_* (\mu) ) \ll h(\mathbf{x}), \end{align*} $$
and we derive from (3.3) that
$$ \begin{align*}h(\mathbf{x}) \ll h_* (\gamma) \ll h_* (\mu). \end{align*} $$
It then follows from Lemma 3.1 that there exist effectively computable, positive real numbers
$c, \kappa $
, depending only on
$\xi $
, such that
$$ \begin{align*}|\xi - \alpha|> c \, H(\alpha)^{- \frac{d}{2} + \kappa }, \end{align*} $$
for every algebraic number
$\alpha $
which is a root of
$x_0 + x_1 X + x_2 X^2$
, as soon as we are in Case 1.
Case 2. Assume now that the embeddings cannot be numbered in such a way that
$\sigma _d (\mathbf {x}) = \mathbf {x} = P_\alpha (\xi )$
. Then, there is a Galois conjugate
$\xi _0$
of
$\xi $
such that
$$ \begin{align*}|P_\alpha (\xi_0)| < |P_\alpha (\xi)| = |P_\alpha ({\overline \xi})|. \end{align*} $$
Consequently, we have
$$ \begin{align*}1 \le m = |\mathrm{Norm}_{K / {\mathbb {Q}}} (P_{\alpha} (\xi))| \ll |P_{\alpha} (\xi)|^3 \cdot H(\alpha)^{d - 3} \ll |\xi - \alpha |^3 \cdot H(\alpha)^d, \end{align*} $$
giving that
$|\xi - \alpha | \gg H(\alpha )^{-d/3} \gg H(\alpha )^{-d/2 + 2/3 }$
, since
$d \ge 4$
.
Conclusion. We have established in both cases that
$$ \begin{align*}|\xi - \alpha| \gg H(\alpha)^{- \frac{d}{2} + \min\{\kappa, 2/3\} }, \end{align*} $$
for every complex nonreal quadratic algebraic number
$\alpha $
. This completes the proof of Theorem 1.1.
