# On a conjecture of Levesque and Waldschmidt

Tobias Hilgart<EMAIL_ADDRESS>
Volker Ziegler<EMAIL_ADDRESS>
University of Salzburg, Austria

##  1 Motivation and Statement

Ever since the work of Baker [Bak68], we know how to effectively solve any
Thue equation, i.e. a diophantine equation $F(X,Y)=m$ over the integers, where
$F$ is an irreducible homogenous polynomial of degree at least $3$. And if we
can solve one, we can solve a finite number of them.

Thomas [Tho90] was amongst the first to study infinitely many (non-binary)
Thue equations by looking at parametrised equations $F_{n}(X,Y)=\pm 1$, where
$F_{n}\in\mathbb{Z}[n][X,Y]$ gives a Thue equation for each $n\in\mathbb{N}$.
He proved that

$F_{n}(X,Y)=X^{3}-(n-1)X^{2}Y-(n+2)XY^{2}-Y^{3}=\pm 1$ (1)

has only a few trivial solutions for large parameters $n$ while giving a
complete list of all solutions for small parameters $n$. Said parameter can
furthermore be taken to be an integer due to the symmetry
$F_{-n-1}(X,Y)=F_{n}(-Y,-X)$.

Levesque and Waldschmidt [LW15] took this equation one step further and
twisted it by an exponential parameter $t$ in the following way: We can write
Thue equations as norm-form equations, e.g. we first factorise the polynomial
$F_{n}(X,1)$ of Equation (1) into its complex roots
$\uplambda_{0},\uplambda_{1},\uplambda_{2}$ (and forego to denote their
dependency on the parameter $n$). Then Equation (1) can be written as

$N_{K/\mathbb{Q}}\left(X-\uplambda_{0}Y\right)=\pm 1,$

where $K=\mathbb{Q}\left(\uplambda_{0}\right)$ and $N_{K/\mathbb{Q}}$ denotes
the norm relative to $K/\mathbb{Q}$. Levesque and Waldschmidt twisted this
equation by an integer $t$ and looked, among other things, at the equation

$N_{K/\mathbb{Q}}\left(X-\uplambda_{0}^{t}Y\right)=\pm 1,$

for which they managed to prove that there exists an effectively computable
number $\upkappa$ such that any solution $(x,y,n,t)$ to the above equation
with $\max\left(\left|x\right|,\left|y\right|\right)\geq 2$ satisfies
$\max\left(\left|x\right|,\left|y\right|,\left|n\right|,\left|t\right|\right)<\upkappa$,
i.e. there are but finitely many solutions and we, at least in theory, could
find them all.

One conjecture Levesque and Waldschmidt posed in their paper is whether a
twist by multiple exponential parameters, i.e. the norm-form equation

$N_{K/\mathbb{Q}}\left(X-\uplambda_{0}^{s}\uplambda_{1}^{t}Y\right)=\pm 1$

can be tackled analogously, working with the existing tools used to study Thue
equations—a conjecture we want to give a partial positive answer for in this
paper. Our theorem is as follows:

######  Theorem 1.

Let
$\uplambda_{0}=\uplambda_{0}(n),\uplambda_{1}=\uplambda_{1}(n),\uplambda_{2}=\uplambda_{2}(n)$
be the roots of the integer polynomial

$f(X)=f(X;\,n)=X^{3}-(n-1)X^{2}-(n+2)X-1$

and

$F(X,Y)=F(X,Y;\,n,s,t)=\left(X-\uplambda_{0}^{s}\uplambda_{1}^{t}Y\right)\left(X-\uplambda_{1}^{s}\uplambda_{2}^{t}Y\right)\left(X-\uplambda_{2}^{s}\uplambda_{0}^{t}Y\right).$

Let $(x,y;n,s,t)\in\mathbb{Z}^{5}$ be a solution to the Thue-Equation

$\left|F(X,Y)\right|=1,$ (2)

with $\left|y\right|\geq 2,n\geq 3$, $st\neq 0$ and let us assume that for a
given $\upvarepsilon>0$ we have

$\min\left(\left|2s-t\right|,\left|2t-s\right|,\left|s+t\right|\right)>\upvarepsilon\cdot\max\left(\left|s\right|,\left|t\right|\right)>2.$
(3)

Then there exists an effectively computable constant $\upkappa>0$, depending
only on $\upvarepsilon$, such that

$\max\left(\left|x\right|,\left|y\right|,n,\left|s\right|,\left|t\right|\right)<\upkappa.$

##  2 Outline of the Proof

We strive to improve the readability of our paper by separating the conceptual
proof from the necessary detail work. Thus we first give an outline of the
proof of Theorem 1, which follows the general strategy to solve Thue equations
via Baker bounds on linear forms in logarithms. At the intermediate steps that
require careful technical consideration, we refer to the corresponding lemmas
in the subsequent section, where we also collect other auxiliary results that
we use in the process.

Let us start by briefly mentioning the degree of restriction we placed on the
theoretical solution $(x,y;\,n,s,t)$ of Equation (2). There are infinitely
many solutions where $\left|y\right|\leq 2$, e.g. $(1,0;\,n,s,t)$, hence the
condition $\left|y\right|\geq 2$ is necessary. Due to the symmetry
$F_{-n-1}(X,Y)=F_{n}(-Y,-X)$, we only have to consider non-negative $n$, while
we drop the $n$ smaller than $3$ for $\log(n)$ and $\log(\log(n))$ to be
positive (and indeed defined at all over the real numbers). But we could
alternatively take $n\geq 3$ to be the absolute value of the integer parameter
and conduct the same proof, where we just have to be mindful of taking
$\uplambda_{0}$ in absolute value as well. Finally, Condition (3) is the
degree to which we could solve Levesque and Waldschmidt’s conjecture and can
be read thus: No quotient of two conjugates of
$\uplambda_{0}^{s}\uplambda_{1}^{t}$ is allowed to be too close to $1$, for
the whole argument falls apart otherwise.

We want to improve the readability of the terms and equations and thus
introduce several short hands. First, we want to hide constants in our
expressions that do not affect their asymptotic without immediately resorting
to $O$-notations. So when we write $z\ll_{\,\updelta}z^{\prime}$, we mean that
there exists an effectively computable constant $c(\updelta)$ which does not
depend on $z$ or $z^{\prime}$, but which may depend on $\updelta$ such that
$z\leq c(\updelta)\,z^{\prime}$. If both $z\ll z^{\prime}$ and $z^{\prime}\ll
z$, we write $z=\uptheta\left(z^{\prime}\right)$.

Let furthermore $\uptau:=\max\left(\left|s\right|,\left|t\right|\right)$ and

$\displaystyle\upalpha_{0}:=\uplambda_{0}^{s}\uplambda_{1}^{t},$
$\displaystyle\upalpha_{1}:=\uplambda_{1}^{s}\uplambda_{2}^{t},$
$\displaystyle\upalpha_{2}:=\uplambda_{2}^{s}\uplambda_{0}^{t},$
$\displaystyle\upbeta_{0}:=x-\upalpha_{0}y,\;\;$
$\displaystyle\upbeta_{1}:=x-\upalpha_{1}y,\;\;$
$\displaystyle\upbeta_{2}:=x-\upalpha_{2}y.$ (4)

In this notation, $\upalpha_{i}$, $\upbeta_{i}$ resp., is the image of
$\upalpha_{0}$, $\upbeta_{0}$ resp., under the automorphism that maps
$\uplambda_{0}$ to $\uplambda_{i}$. The numeration of the roots
$\uplambda_{0},\uplambda_{1},\uplambda_{2}$ is chosen such that
$\uplambda_{0}$ is close to $n$, $\uplambda_{1}$ is close to $0$ and
$\uplambda_{2}$ is close to $-1$. The exact inequalities used are given here
in Lemma 1, from Lemma 10 of [LPV99]

One particular $\upbeta_{i}$ stands out as being very small, or $\frac{x}{y}$
is a good rational approximation to one particular $\upalpha_{i}$
respectively. We denote this as $\upbeta_{j}$, i.e.
$\left|\upbeta_{j}\right|=\min\left(\left|\upbeta_{0}\right|,\left|\upbeta_{1}\right|,\left|\upbeta_{2}\right|\right)$
and the remaining two, in no particular order, as $\upbeta_{k}$ and
$\upbeta_{l}$.

By the triangle inequality, we have that

$2\left|\upbeta_{k}\right|\geq\left|\upbeta_{k}-\upbeta_{j}\right|=\left|y\left(\upalpha_{j}-\upalpha_{k}\right)\right|,$
(5)

same for $\upbeta_{l}$. We have to be mindful of the difference between two of
the algebraic numbers, e.g. $\upalpha_{j},\upalpha_{k}$, not becoming too
small, which we do in Lemma 2. Having done that, we can use the above
inequality in combination with
$\left|\upbeta_{j}\upbeta_{k}\upbeta_{l}\right|=\left|F(x,y)\right|=1$ to
derive for an effectively computable constant $c_{1}>0$ the upper bound

$\left|\upbeta_{j}\right|\leq\frac{1}{\left|\upbeta_{k}\upbeta_{l}\right|}\ll\frac{1}{\left|y\right|^{2}\left|\upalpha_{j}-\upalpha_{k}\right|\left|\upalpha_{j}-\upalpha_{l}\right|}\overset{\text{Lemma~{}\ref{lem:
alphadiff}}}{\ll}\frac{1}{\left|y\right|^{2}\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{k}\right|\right)\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{l}\right|\right)}.$

Using Lemma 3, the simple property that the product of pairwise maxima of
positive real numbers multiplying to $1$ is bounded from below by an
appropriate root of the global maximum, we get

$\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{k}\right|\right)\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{l}\right|\right)\geq\sqrt{\max\left(\left|\upalpha_{0}\right|,\left|\upalpha_{1}\right|,\left|\upalpha_{2}\right|\right)}.$

We ascertain that at least the largest of the algebraic number $\upalpha_{i}$
behaves as expected, and indeed, by Lemma 4, the above square root is at least
$n^{\frac{c_{1}}{2}\uptau}$. We thus get that our bound for
$\left|\upbeta_{j}\right|$ is indeed small, namely

$\left|\upbeta_{j}\right|\ll\frac{1}{\left|y\right|^{2}\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{k}\right|\right)\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{l}\right|\right)}\ll\frac{1}{\left|y\right|^{2}n^{\frac{c_{1}}{2}\,\uptau}}.$
(6)

We then use this information to rewrite Siegel’s Identity into a unit equation
where we now know one addend to be very small. Stating the identity

$\upbeta_{j}\left(\upalpha_{k}-\upalpha_{l}\right)+\upalpha_{l}\left(\upalpha_{j}-\upalpha_{k}\right)+\upbeta_{k}\left(\upalpha_{l}-\upalpha_{j}\right)=0$
(7)

and dividing by the last addend with flipped sign gives

$\frac{\upbeta_{j}}{\upbeta_{k}}\frac{\upalpha_{k}-\upalpha_{l}}{\upalpha_{j}-\upalpha_{l}}+\frac{\upbeta_{l}}{\upbeta_{k}}\frac{\upalpha_{j}-\upalpha_{k}}{\upalpha_{j}-\upalpha_{l}}=1.$
(8)

The first addend on the left-hand side is very small: We use the inequality
from Equation (5) again, in combination with the bound from Inequality (6)
derived above to get

$\displaystyle\left|\frac{\upbeta_{j}}{\upbeta_{k}}\frac{\upalpha_{k}-\upalpha_{l}}{\upalpha_{j}-\upalpha_{l}}\right|\ll\frac{\left|\upalpha_{k}-\upalpha_{l}\right|}{\left|y\right|^{3}n^{\frac{c_{1}}{2}\uptau}\left|\upalpha_{j}-\upalpha_{k}\right|\left|\upalpha_{j}-\upalpha_{l}\right|}\ll\frac{\max\left(\left|\upalpha_{k}\right|,\left|\upalpha_{l}\right|\right)}{\left|y\right|^{3}n^{\frac{c_{1}}{2}\uptau}\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{k}\right|\right)\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{l}\right|\right)}.$

We differentiate the cases for whether $\upalpha_{j}$ has the maximal absolute
value or not in Lemma 5 and resolve the quotients of the various maxima with
an upper bound of $2\left|y\right|$. We thus have

$\left|\frac{\upbeta_{j}}{\upbeta_{k}}\frac{\upalpha_{k}-\upalpha_{l}}{\upalpha_{j}-\upalpha_{l}}\right|\ll\frac{1}{\left|y\right|^{2}n^{\frac{c_{1}}{2}\uptau}},$

and unless both $n$ and $\uptau$ are small, we can assume this to be smaller
than $\frac{1}{2}$.

Note that neither of the two addends in Equation (8) is zero: None of the
$\upbeta_{i}$ is since each $\upalpha_{i}$ is irrational, and any two
$\upalpha_{p},\upalpha_{q}$ are distinct. The latter fact of which can be seen
by looking at Equation (11) from the proof of Lemma 2, together with the fact
that $\log\uplambda_{0},\log\left|\uplambda_{2}\right|$ are
$\mathbb{Q}$-linearly independent, since $\uplambda_{0},\uplambda_{2}$ give a
system of fundamental units for $\mathbb{Z}[\uplambda_{0}]$ by Proposition 2,
and $st\neq 0$.

This gives for Equation (8) that

$\frac{\upbeta_{l}}{\upbeta_{k}}\frac{\upalpha_{j}-\upalpha_{k}}{\upalpha_{j}-\upalpha_{l}}=1-\frac{\upbeta_{j}}{\upbeta_{k}}\frac{\upalpha_{k}-\upalpha_{l}}{\upalpha_{j}-\upalpha_{l}}\neq
1,$

and taking the logarithm, using the relation
$\left|\log\left(1-v\right)\right|\leq 2\left|v\right|$ for
$\left|v\right|\leq\frac{1}{2}$, yields

$0<\left|\underbrace{\log\left|\frac{\upbeta_{l}}{\upbeta_{k}}\right|+\log\left|\frac{\upalpha_{j}-\upalpha_{k}}{\upalpha_{j}-\upalpha_{l}}\right|}_{=:\Uplambda}\right|\leq
2\left|\frac{\upbeta_{j}}{\upbeta_{k}}\frac{\upalpha_{k}-\upalpha_{l}}{\upalpha_{j}-\upalpha_{l}}\right|\ll\frac{1}{\left|y\right|^{2}n^{\frac{c_{1}}{2}\uptau}}.$
(9)

With some extra effort exerted in Lemma 7, this allows us to directly derive
an upper bound for $\log\left|y\right|$, without troubling, say, Bugeaud and
Győry’s result [BG96], which holds much more generally. We get that

$\log\left|y\right|\ll\,\uptau\,\left(\log
n\right)^{3}\left(\log\uptau+\log\log n\right).$

Returning to the linear form in logarithms $\Uplambda$, we managed to derive
the bound for $\log\left|y\right|$ by shifting the dependence on the exponents
$s,t$ from the logarithm of $\left|\frac{\upbeta_{l}}{\upbeta_{k}}\right|$
into its coefficient. If we manage to do the same for the second logarithm
$\log\left|\frac{\upalpha_{j}-\upalpha_{k}}{\upalpha_{j}-\upalpha_{l}}\right|$
as well, then the tools we used before will give us, also using the bound for
$\log\left|y\right|$, an absolute bound for the parameters $n$ and $\uptau$,
thus proving the theorem.

To that end, we extract the larger of $\upalpha_{j}$ and $\upalpha_{k}$, which
we call $\upalpha_{u}$, from the denominator, and $\upalpha_{v}$, defined the
same, from the numerator. We can then separate the factor
$\log\left|\frac{\upalpha_{u}}{\upalpha_{v}}\right|$, and the remaining
logarithm is at most $4n^{-c_{1}\uptau}$ by Lemma 2. We shift this into the
upper bound and get a new small linear form in logarithms
$\Uplambda^{\prime}$, where

$\left|\Uplambda^{\prime}\right|=\left|\log\left|\frac{\upbeta_{l}}{\upbeta_{k}}\right|+\log\left|\frac{\upalpha_{u}}{\upalpha_{v}}\right|\right|\ll\frac{1}{n^{\frac{c_{1}}{2}\uptau}}.$
(10)

Note that we can no longer argue that $\Uplambda^{\prime}$ is non-zero by
construction. We must consider that it is, which we do at length in Lemma 8.

We end up either way with a (possibly different) linear form
$\Uplambda^{\prime\prime}$ in logarithms of
$\upbeta_{k},\upbeta_{l},\upalpha_{j},\upalpha_{k},\upalpha_{l}$ and $2$, for
which inequalities of the form

$0<\left|\Uplambda^{\prime\prime}\right|<\frac{1}{n^{\frac{c_{1}}{2}\uptau}}$

hold again. We can now again explicitly shift the dependency on $s,t$ into the
coefficients and write
$\Uplambda^{\prime\prime}=A\log\left|\uplambda_{0}\right|+B\log\left|\uplambda_{2}\right|-C\log
2$, where $A,B$ are linear-combinations of $s,t$ and $C$ is $0$ or $1$. We can
use the bounds for the coefficients, heights and $\log\left|y\right|$ we
already established in Lemmas 6-7 to then use bounds for linear forms in
logarithms again and derive in Lemma 9 that

$\uptau<c_{3}\log n\log\log n.$

With this information, we can say that in all our decompositions of
$\upalpha_{i}$ and $\upbeta_{i}$ into powers of $\uplambda_{0},\uplambda_{2}$,
the second power is of no consequence unless $n$ is already sufficiently
”small”, i.e. $n<\upkappa$, since
$\log\left|\uplambda_{2}\right|=\uptheta\left(\frac{1}{n}\right)$ goes to
zero, even pitted against the logarithmic bound for the exponents $s,t$.

Carrying out a rigorous case-differentiation for the type $j$ and the order of
the $\left|\upalpha_{i}\right|$ in Lemma 7 gives a contradiction to
$\left|y\right|\geq 2$ in each case under the assumption that $n\geq\upkappa$
is sufficiently large for the powers of $\uplambda_{2}$ to be of no
consequence.

This gives $n<\upkappa$, which gives a bound first for $\uptau$ by Lemma 9 and
then $\log\left|y\right|$ by Lemma 7 and concludes the proof of Theorem 1.

##  3 Auxilary Results

We first define the absolute Weil height and Mahler’s measure for the sake of
completeness—see, e.g., [Bom82] or [Smy08]. If $K$ is a number field of degree
$d=[K:\mathbb{Q}]$, and for every place $\upnu$, write
$d_{\upnu}=[K_{\upnu}:\mathbb{Q}_{\upnu}]$ for the completions
$K_{\upnu},\mathbb{Q}_{\upnu}$ with respect to $\upnu$, we normalise the
absolute value $\left|\cdot\right|_{\upnu}$ so that

  1. 1.

if $\upnu|p$ for a prime number $p$, then
$\left|p\right|_{\upnu}=p^{-d_{\upnu}/d}$,

  2. 2.

if $\upnu|\infty$ and $\upnu$ is real, then
$\left|x\right|_{\upnu}=\left|x\right|^{1/d}$,

  3. 3.

if $\upnu|\infty$ and $\upnu$ is complex, then
$\left|x\right|_{\upnu}=\left|x\right|^{2/d}$,

where $\left|x\right|$ denotes the Euclidian absolute value in $\mathbb{R}$ or
$\mathbb{C}$. Given this normalisation, the product formula

$\prod_{\upnu}\left|\upalpha\right|_{\upnu}=1$

holds for every $\upalpha\in K^{*}$. The absolute height of $\upalpha\in K$ is
then defined as

$\mathscr{H}(\upalpha)=\prod_{\upnu}\max\left(1,\left|\upalpha\right|_{\upnu}\right),$

and the absolute logarithmic height as
$\mathscr{h}(\upalpha)=\log\mathscr{H}(\upalpha)$. The absolute height is then
equal to the Mahler measure $\mathscr{M}(m_{\upalpha})$ of its minimal
polynomial $m_{\upalpha}$, i.e. if
$m_{\upalpha}(X)=a_{d}\prod_{i=1}^{d}\left(X-\upalpha_{i}\right)\in\mathbb{Z}[X]$
is the minimal polynomial of $\upalpha\in K$, then

$\mathscr{h}(\upalpha)=\frac{1}{d}\log\mathscr{M}(m_{\upalpha})=\frac{1}{d}\left(\log\left|a_{d}\right|+\sum_{i=1}^{d}\log\max\left(1,\left|\upalpha_{i}\right|\right)\right).$

######  Proposition 1 (Baker, Wüstholz; [BW93]).

Let $\upgamma_{1},\dots,\upgamma_{t}$ be algebraic numbers not $0$ or $1$ in
$K=\mathbb{Q}\left(\upgamma_{1},\dots,\upgamma_{t}\right)$, which is of degree
$D$. Let $b_{1},\dots,b_{t}\in\mathbb{Z}$ and

$\Uplambda=b_{1}\log\upgamma_{1}+\cdots+b_{t}\log\upgamma_{t}\neq 0.$

Then

$\log\left|\Uplambda\right|\geq-C\cdot h_{1}\cdots h_{t}\cdot\log B,$

where $C=18(t+1)!t^{t+1}(32D)^{t+2}\log(2tD)$,
$B\geq\max\left(3,\left|b_{1}\right|,\cdots,\left|b_{t}\right|\right)$ and

$h_{i}\geq\max\left(\mathscr{h}(\upgamma_{i}),\log\left|\upgamma_{i}\right|\,D^{-1},0.16\,D^{-1}\right)$

for $i\in\left\\{1,\dots,t\right\\}$.

After stating the notion of height we use and the fundamental tool–lower
bounds for linear form in logarithms–to solve Thue equations, we now start
with a few observations that concern our Thue Equations (2).

######  Lemma 1.

The roots $\uplambda_{0},\uplambda_{1},\uplambda_{2}$ of the polynomial
$f(X)=X^{3}-(n-1)X^{2}-(n+2)X-1$ satisfy the following inequalities:

$\displaystyle n+\frac{1}{n}$ $\displaystyle<\uplambda_{0}$
$\displaystyle<n+\frac{2}{n}$ $\displaystyle-\frac{1}{n+1}$
$\displaystyle<\uplambda_{1}=-\frac{1}{\uplambda_{0}+1}$
$\displaystyle<-\frac{1}{n+2}$ $\displaystyle-1-\frac{1}{n}$
$\displaystyle<\uplambda_{2}=-\frac{\uplambda_{0}+1}{\uplambda_{0}}$
$\displaystyle<-1-\frac{1}{n+1}.$

###### Proof.

See, for instance, Lemma 10 in [LPV99] ∎

######  Corollary 1.

The logarithms of the roots of the polynomial $f(X)=X^{3}-(n-1)X^{2}-(n+2)X-1$
satisfy the following inequalities:

$\displaystyle\log n$ $\displaystyle<\log\uplambda_{0}$ $\displaystyle<\log
n+\frac{1}{n^{2}}$ $\displaystyle-\log n-\frac{2}{n}$
$\displaystyle<\log\left|\uplambda_{1}\right|$ $\displaystyle<-\log
n-\frac{1}{2n}$ $\displaystyle\frac{1}{n}-\frac{2}{n^{2}}$
$\displaystyle<\log\left|\uplambda_{2}\right|$
$\displaystyle<\frac{1}{n}+\frac{1}{n^{2}}.$

######  Proposition 2 (Thomas; [Tho79]).

Let $\uplambda_{0},\uplambda_{1},\uplambda_{2}$ be the roots of the polynomial
$f(X)=X^{3}-(n-1)X^{2}-(n+2)X-1$ and $K$ be the number field generated by
them. Then $\uplambda_{0},\uplambda_{2}$ give a set of fundamental units for
the order $\mathbb{Z}[\uplambda_{0}]$.

We now show that any two $\upalpha_{p},\upalpha_{q}$, as defined in Equation
(2) are separated.

######  Lemma 2.

There is an effectively computable constant $c_{1}=c_{1}(\upvarepsilon)>0$
which, for sufficiently large $n$, can be chosen to be arbitrarily close to
$\upvarepsilon$, so that, for two distinct $\upalpha_{p},\upalpha_{q}$ with
$\left|\upalpha_{q}\right|>\left|\upalpha_{p}\right|$ we have
$\left|\frac{\upalpha_{p}}{\upalpha_{q}}\right|<n^{-c_{1}\uptau}\leq\frac{1}{2}$
and
$\left|\upalpha_{p}-\upalpha_{q}\right|>\left(1-n^{-c_{1}\uptau}\right)\left|\upalpha_{q}\right|>\frac{1}{2}\left|\upalpha_{q}\right|$.

###### Proof.

We have
$\left|\upalpha_{p}-\upalpha_{q}\right|=\left|\upalpha_{q}\right|\left(1-\left|\frac{\upalpha_{p}}{\upalpha_{q}}\right|\right)$,
i.e. we have to prove that the fraction is at most $\frac{1}{2}$. To that end,
we look at the logarithm $-\log\left|\frac{\upalpha_{p}}{\upalpha_{q}}\right|$
and bound it from below.

By Lemma 1, we can express everything and especially logarithms of quotients
of $\upalpha_{i}$ in terms of $\log\uplambda_{0}$ and
$\log\left|\uplambda_{2}\right|$.

$\displaystyle\log\left|\frac{\upalpha_{p}}{\upalpha_{q}}\right|=\left\\{\begin{aligned}
&&\left(2s-t\right)\,&\log\uplambda_{0}&-\left(2t-s\right)\,&\log\left|\uplambda_{2}\right|\;\;&\text{
if }\left\\{p,q\right\\}=\left\\{0,1\right\\}\\\
&&\left(s-2t\right)\,&\log\uplambda_{0}&-\left(s+t\right)\,&\log\left|\uplambda_{2}\right|\;\;&\text{
if }\left\\{p,q\right\\}=\left\\{0,2\right\\}\\\
&&\left(s+t\right)\,&\log\uplambda_{0}&-\left(t-2s\right)\,&\log\left|\uplambda_{2}\right|\;\;&\text{
if }\left\\{p,q\right\\}=\left\\{1,2\right\\}\end{aligned}\right.$ (11)

We use the reverse triangle inequality to bound the positive
$-\log\left|\frac{\upalpha_{p}}{\upalpha_{q}}\right|$. The absolute value of
the coefficient of $\log\left|\uplambda_{2}\right|$ is at most $3\uptau$,
whereas Condition (3) guarantees that the absolute value of the coefficient of
$\log\uplambda_{0}$ is at least $\upvarepsilon\uptau$. If we put everything
together and use Corollary 1 to estimate the logarithms, we get

$-\log\left|\frac{\upalpha_{p}}{\upalpha_{q}}\right|>\upvarepsilon\uptau\log\uplambda_{0}-3\uptau\log\left|\uplambda_{2}\right|>\upvarepsilon\uptau\log
n-\frac{3\uptau}{n}-\frac{3\uptau}{n^{2}},$

and unless both $\uptau$ and $n$ are already small, there exists an
effectively computable constant $c_{1}>0$, dependent only on $\upvarepsilon$,
and for sufficiently large $n$ arbitrarily close to $\upvarepsilon$, such that

$\upvarepsilon\uptau\log
n-\frac{3\uptau}{n}-\frac{3\uptau}{n^{2}}>c_{1}\uptau\log n\geq\log 2.$

∎

######  Lemma 3.

Let $a,b,c$ be positive real numbers with $abc=1$, then

$\max\left(a,b\right)\max\left(a,c\right)\geq\sqrt{\max\left(a,b,c\right)}.$

###### Proof.

Let $a=\max\left(a,b,c\right)$, then $abc=1$ implies that $a\geq 1$ and thus
$\max\left(a,b\right)\max\left(a,c\right)=a^{2}\geq\sqrt{a}$ holds. If
instead, after possibly renaming $b$ and $c$, the maximum is
$b=\max\left(a,b,c\right)$ then $a\geq\frac{1}{\sqrt{b}}$ or
$c\geq\frac{1}{\sqrt{b}}$, with equality in the case that $a=c$ for $abc=1$ to
hold. This in turn means that
$\max\left(a,b\right)\max\left(a,c\right)\geq\frac{b}{\sqrt{b}}=\sqrt{b}$. ∎

######  Lemma 4.

We have

$n^{3\uptau}\geq\max\left(\left|\upalpha_{0}\right|,\left|\upalpha_{1}\right|,\left|\upalpha_{2}\right|\right)\geq
n^{c_{1}\uptau}.$

###### Proof.

Analogously to the proof of Lemma 2, we have for the maximum $\upalpha$ that

$\displaystyle\log\left|\upalpha\right|=\left\\{\begin{aligned}
&&(s-t)\,&\log\uplambda_{0}\;&-t\,&\log\left|\uplambda_{2}\right|\;\;\text{ if
}\upalpha=\upalpha_{0}\\\
&&-s\,&\log\uplambda_{0}\;&+\left(t-s\right)\,&\log\left|\uplambda_{2}\right|\;\;\text{
if }\upalpha=\upalpha_{1}\\\
&&t\,&\log\uplambda_{0}\;&+s\,&\log\left|\uplambda_{2}\right|\;\;\text{ if
}\upalpha=\upalpha_{2}.\end{aligned}\right.$

Since $\left|\upalpha\right|$ is the maximum of
$\left|\upalpha_{0}\right|,\left|\upalpha_{1}\right|,\left|\upalpha_{2}\right|$,
its logarithm is the largest of the three expressions given above. Since it is
either $\left|-s\right|=\uptau$ or $\left|t\right|=\uptau$, we can infer

$3\uptau\log
n>\log\left|\upalpha\right|\geq\uptau\log\uplambda_{0}-2\uptau\log\left|\uplambda_{2}\right|>\uptau\log
n-2\uptau\left(\frac{1}{n}+\frac{1}{n^{2}}\right)$

in each of the three cases for $\upalpha$. We can certainly bound the right-
hand side by $c_{1}\uptau\log n$, the same expression as used in Lemma 2 but
could also use a constant that does not depend on $\upvarepsilon$, such as
$0.19$: Since $\uptau\log
n-2\uptau\left(\frac{1}{n}+\frac{1}{n^{2}}\right)>0.19\,\uptau\log n$ if and
only if $1-\left(\frac{2}{n\log n}+\frac{2}{n^{2}\log n}\right)>0.19$ which is
true for $n\geq 3$. ∎

######  Lemma 5.

We have

$\frac{\max\left(\left|\upalpha_{k}\right|,\left|\upalpha_{l}\right|\right)}{\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{k}\right|\right)\,\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{l}\right|\right)}<2\left|y\right|.$

###### Proof.

Irrespective of whether $\upalpha_{j}$ has the largest absolute value of the
three or not, we can bound the quotient by its inverse, i.e.

$\frac{\max\left(\left|\upalpha_{k}\right|,\left|\upalpha_{l}\right|\right)}{\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{k}\right|\right)\,\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{l}\right|\right)}<\frac{1}{\left|\upalpha_{j}\right|}.$

If
$\left|\upalpha_{j}\right|=\max\left(\left|\upalpha_{0}\right|,\left|\upalpha_{1}\right|,\left|\upalpha_{2}\right|\right)$,
then the right-hand side is at most $n^{-c_{1}\uptau}$, by Lemma 4. And unless
$n$ and $\uptau$ are already absolutely bounded, we can assume this to be
smaller than any constant, say, $4\leq 2\left|y\right|$.

Otherwise, we use Inequality (6), which by the same argument is smaller still
than, say $\frac{1}{2}$, and get

$\frac{1}{2}>\left|x-\upalpha_{j}y\right|\geq\left|x\right|-\left|\upalpha_{j}\right|\left|y\right|.$

Since $x\neq 0$—the only solutions with $x=0$ have $y=\pm 1$—we have
$\left|x\right|\geq 1$ and thus the above inequality can be read as

$\left|\upalpha_{j}\right|>\frac{1}{2\left|y\right|},$ (12)

which proves the assertion. ∎

######  Lemma 6.

Let $\upbeta_{0}=\pm\uplambda_{0}^{a}\uplambda_{2}^{b}$ be the decomposition
of the unit $\upbeta_{0}\in\mathbb{Z}[\uplambda_{0}]$ into powers of the
fundamental units $\uplambda_{0},\uplambda_{2}$ of the mentioned order. Then
we have

$\max\left(\left|a\right|,\left|b\right|\right)=\uptheta\left(\frac{\log\left|y\right|}{\log
n}+\uptau\right).$

###### Proof.

By conjugation, we have that
$\upbeta_{0}=\pm\uplambda_{0}^{a}\uplambda_{2}^{b},\upbeta_{1}=\pm\uplambda_{1}^{a}\uplambda_{0}^{b}$,
and $\upbeta_{2}=\pm\uplambda_{2}^{a}\uplambda_{1}^{b}$. We take the
decompositions of $\upbeta_{k}$ and $\upbeta_{l}$ and take the logarithm. This
gives the system of linear equations

$M\begin{pmatrix}a\\\
b\end{pmatrix}=\begin{pmatrix}\log\left|\upbeta_{k}\right|\\\
\log\left|\upbeta_{l}\right|\end{pmatrix},$ (13)

where, depending on $j$,

$M\in\left\\{\begin{pmatrix}\log\left|\uplambda_{1}\right|&\log\left|\uplambda_{0}\right|\\\
\log\left|\uplambda_{2}\right|&\log\left|\uplambda_{1}\right|\end{pmatrix},\begin{pmatrix}\log\left|\uplambda_{0}\right|&\log\left|\uplambda_{1}\right|\\\
\log\left|\uplambda_{1}\right|&\log\left|\uplambda_{0}\right|\end{pmatrix},\begin{pmatrix}\log\left|\uplambda_{0}\right|&\log\left|\uplambda_{2}\right|\\\
\log\left|\uplambda_{1}\right|&\log\left|\uplambda_{0}\right|\end{pmatrix}\right\\}.$

Since $\log\uplambda_{0}=\uptheta\left(\log
n\right),\log\uplambda_{1}=\uptheta\left(-\log n\right)$ and
$\log\uplambda_{2}=\uptheta\left(\frac{1}{n}\right)$ by Corollary 1, we have
that $\left|\det M\right|=\uptheta\left(\left(\log n\right)^{2}\right)$ in
either case. In particular, the determinant is non-zero, the matrix thus
invertible.

We then multiply Equation (13) with the inverse $M^{-1}$ and take the column-
wise-maximum norm. We can calculate it for $M$ by taking the row-wise-maximum
norm of $M$ and dividing by $\left|\det M\right|$, which gives a
$\uptheta\left(\frac{1}{\log n}\right)$. We thus get, with the consistency
$\lVert M^{-1}\vec{\upbeta}\rVert\leq\lVert
M^{-1}\rVert\cdot\lVert\vec{\upbeta}\rVert$,

$\max\left(\left|a\right|,\left|b\right|\right)=\uptheta\left(\frac{\max\left(\left|\log\left|\upbeta_{k}\right|\right|,\left|\log\left|\upbeta_{l}\right|\right|\right)}{\log
n}\right).$

Furthermore, we have

$\displaystyle\log\left|\upbeta_{k}\right|$
$\displaystyle=\log\left|x-\upalpha_{k}y+\upalpha_{j}y-\upalpha_{j}y\right|$
$\displaystyle=\log\left|y\right|+\log\left|\upalpha_{j}-\upalpha_{k}\right|+\log\left|1+\left|\frac{\upbeta_{j}}{y\left|\upalpha_{j}-\upalpha_{k}\right|}\right|\right|,$
(14)

and since
$\left|\upalpha_{j}-\upalpha_{k}\right|\geq\frac{1}{2}\left|\upalpha_{j}\right|\geq\frac{1}{4}\left|y\right|^{-1}$
by Equation (12), while
$\left|\upbeta_{j}\right|\ll\left|y\right|^{-2}n^{-\frac{c_{1}}{2}\uptau}$ by
Inequality (6), the last logarithm is very small and can certainly be bounded
by $1$. Thus,

$\max\left(\left|\log\left|\upbeta_{k}\right|\right|,\left|\log\left|\upbeta_{l}\right|\right|\right)=\uptheta\left(\log\left|y\right|+\log\max\left(\left|\upalpha_{j}-\upalpha_{k}\right|,\left|\upalpha_{j}-\upalpha_{l}\right|\right)\right).$

Finally, we have by Lemma 2 that $\left|\upalpha_{j}-\upalpha_{k}\right|$ is
at least
$\frac{1}{2}\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{k}\right|\right)$,
and it is at most $2$ times the maximum. That is to say,

$\log\max\left(\left|\upalpha_{j}-\upalpha_{k}\right|,\left|\upalpha_{j}-\upalpha_{l}\right|\right)=\uptheta\left(\log\max\left(\left|\upalpha_{j}\right|,\left|\upalpha_{k}\right|,\left|\upalpha_{l}\right|\right)\right),$

which by Lemma 4 is a $\uptheta\left(\uptau\log n\right)$. This gives us that

$\max\left(\left|\log\left|\upbeta_{k}\right|\right|,\left|\log\left|\upbeta_{l}\right|\right|\right)=\uptheta\left(\log\left|y\right|+\uptau\log
n\right)$

and thus

$\max\left(\left|a\right|,\left|b\right|\right)=\uptheta\left(\frac{\log\left|y\right|}{\log
n}+\uptau\right),$

which proves the assertion. ∎

##  4 Proof of Theorem 1

With most of the technical considerations done in the previous section, we can
formally finish the proof of Theorem 1, as outlined in Section 2. That is to
say, we now use bounds for linear forms in logarithms, namely Proposition 1,
to deduce bounds for the size of the parameters of our Thue equations.

######  Lemma 7.

We have, for some effectively computable constant $c_{2}>0$,

$\log\left|y\right|<c_{2}\,\uptau\,\left(\log
n\right)^{3}\left(\log\uptau+\log\log n\right).$

###### Proof.

We take the decomposition $\upbeta_{0}=\pm\uplambda_{0}^{a}\uplambda_{2}^{b}$
and conjugate the expression to get decompositions for $\upbeta_{1}$ and
$\upbeta_{2}$. Only this time, we express everything in powers of
$\uplambda_{0}$ and $\uplambda_{2}$ using the connections from Lemma 1. This
gives that $\upbeta_{0}=\pm\uplambda_{0}^{a}\uplambda_{2}^{b}$,
$\upbeta_{1}=\pm\uplambda_{0}^{-a+b}\uplambda_{2}^{-a}$, and
$\upbeta_{2}=\pm\uplambda_{0}^{-b}\uplambda_{2}^{a-b}$.

We can thus write for the logarithm of the quotient,
$\log\left|\frac{\upbeta_{l}}{\upbeta_{k}}\right|=a^{\prime}\log\uplambda_{0}+b^{\prime}\log\left|\uplambda_{2}\right|$,
where both $a^{\prime},b^{\prime}$ are the respective linear combination of
$a,b$ and can write the linear form in logarithms $\Uplambda$ from Equation
(9) as

$0<\left|\tilde{a}\log\uplambda_{0}+\tilde{b}\log\left|\uplambda_{2}\right|+\log\left|\frac{\upalpha_{j}-\upalpha_{k}}{\upalpha_{j}-\upalpha_{l}}\right|\right|\ll\frac{1}{\left|y\right|^{2}n^{\frac{c_{1}}{2}\uptau}}.$

We now use lower bounds for this linear form in logarithms, Proposition 1.

By Lemma 6, we have that
$\max\left(\left|a^{\prime}\right|,\left|b^{\prime}\right|\right)\ll\frac{\log\left|y\right|}{\log
n}+\uptau$. Due to the sub-additivity and sub-multiplicativity of the absolute
logarithmic height, we have
$\mathscr{h}\left(\frac{\upalpha_{j}-\upalpha_{k}}{\upalpha_{j}-\upalpha_{l}}\right)\ll\uptau\,\mathscr{h}\left(\uplambda_{0}\right)$.
And switching to the Mahler measure of the minimal polynomial $f$ of
$\uplambda_{0}$ gives

$\mathscr{h}\left(\uplambda_{0}\right)=\mathscr{M}\left(f\right)=\frac{1}{3}\left(\log\left(n+2\right)+\log\left|\uplambda_{0}\right|+\log\left|\uplambda_{2}\right|\right)=\uptheta\left(\log
n\right).$ (15)

We plug everything into Proposition 1 and get that for some large effective
constant $c_{2}>0$, since the constant in Proposition 1 is already larger than
$10^{14}$, that

$-c_{2}\,\uptau\left(\log n\right)^{3}\log\left(\frac{\log\left|y\right|}{\log
n}+\uptau\right)<-\log\left|y\right|$

holds, which implies the assertion. ∎

######  Lemma 8.

There exists a non-zero linear form $\Uplambda^{\prime\prime}$ in logarithms
of $\upbeta_{k},\upbeta_{l},\upalpha_{0},\upalpha_{1},\upalpha_{2},2$ with
coefficients $0$ or $\pm 1$, such that

$0<\left|\Uplambda^{\prime\prime}\right|<\frac{1}{n^{\frac{c_{1}}{2}\uptau}}$

holds.

###### Proof.

If the linear form $\Uplambda^{\prime}$ from Equation (10) is already non-
zero, the purported upper bound follows from factoring
$\frac{\upalpha_{u}}{\upalpha_{v}}$ in Equation (9) and shifting the remainder
into the upper bound, which gives

$\left|\Uplambda^{\prime}\right|\ll\frac{1}{\left|y\right|^{2}n^{\frac{c_{1}}{2}\uptau}}+\log\left|\frac{1-\frac{\upalpha_{u^{\prime}}}{\upalpha_{u}}}{1-\frac{\upalpha_{v^{\prime}}}{\upalpha_{v}}}\right|.$

Both quotients of two $\upalpha_{i}$ respectively on the right-hand side are
smaller than $n^{-c_{1}\uptau}$ by Lemma 2, which gives the purported upper
bound for $\left|\Uplambda^{\prime}\right|$, since $\log\left|1\pm
n^{-c_{1}\uptau}\right|<n^{-c_{1}\uptau}$.

Assume instead that $\Uplambda^{\prime}=0$, which holds if and only if
$\left|\upbeta_{l}\upalpha_{u}\right|=\left|\upbeta_{k}\upalpha_{v}\right|$.
We now differentiate the cases for $(u,v)$ and the sign.

_Case
$\left|\upalpha_{j}\right|>\max\left(\left|\upalpha_{k}\right|,\left|\upalpha_{l}\right|\right)$:_
In this case, $\upalpha_{u}=\upalpha_{v}=\upalpha_{j}$ cancels out and we have
$\left|\upbeta_{l}\right|=\left|\upbeta_{k}\right|$. They cannot be equal,
since $\upbeta_{l}=\upbeta_{k}$ implies $\upalpha_{l}=\upalpha_{k}$, which
gives $s=t=0$ due to the multiplicative independence of
$\uplambda_{0},\uplambda_{2}$. Thus, $\upbeta_{l}=-\upbeta_{k}$.

We substitute $\upbeta_{l}$ for $-\upbeta_{k}$ in Siegel’s Identity (7) and
obtain

$\upbeta_{j}\left(\upalpha_{k}-\upalpha_{l}\right)+\upbeta_{k}\upalpha_{l}+2\upbeta_{k}\upalpha_{j}-\upbeta_{k}\upalpha_{l}=0,$

which we transform into

$\frac{\upbeta_{j}\left(\upalpha_{k}-\upalpha_{l}\right)}{2\upbeta_{k}\upalpha_{j}}+\frac{\upalpha_{l}}{2\upalpha_{j}}+1=\frac{\upalpha_{k}}{2\upalpha_{j}}.$

Since $\left|\upalpha_{j}\right|$ is the largest, the right-hand side is not
$1$. We can bound, by Lemma 2,
$\left|\frac{\upalpha_{l}}{2\upalpha_{j}}\right|\ll
n^{-\frac{c_{1}}{2}\uptau}$. By a combination of Inequality (5)-(6), and Lemma
2-4, we can bound

$\left|\frac{\upbeta_{j}\left(\upalpha_{k}-\upalpha_{l}\right)}{2\upbeta_{k}\upalpha_{j}}\right|\ll\frac{\max\left(\left|\upalpha_{k}\right|,\left|\upalpha_{l}\right|\right)}{\left|y\right|^{3}n^{\frac{c_{1}}{2}\uptau}\left|\upalpha_{j}\right|^{2}}\ll\frac{1}{\left|y\right|^{3}n^{\frac{3c_{1}}{2}\uptau}}.$

We thus have that

$0<\left|\log\left|\upalpha_{k}\right|-\log\left|\upalpha_{j}\right|-\log
2\right|\ll\frac{1}{n^{\frac{c_{1}}{2}\uptau}}.$

_Case
$\left|\upalpha_{k}\right|>\left|\upalpha_{l}\right|>\left|\upalpha_{j}\right|$:_
In this case, we have $\upalpha_{u}=\upalpha_{k}$ and
$\upalpha_{v}=\upalpha_{l}$ and thus
$\left|\upbeta_{l}\upalpha_{k}\right|=\left|\upbeta_{k}\upalpha_{l}\right|$.
If the absolute value cancels without a sign change, the term
$\upbeta_{k}\upalpha_{l}-\upbeta_{l}\upalpha_{k}$ vanishes in Siegel’s
Identity (7). What remains is

$\upbeta_{j}\left(\upalpha_{k}-\upalpha_{l}\right)-\upbeta_{k}\upalpha_{j}+\upbeta_{l}\upalpha_{j}=0,$

or

$-\frac{\upbeta_{j}\left(\upalpha_{k}-\upalpha_{l}\right)}{\upbeta_{k}\upalpha_{j}}+1=\frac{\upbeta_{l}}{\upbeta_{k}}.$

The right-hand side is again not $1$, since $\upalpha_{l}\neq\upalpha_{k}$.
Analogously to the above case and then using Inequality (12), we have that

$\left|\frac{\upbeta_{j}\left(\upalpha_{k}-\upalpha_{j}\right)}{\upbeta_{k}\upalpha_{j}}\right|\ll\frac{\left|\upalpha_{k}\right|}{\left|y\right|^{3}n^{\frac{c_{1}}{2}\uptau}\left|\upalpha_{k}-\upalpha_{j}\right|\left|\upalpha_{j}\right|}\ll\frac{1}{\left|y\right|^{2}n^{\frac{c_{1}}{2}\uptau}},$

and thus

$0<\left|\log\left|\upbeta_{l}\right|-\log\left|\upbeta_{k}\right|\right|<\frac{1}{n^{\frac{c_{1}}{2}\uptau}}.$

If the absolute value cancels with a sign change, then
$\upbeta_{k}\upalpha_{l}-\upbeta_{l}\upalpha_{k}=-2\upbeta_{l}\upalpha_{k}$.
Plugging this into Siegel’s identity yields

$-\frac{\upbeta_{j}\left(\upalpha_{k}-\upalpha_{l}\right)}{2\upbeta_{l}\upalpha_{k}}-\frac{\upalpha_{j}}{2\upalpha_{k}}+1=-\frac{\upbeta_{k}\upalpha_{j}}{2\upbeta_{l}\upalpha_{k}}.$

The right-hand side is not $1$: On the one hand, we have that
$\left|\upbeta_{k}\upalpha_{j}\right|\ll\left|y\right|\left|\upalpha_{k}-\upalpha_{j}\right|\left|\upalpha_{j}\right|\ll\left|y\right|\left|\upalpha_{k}\right|\left|\upalpha_{j}\right|$,
and on the other hand that
$\left|y\right|\left|\upalpha_{l}\right|\left|\upalpha_{k}\right|\ll\left|y\right|\left|\upalpha_{l}-\upalpha_{j}\right|\left|\upalpha_{k}\right|\ll\left|2\upbeta_{l}\upalpha_{k}\right|$.
Thus, the right-hand side is effectively bounded by
$\left|\frac{\upalpha_{j}}{\upalpha_{l}}\right|$, which in this case is
smaller than $1$, even taking into account the implied constants. We can
similarly effectively bound the terms on the left-hand side by
$n^{-\frac{c_{1}}{2}\uptau}$, from which it follows that

$0<\left|\log\left|\upbeta_{k}\right|-\log\left|\upbeta_{l}\right|+\log\left|\upalpha_{j}\right|-\log\left|\upalpha_{k}\right|-\log
2\right|<\frac{1}{n^{\frac{c_{1}}{2}\uptau}}.$

The case where
$\left|\upalpha_{k}\right|>\left|\upalpha_{j}\right|>\left|\upalpha_{l}\right|$
follows analogously. ∎

######  Lemma 9.

There exists an effectively computable constant $c_{3}>0$ such that

$\uptau<c_{3}\log n\log\log n.$

###### Proof.

By Lemma 8, we have a linear form in logarithms of
$\upbeta_{k},\upbeta_{l},\upalpha_{j},\upalpha_{k},\upalpha_{l},2$ with
coefficients in $\left\\{-1,0,1\right\\}$, for which

$0<\left|\Uplambda^{\prime\prime}\right|<\frac{1}{n^{\frac{c_{1}}{2}\uptau}}$

holds. We proceed analogously to the proof of Lemma 7 and write
$\upbeta_{0}=\pm\uplambda_{0}^{a}\uplambda_{2}^{b}$ and so on. We do the same
for the $\upalpha_{i}$, i.e.
$\upalpha_{0}=\uplambda_{0}^{s}\uplambda_{1}^{t}=\uplambda_{0}^{s-t}\uplambda_{2}^{-t}$.
We can then write $\Uplambda^{\prime\prime}$ alternatively as a linear form in
logarithms of $\uplambda_{0},\uplambda_{2},2$.

The coefficients are then linear combinations of $a,b,s,t$ and thus
effectively bounded by $\frac{\log\left|y\right|}{\log n}+\uptau$ by Lemma 6.
Applying the bound for $\log\left|y\right|$ from Lemma 7 gives, up to
effective constants, the bound

$\uptau\left(\log n\right)^{2}\left(\log\uptau+\log\log n\right).$

The logarithmic heights of $\uplambda_{0},\uplambda_{2}$ are effectively
bounded by $\log n$, and the logarithmic height of $2$ is $\log 2$.

We plug this into Proposition 1 and get

$\uptau\log n\ll\left(\log n\right)^{2}\log\left(\uptau\left(\log
n\right)^{2}\left(\log\uptau+\log\log n\right)\right).$

If $\uptau\ll\log n$, then the proposed bound holds in particular. If $\log
n\ll\uptau$ instead, then $\log\log n\ll\log\uptau$ as well and the right-hand
side of the above inequality becomes

$\uptau\ll\log n\log\left(\uptau^{3}\log\uptau\right)\ll\log n\log\uptau,$

from which $\uptau\ll\log n\log\log n$ follows.

∎

If we use the bound $\uptau<c_{3}\log n\log\log n$ and plug it into Lemma 7,
we get that

$\log\left|y\right|<c_{4}\left(\log n\right)^{4}\left(\log\log n\right)^{2},$
(16)

and plugging both bounds into Lemma 6, we get

$\max\left(\left|a\right|,\left|b\right|\right)<c_{5}\left(\log
n\right)^{3}\left(\log\log n\right)^{2}.$ (17)

All we have left to do is to bound $n$ by an absolute constant, which we do in
the following

######  Lemma 10.

There exists an effectively computable constant $\upkappa$ such that
$n<\upkappa$.

###### Proof.

We assume $n\geq\upkappa$ is sufficiently large, such that the powers of
$\uplambda_{2}$ do not influence the following arguments.

We first recall the forms of the $\upbeta_{i},\upalpha_{i}$ in terms of
$\uplambda_{0},\uplambda_{2}$, i.e. that
$\upbeta_{0}=\uplambda_{0}^{a}\uplambda_{2}^{b},\upbeta_{1}=\uplambda_{0}^{-a+b}\uplambda_{2}^{-a}$
and $\upbeta_{2}=\uplambda_{0}^{-b}\uplambda_{2}^{a-b}$, while
$\upalpha_{0}=\uplambda_{0}^{s-t}\uplambda_{2}^{-t},\upalpha_{1}=\uplambda_{0}^{-s}\uplambda_{2}^{-s+t}$
and $\upalpha_{2}=\uplambda_{0}^{t}\uplambda_{2}^{s}$. We differentiate the
cases for $j$, i.e. which algebraic number $\upalpha_{i}$ gets approximated
the best by $\frac{x}{y}$, resp. which $\upbeta_{i}$ is the smallest in
absolute value, and $u,v$, i.e. for fixed $j$, which algebraic number is the
larger in absolute value between $\upalpha_{j},\upalpha_{k}$ or
$\upalpha_{j},\upalpha_{l}$ respectively. We start with the case $j=0$ and
choose $(k,l)=(1,2)$.

_Case $u=v=0$:_ In this case,
$\left|\upalpha_{0}\right|=\max\left(\left|\upalpha_{0}\right|,\left|\upalpha_{1}\right|,\left|\upalpha_{2}\right|\right)>1$
in particular. Taking the logarithm yields
$\left(s-t\right)\log\uplambda_{0}-t\log\left|\uplambda_{2}\right|>0$. But we
have $\left|t\right|<c_{3}\log n\log\log n$ by Lemma 9, and
$\log\uplambda_{2}=\uptheta\left(\frac{1}{n}\right)$ by Corollary 1. For
$n\geq\upkappa$, the second summand thus does not influence the sign of the
expression—we derive $s-t>0$ or $-s+t<0$.

It is also the case that
$\log\left|\frac{\upalpha_{u}}{\upalpha_{v}}\right|=0$, since
$\upalpha_{u}=\upalpha_{v}=\upalpha_{0}$. The absolute value of the linear
form $\Uplambda^{\prime}$ of Equation (10) is thus

$\left|\Uplambda^{\prime}\right|=\left|\log\left|\frac{\upbeta_{1}}{\upbeta_{2}}\right|\right|=\left|\left(a-2b\right)\log\uplambda_{0}+\left(2a-b\right)\log\uplambda_{2}\right|<\frac{1}{n^{\frac{c_{1}}{2}\uptau}}.$
(18)

Either $a-2b=0$, or we can deduce from the above equation, with
$\left|a-2b\right|\geq 1$ and $\log\uplambda_{0}>\log n$, that

$\log
n<\left|a-2b\right|\log\uplambda_{0}<\left|2a-b\right|\log\left|\uplambda_{2}\right|+\frac{1}{n^{\frac{c_{1}}{2}\uptau}}.$

But $\log\left|\uplambda_{2}\right|<\frac{1}{n}+\frac{1}{n^{2}}<\frac{2}{n}$
by Corollary 1, and thus

$\log
n<\left|2a-b\right|\log\left|\uplambda_{2}\right|+\frac{1}{n^{\frac{c_{1}}{2}\uptau}}<\frac{2\left|2a-b\right|}{n}+\frac{1}{n^{\frac{c_{1}}{2}\uptau}}.$

We compare the left-hand and right-hand sides of this inequality and since by
Inequality (17)

$\left|2a-b\right|<3\max\left(\left|a\right|,\left|b\right|\right)<3c_{5}\left(\log
n\right)^{3}\left(\log\log n\right)^{2}$

find, that it cannot hold if $n\geq\upkappa$ is sufficiently large. Thus,
$a-2b=0$.

With $a-2b=0$, Equation (18) becomes

$\left|2a-b\right|\log\left|\uplambda_{2}\right|<\frac{1}{n^{\frac{c_{1}}{2}\uptau}},$

and by Corollary 1, if $2a-b\neq 0$ this then implies

$\frac{1}{n}-\frac{2}{n^{2}}<\frac{1}{n^{\frac{c_{1}}{2}\uptau}}.$

This gives a contradiction and thus $2a-b=0$, unless
$\frac{c_{1}}{2}\uptau\leq 1$. However, if $n\geq\upkappa$ is sufficiently
large, the constant $c_{1}$ can be chosen to be arbitrarily close to
$\upvarepsilon$ by Lemma 2. So $\frac{c_{1}}{2}\uptau\leq 1$ implies
$\upvarepsilon\uptau\leq 2$ and contradicts Condition (3).

We thus have both $a-2b=0$ and $2a-b=0$, which implies $a=b=0$. But this in
turn means that $\upbeta_{0}=\uplambda_{0}^{a}\uplambda_{2}^{b}=1$ and thus
$x-\upalpha_{0}y=1$. Since $x,y$ are integers and $\upalpha_{0}$ irrational,
this implies $x=1,y=0$ and contradicts $\left|y\right|\geq 2$. Our assumption
that $n\geq\upkappa$ was thus false.

_Case $u=0,v=2$:_ Here, the relation between the sizes of the algebraic
numbers $\upalpha_{i}$ is

$1<\left|\upalpha_{2}\right|\text{ and
}\left|\upalpha_{2}\right|>\left|\upalpha_{0}\right|>\left|\upalpha_{1}\right|.$

Taking the logarithm and using that for $n\geq\upkappa$, the summand with
$\log\left|\uplambda_{2}\right|$ is small enough not to influence the
inequality relations, we get from this that $0<t$ and $t>s-t>-s$. This implies
$s>0$ in particular.

Furthermore,
$\log\left|\frac{\upalpha_{u}}{\upalpha_{v}}\right|=\log\left|\frac{\upalpha_{0}}{\upalpha_{2}}\right|=\left(s-2t\right)\log\uplambda_{0}+\left(-s-t\right)\log\left|\uplambda_{2}\right|$.
We follow the same process as in the previous case, i.e. first assume that the
coefficient of $\log\left|\uplambda_{0}\right|$ is non-zero and rewrite

$\left|\Uplambda^{\prime}\right|=\left|\log\left|\frac{\upbeta_{1}}{\upbeta_{2}}\right|+\log\left|\frac{\upalpha_{0}}{\upalpha_{2}}\right|\right|=\left|\left(a-2b+s-2t\right)\log\left|\uplambda_{0}\right|+\left(2a-b-
s-t\right)\log\left|\uplambda_{2}\right|\right|<\frac{1}{n^{\frac{c_{1}}{2}\uptau}}$

into

$\log n<\frac{2\left|2a-b-s-t\right|}{n}+\frac{1}{n^{\frac{c_{1}}{2}\uptau}}.$

But this cannot hold, since by Lemma 9 and Equation (17),

$\left|2a-b-
s-t\right|\ll\max\left(\left|a\right|,\left|b\right|\right)+\uptau\ll\left(\log
n\right)^{3}\left(\log\log n\right)^{2}.$

The assumption $a-2b+s-2t\neq 0$ was thus false, so the term with
$\log\uplambda_{0}$ vanishes in $\Uplambda^{\prime}$.

We then assume that the coefficient $2a-b-s-t$ of
$\log\left|\uplambda_{2}\right|$ does not vanish and rewrite
$\left|\Uplambda^{\prime}\right|<n^{-\frac{c_{1}}{2}\uptau}$ into

$\frac{1}{n}-\frac{2}{n^{2}}<\left|2a-b-
s-t\right|\log\left|\uplambda_{2}\right|<\frac{1}{n^{\frac{c_{1}}{2}\uptau}}$

and comparing the left-hand and right-hand sides gives the contradiction, thus
$2a-b-s-t=0$ as well. We rewrite the equations $a-2b+s-2t=2a-b-s-t=0$ to
express $a,b$ in terms of $s,t$ and get $a=s,b=s-t$.

We again use this information on $a,b$ to contradict $\left|y\right|\geq 2$.
Recall that by Equation (3), we can approximate $\log\left|y\right|$, up to an
error of order $O\left(n^{-\frac{c_{1}}{2}\uptau}\right)$, very well by either
$\log\left|\upbeta_{k}\right|-\log\left|\upalpha_{j}-\upalpha_{k}\right|$ or
$\log\left|\upbeta_{l}\right|-\log\left|\upalpha_{j}-\upalpha_{l}\right|$. We
are in the case $j=0$ and take a closer look at the approximation for $l=2$,
i.e.
$\log\left|\upbeta_{2}\right|-\log\left|\upalpha_{0}-\upalpha_{2}\right|$. By
Equation (1), using Lemma 2 gives
$\left|\upalpha_{0}-\upalpha_{2}\right|=\left|\upalpha_{2}\right|\left(1+O\left(n^{-c_{1}\uptau}\right)\right)$
and thus

$\displaystyle\log\left|y\right|$
$\displaystyle=\log\left|\upbeta_{2}\right|-\log\left|\upalpha_{2}\right|+O\left(n^{-\frac{c_{1}}{2}\uptau}\right)$
$\displaystyle=\left(-b-t\right)\log\uplambda_{0}+\left(a-b-s\right)\log\left|\uplambda_{2}\right|+O\left(n^{-\frac{c_{1}}{2}\uptau}\right)$
$\displaystyle=-s\log\uplambda_{0}+\left(-s+t\right)\log\left|\uplambda_{2}\right|+O\left(n^{-\frac{c_{1}}{2}\uptau}\right).$

But $-s<0$ in this case, so if $n\geq\upkappa$ is sufficiently large such that
neither the second addend nor the omitted error terms influence the sign, this
reads as $\log\left|y\right|<0$ and contradicts $\left|y\right|\geq 2$.

_Case $u=1,v=0$:_ The relation between the sizes of the algebraic numbers, and
thus between the exponents under the condition that $n\geq\upkappa$, is

$1<\left|\upalpha_{1}\right|\text{ and
}\left|\upalpha_{1}\right|>\left|\upalpha_{0}\right|>\left|\upalpha_{2}\right|\iff
0<-s\text{ and }-s>s-t>t,$

which implies $t<0$.

We also write
$\log\left|\frac{\upalpha_{u}}{\upalpha_{v}}\right|=\log\left|\frac{\upalpha_{1}}{\upalpha_{0}}\right|=\left(-2s+t\right)\log\uplambda_{0}+\left(-s+2t\right)\log\left|\uplambda_{2}\right|$.
Setting up $\Uplambda^{\prime}$ and its upper bound and assuming the
coefficient of $\log\uplambda_{0}$ yields

$\log
n<\left|a-2b-2s+t\right|\log\left|\uplambda_{0}\right|<\frac{2\left|2a-b-s+2t\right|}{n}+\frac{1}{n^{\frac{c_{1}}{2}\uptau}}$

and gives the contradiction for sufficiently large $n\geq\upkappa$, hence
$a-2b-2s+t=0$.

Doing the same for the coefficient of $\log\left|\uplambda_{2}\right|$ gives

$\frac{1}{n}-\frac{2}{n^{2}}<\left|2a-b-s+2t\right|\log\left|\uplambda_{2}\right|<\frac{1}{n^{\frac{c_{1}}{2}\uptau}}$

and the contradiction to assuming $2a-b-s+2t\neq 0$. Hence both terms equal
$0$, and we again express $a,b$ in terms of $s,t$ and get $a=t,b=-s$.

We again approximate $\log\left|y\right|$ by
$\log\left|\upbeta_{2}\right|-\log\left|\upalpha_{0}-\upalpha_{2}\right|$.
Since $v=0$, we approximate $\log\left|\upalpha_{0}-\upalpha_{2}\right|$ by
$\log\left|\upalpha_{0}\right|$ this time, which gives

$\displaystyle\log\left|y\right|$
$\displaystyle=\log\left|\upbeta_{2}\right|-\log\left|\upalpha_{0}\right|+O\left(n^{-\frac{c_{1}}{2}\uptau}\right)$
$\displaystyle=\left(-b-s+t\right)\log\uplambda_{0}+\left(a-b+t\right)\log\left|\uplambda_{2}\right|+O\left(n^{-\frac{c_{1}}{2}\uptau}\right)$
$\displaystyle=t\log\uplambda_{0}+s\log\left|\uplambda_{2}\right|+O\left(n^{-\frac{c_{1}}{2}\uptau}\right),$

if we plug in our expressions for $a,b$. But since $t<0$, this gives
$\log\left|y\right|<0$ and contradicts $\left|y\right|\geq 2$.

_Case $u=1,v=2$:_ In this case, we get no chain of inequalities between the
algebraic numbers $\upalpha_{0},\upalpha_{1},\upalpha_{2}$, but as it turns
out, we won’t need it.

We write
$\log\left|\frac{\upalpha_{u}}{\upalpha_{v}}\right|=\log\left|\frac{\upalpha_{1}}{\upalpha_{2}}\right|=\left(-s-t\right)\log\uplambda_{0}+\left(-2s+t\right)\log\left|\uplambda_{2}\right|$
and plug this into our linear form $\Uplambda^{\prime}$. We again first argue
that $a-2b-s-t=0$ and then do the same for the coefficient of
$\log\left|\uplambda_{2}\right|$, i.e. $2a-b-2s+t=0$. We deduce $a=s-t,b=-t$.

Choosing again to approximate $\log\left|y\right|$ by
$\log\left|\upbeta_{2}\right|-\log\left|\upalpha_{0}-\upalpha_{2}\right|$, we
have
$\log\left|\upalpha_{0}-\upalpha_{2}\right|\approx\log\left|\upalpha_{2}\right|$
and thus

$\displaystyle\log\left|y\right|$
$\displaystyle=\log\left|\upbeta_{2}\right|-\log\left|\upalpha_{0}-\upalpha_{2}\right|+O\left(n^{-\frac{c_{1}}{2}\uptau}\right)$
$\displaystyle=\log\left|\upbeta_{2}\right|-\log\left|\upalpha_{2}\right|+O\left(n^{-\frac{c_{1}}{2}\uptau}\right)=\underbrace{\left(-b-t\right)}_{=0}\log\uplambda_{0}+\underbrace{\left(a-b-s\right)}_{=0}\log\uplambda_{2}+O\left(n^{-\frac{c_{1}}{2}\uptau}\right)$
$\displaystyle=O\left(n^{-\frac{c_{1}}{2}\uptau}\right).$

Thus $\log\left|y\right|$ is entirely described by the error
$O\left(n^{-\frac{c_{1}}{2}\uptau}\right)$ that we made. But if
$n\geq\upkappa$ is sufficiently large, this error is bounded by any constant,
say, $\log 2$, and we can thus deduce $\log\left|y\right|<\log 2$, which
contradicts $\left|y\right|\geq 2$.

The other cases, where $j=1$ or $j=2$, follow analogously. The sub-cases for
$u,v$ depend on the particular $j$ and choice for $(k,l)$. ∎

By this Lemma 10, we have deduced an effectively computable constant
$\upkappa$, which depends implicitly only on $\upvarepsilon$ (since the
constant $c_{1}$ in Lemma 2 does), such that $n<\upkappa$. Plugging this into
Lemma 9 gives fixed upper bound for $\uptau$, and into Inequality (16) a bound
for $\log\left|y\right|$, and thus $\left|y\right|$. And if everything else is
bounded, so must $\left|x\right|$. By abuse of notation, we shall call
$\upkappa$ the bound that holds for all these parameters, thus concluding the
proof of Theorem 1.

## References

  * [Bak68] A. Baker  “Contributions to the theory of diophantine equations. I: On the representation of integers by binary forms”  In _Philos. Trans. R. Soc. Lond., Ser. A_ 263 The Royal Society of London, London, 1968, pp. 173–191  DOI: 10.1098/rsta.1968.0010
  * [BW93] A. Baker and G. Wüstholz  “Logarithmic forms and group varieties”  In _J. Reine Angew. Math._ 442 De Gruyter, Berlin, 1993, pp. 19–62  DOI: 10.1515/crll.1993.442.19
  * [Bom82] E. Bombieri  “On the Thue-Siegel-Dyson theorem”  In _Acta Math_ 148, 1982, pp. 255–296  DOI: 10.1007/BF02392731
  * [BG96] Yann Bugeaud and Kálmán Győry  “Bounds for the solutions of Thue-Mahler equations and norm form equations”  In _Acta Arith._ 74.3 Polish Academy of Sciences (Polska Akademia Nauk - PAN), Institute of Mathematics (Instytut Matematyczny), Warsaw, 1996, pp. 273–292  DOI: 10.4064/aa-74-3-273-292
  * [LPV99] Günter Lettl, Attila Pethő and Paul Voutier  “Simple Families of Thue Inequalities”  In _Transactions of the American Mathematical Society_ 351.5 American Mathematical Society, 1999, pp. 1871–1894  URL: http://www.jstor.org/stable/117836
  * [LW15] Claude Levesque and Michel Waldschmidt  “A family of Thue equations involving powers of units of the simplest cubic fields”  In _J. Théor. Nombres Bordx._ 27.2, 2015, pp. 537–563  DOI: 10.5802/jtnb.913
  * [Smy08] Chris Smyth  “The Mahler measure of algebraic numbers: a survey”  In _Number Theory and Polynomials_ , London Mathematical Society Lecture Note Series  Cambridge University Press, 2008, pp. 322–349  DOI: 10.1017/CBO9780511721274.021
  * [Tho79] Emery Thomas  “Fundamental units for orders in certain cubic number fields”  In _J. Reine Angew. Math._ 310 De Gruyter, Berlin, 1979, pp. 33–55
  * [Tho90] Emery Thomas  “Complete solutions to a family of cubic Diophantine equations”  In _J. Number Theory_ 34.2, 1990, pp. 235–250  DOI: 10.1016/0022-314X(90)90154-J


We fix $n \in \N$ large enough so that $\lambda_\mu \geq 5/n$ (recall that the Lyapunov exponent $\lambda_\mu$ is  positive, a well-known result of Furstenberg), fix $\varepsilon_1$, $\varphi$ and $\omega$ as in the statement. We will estimate the left-hand side of (<ref>) separately for $\sigma_\chi(g,\z )$ inside three different intervals $[(\lambda_\mu-\eps_1)n,\infty)$, $[\eps_1n,(\lambda_\mu-\eps_1)n]$ and $(-\infty,\eps_1n]$. The second interval will give us the main term, other intervals will yield the error term. Take a smooth cutoff $\chi$ which equals $1$ on $[\eps_1n,(\lambda_\mu -\eps_1)n]$ and equals $0$ outside of $[\eps_1n-1,(\lambda_\mu -\eps_1)n+1]$ so that we have $\mathds{1}-\chi\leq\mathds{1}_{s<\eps_1n}+\mathds{1}_{s>(\lambda_\mu-\eps_1)n} $. Then, we can write

\begin{equation}\label{eq:split}
\begin{split}
&\left|\frac{1}{n}\sum_{k=1}^n\int \varphi(g\z ,\sigma_\chi(g,\z ))\ d\mu^{*k}(g)-\frac{1}{n}\sum_{k=1}^n\int \varphi(g\z ,\sigma_\chi(g,\z ))\chi(\sigma_\chi(g,\z ))\ d\mu^{*k}(g)\right|\\
\leq & \frac{1}{n}\left|\sum_{k=1}^n\int \varphi(g\z ,\sigma_\chi(g,\z ))\mathds{1}_{\sigma_\chi(g,\z )<\eps_1n}\ d\mu^{*k}(g)\right|\\
&+\frac{1}{n}\left|\sum_{k=1}^n\int \varphi(g\z ,\sigma_\chi(g,\z ))\mathds{1}_{\sigma_\chi(g,\z )>(\lambda_\mu-\eps_1)n}\ d\mu^{*k}(g)\right|.
\end{split}
\end{equation}

Main term:
Let $t=n\lambda_\mu$ and $f(\z ,s)=\varphi(\z ,s+t)\chi(s+t)$. Then by (<ref>)
\begin{equation}\label{eq:main-renewal}
\begin{split}
&\frac{1}{n}\sum_{k=1}^\infty\int \varphi(g\z ,\sigma_\chi(g,\z ))\chi(\sigma_\chi(g,\z ))\ d\mu^{*k}(g)\\
%	&=\frac{1}{n}(\sum_{k=1}^n\int f(g\z ,\sigma_\chi(g,\z )-t)\chi(\sigma_\chi(g,\z ))\mu^{*k}(g))\\
=&\frac{1}{n}\sum_{k=1}^\infty\int f(g\z ,\sigma_\chi(g,\z )-t)\mu^{*k}(g)\\
=&\frac{1}{t}\int_{\bbS}\int_{-t}^\infty f\ dLeb\ d\nu_\z +\frac{1}{n}O\big(e^{-\eta(t-|\Supp f|)}|f|_{W^{1,2}C^\gamma}\big),
\end{split}
\end{equation}

where in the error term, we have $t-|\Supp f|=t-(t-\eps_1 n)=\eps_1n$. For the main term, using the formula of $f$, we have

\begin{align*}
&\frac{1}{n\lambda_\mu}\int_{\bbS}\int_0^\infty \varphi(y,s)\chi(s)\ dLeb(s)d\nu_\z(y)\\
=&\frac{1}{n\lambda_\mu}\int_{\bbS}\int_{\eps_1n}^{(\lambda_\mu-\eps_1)n} \varphi(y,s)\chi(s)\ dLeb(s)d\nu_\z(y) +|\varphi|_\infty\frac{2}{n\lambda_\mu}\\
=&\frac{1}{t}\int_{\bbS}\int_0^{t}\varphi(y,s)\ dLeb(s)\ d\nu_\z (y)+|\varphi|_\infty O(\eps_1+\frac{1}{n}).
\end{align*}

For the error term in (<ref>), we have

\[ \frac{1}{n}|f|_{W^{1,2}C^\gamma}\leq \sup_s\{|\varphi|_{C^\gamma}\}+\sup_s\{ |\partial_{ss}\varphi|_{C^\gamma}\}\leq |\varphi|_{C^3}.\]

Now, we give an upper bound of the sum over  ${k> n}$:

\begin{align*} &\frac{1}{n}\sum_{k>n}^\infty\int \varphi(g\z ,\sigma_\chi(g,\z ))\chi(\sigma_\chi(g,\z ))d\mu^{*k}(g) \\
\leq & |\varphi|_\infty \frac{1}{n}\sum_{k>n}^\infty\mu^{*k}(\{g,\ \sigma_\chi(g,\z )<(\lambda_\mu-\eps_1)n+1\}).
\end{align*}

Due to the assumption $\eps_1 n\geq 2$, we obtain

\[\sigma_\chi(g,\z )-\lambda_\mu n\leq -\eps_1n+1\leq -\eps_1n/2. \]

We use the large deviation estimate (Theorem <ref>) to obtain

\begin{equation*}
\frac{1}{n}\sum_{k>n}^\infty\mu^{*k}(\{g : \sigma_\chi(g,\z )<(\lambda_\mu-\eps_1)n+1\})\leq \frac{1}{n}\sum_{k>n}Ce^{-ck}=\frac{Ce^{-cn}}{n(1-e^{-c})},
\end{equation*}

where the constants $C,c$ depend on $\eps_1$.

Collecting above estimates, we obtain

\begin{equation}\label{eq:second}
\begin{split}
\frac{1}{n}\sum_{k=1}^n\int \varphi(g\z ,\sigma_\chi(g,\z ))\chi(\sigma_\chi(g,\z ))d\mu^{*k}(g)=&\frac{1}{t}\int_{\bbS}\int_0^t \varphi(y,s)\ dLeb(s)\ d\nu_\z (y)\\
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +O\left(e^{-\eta\eps_1 n}|\varphi|_{C^3}+|\varphi|_\infty  \left(\eps_1+\frac{1}{n}+\frac{Ce^{-cn}}{n(1-e^{-c})}\right)\right).
\end{split}
\end{equation}

Error term I:
For $k<n_0:=\frac{\lambda_\mu-\eps_1}{\lambda_\mu+\eps_1}n$, we have $k(\lambda_\mu+\eps_1)<(\lambda_\mu-\eps_1)n$.
By the large deviation estimates (Theorem <ref>), we have

\begin{align*}
&\frac{1}{n}\sum_{k=1}^{n_0}\int \varphi(g\z ,\sigma_\chi(g,\z ))\mathds{1}_{\sigma_\chi(g,\z )\geq (\lambda_\mu-\eps_1)n}d\mu^{*k}(g)\\
\leq&  |\varphi|_\infty \frac{1}{n}\sum_{k=1}^{n_0}\mu^{*k}(\{g\in G : \sigma_\chi(g,\z )>(\lambda_\mu+\eps_1)k \})\\
\leq & |\varphi|_\infty C\frac{1}{n}\sum_{k=1}^{n_0} e^{-ck}\leq \frac{|\varphi|_\infty C}{n(1-e^{-c})}.
\end{align*}

For the part $n_0\leq k\leq n$, we use the absolute value to bound

\[\frac{1}{n}\sum_{k=n_0}^{n}\int \varphi(g\z ,\sigma_\chi(g,\z ))\mathds{1}_{\sigma_\chi(g,\z )\geq (\lambda_\mu-\eps_1)n}d\mu^{*k}(g)\leq |\varphi|_\infty\frac{n-n_0}{n}=|\varphi|_\infty\frac{2\eps_1}{\lambda_\mu+\eps_1}. \]

Thus, we have

\begin{equation}\label{eq:first}
\frac{1}{n}\sum_{k=1}^{n}\int \varphi(g\z ,\sigma_\chi(g,\z ))\mathds{1}_{\sigma_\chi(g,\z )\geq (\lambda_\mu-\eps_1)n}d\mu^{*k}(g)\leq  \frac{|\varphi|_\infty C}{n(1-e^{-c})}+|\varphi|_\infty\frac{2\eps_1}{\lambda_\mu+\eps_1}.
\end{equation}

Error term II: If $k>n_1:=\eps_1n/(\lambda_\mu-\eps_1)$, then we have $\eps_1n<k(\lambda_\mu-\eps_1)$ and hence we can apply the large deviation estimate to obtain

\begin{align*}
&\frac{1}{n}\sum_{k=n_1}^n\int \varphi(g\z ,\sigma_\chi(g,\z ))\mathds{1}_{\sigma_\chi(g,\z )\leq \eps_1 n}d\mu^{*k}(g)\\
\leq & |\varphi|_\infty \frac{1}{n}\sum_{k=n_1}^n\mu^{*k}(\{g\in G : \sigma_\chi(g,\z )<(\lambda_\mu-\eps_1)k \})\leq \frac{C|\varphi|_\infty e^{-cn_1}}{n(1-e^{-c})}.
\end{align*}

For the part $k\leq n_1$,

\begin{align*}
\frac{1}{n}\sum_{k=1}^{n_1}\int \varphi(g\z ,\sigma_\chi(g,\z ))\mathds{1}_{\sigma_\chi(g,\z )\leq \eps_1 n}d\mu^{*k}(g)\leq|\varphi|_\infty\frac{n_1}{n}=|\varphi|_\infty\frac{\eps_1}{\lambda_\mu-\eps_1}.
\end{align*}

Thus, we have

\begin{equation}\label{eq:thrid}
\frac{1}{n}\sum_{k=1}^{n}\int \varphi(g\z ,\sigma_\chi(g,\z ))\mathds{1}_{\sigma_\chi(g,\z )\leq \eps_1 n}d\mu^{*k}(g)\leq|\varphi|_\infty\frac{\eps_1}{\lambda_\mu-\eps_1}+\frac{C|\varphi|_\infty e^{-cn_1}}{n(1-e^{-c})}.
\end{equation}

Finally, combining (<ref>), (<ref>), (<ref>) and (<ref>), we obtain

\begin{align*}
\frac{1}{n}\sum_{k=1}^\infty\int \varphi(g\z ,\sigma_\chi(g,\z ))\ d\mu^{*k}(g)=\frac{1}{t}\int_0^t \varphi(y,s)\ dLeb(s)\ d\nu_\z (y)\\
+O\left(e^{-\eta\eps_1 n}|\varphi|_{C^3}+|\varphi|_\infty\frac{\eps_1}{\lambda_\mu-\eps_1}+\frac{C|\varphi|_\infty}{n(1-e^{-c})}\right).
\end{align*}

§.§.§ Equidistribution on $X_\calC$

We now use the equidistribution on $K\times \R$ (Proposition <ref>) to deduce the equidistribution on $X_\calC$, that is, to give

Let $K\times_\sigma D$ be the fiber bundle with $H$ action, the action of $H$ is given by $h(k,d)=(hk,\sigma(h,k)d)$, where we identify $K \simeq \mathbb{S}^1 \simeq H/AN$. We define a map $p$ from $K\times_\sigma D$ to $H/Q_H\times_\alpha D^\pm$ by

\[ p(k,d)=(kM,\sg(k)d), \]

where $\sg(k)$ is the sign element in $M$. By Lemma <ref>, we have

The map $p$ is an $H$-equivariant map from $K\times_\sigma D$ to $H/Q_H\times_\alpha D^\pm$.

We denote by $\calG(r,\sg(w))$ the element $(e^r,\sg(w))\in D^\pm$ and we use additive parameter $r\in \R$. Under this parametrization, the equidistribution of $D$ or $D^\pm$-orbits of $z \in S/\Lambda$ to some measure $m$ means, respectively, that the measure $\frac{1}{t}\int_0^t\delta_{\calG(r,1)z}\ dr$ or the measure $\frac{1}{2t}\int_0^t\delta_{\calG(r,1)z}+\delta_{\calG(r,-1)z}\ dr$ converges to $m$ as $t \to \infty$.

Let $\psi$ be $C^3$ function on $X_\mathcal{C} \simeq H/Q_H \times S /\Lambda$ and $z \in S/\Lambda$. Set $\varphi(w,r):=\psi(\eta,\calG(r,\sg(w))z)$ where $\eta$ is the projection of $w$ on $H/Q_H \simeq K/M$. Thanks to Lemma <ref>, we have the relation

\begin{equation}
\varphi(g s(\eta),\sigma_\chi(g,\eta))=\psi(g\eta,\alpha(g,\eta)z)=\psi(gx),
\end{equation}

for $x=(\eta,z) \in X_\mathcal{C}$ and where, we recall, $s:K/M \to K$ is the section. Therefore, we have

\begin{equation}\label{eq.corresp.phi.psi}
\frac{1}{n}\sum_{1\leq k\leq n}\mu^{*k}*\delta_x(\psi)=\frac{1}{n}\sum_{k=1}^n\int \varphi(g s(\eta) ,\sigma_\chi(g,\eta))\ d\mu^{*k}(g).
\end{equation}

We know that for any $\psi\in C^3(X_\calC)$ with bounded $C^3$ norm, with suitable choice of $\eps_1$ depending on $n$, thanks to Proposition <ref> and the relation (<ref>), for $t=\lambda_\mu n$, we have

\[ \frac{1}{n}\sum_{1\leq k\leq n}\mu^{*k}*\delta_x(\psi)- \frac{1}{t}\int_{\bbS}\int_0^t\varphi(k',r)\ dr\ d\nu_{s(\eta)}(k')\rightarrow 0 \]

as $n$, equivalently $t$, tends to $\infty$.

On the other hand, by construction of the function $\varphi(\cdot,\cdot)$ and the measures $\nu_w$ for $w \in \mathbb{S}^1$, we have

\begin{align*}
\int_{\mathbb{S}^1} \varphi(k',r)\ d\nu_w(k')=& p_1(w)\int_{\bbS} \psi(k'M,\calG(r,\sg(k'))z)\ d\nu_1(k')\\
&+p_2(w)\int_{\mathbb{S}^1} \psi(k'M,\calG(r,\sg(k'))z) d\nu_2(k')\\
=&p_1(w)\int \psi(\eta,\calG(r,1)z)\ d\overline{\nu}_F(\eta)+p_2(w)\int \psi(\eta,\calG(r,-1)z)\ d\overline{\nu}_F(\eta).
\end{align*}

We get that

\begin{equation*}
\frac{1}{n}\sum_{1\leq k\leq n}\mu^{*k}*\delta_x(\psi)- \frac{1}{t}\int \int_0^t p_1(s(\eta))\psi(\eta,\calG(r,1)z)+p_2(s(\eta))\psi(\eta,\calG(r,-1)z)\ dr\ d \overline{\nu}_F(\eta)\rightarrow 0.
\end{equation*}

Recall that for Case 2.2.a, the section $s$ is chosen so that its image contains the support of $\nu_1$. In particular, $p_1(s(\eta))=1$ and $p_2(s(\eta))=0$ for every $\eta$ in the support of the Furstenberg measure $\overline{\nu}_F$. Therefore, in Case 2.2.a, if the $\eta$ coordinate of $x=(\eta,z)$ belongs to the support of $\overline{\nu}_F$, then we have

\begin{equation}\label{eq.conv1}
\frac{1}{n}\sum_{1\leq k\leq n}\mu^{*k}*\delta_x(\psi)- \frac{1}{t}\int \int_0^t \psi(\eta,\calG(r,1)z)\ dr\ d\bar\nu_F(\eta)\rightarrow 0.
\end{equation}

In Case 2.2.b, by a similar computation and using the fact that the unique measure $\nu_K$ on $\mathbb{S}^1$ writes as $\nu_K=\frac{1}{2} \int  (\delta_w + \delta_{-w}) d\overline{\nu}_F(\R w)$, we have

\begin{equation}\label{eq.conv2}
\frac{1}{n}\sum_{1\leq k\leq n}\mu^{*k}*\delta_x(\psi)- \frac{1}{2t}\int \int_0^t (\psi(\eta,\calG(r,1)z)+\psi(\eta,\calG(r,-1)z)) \ dr\ d\overline{\nu}_F(\eta)\rightarrow 0.
\end{equation}

By density of $C^3(X_\calC)$ in $C(X_\calC)$, we deduce from (<ref>) and (<ref>) that $ \frac{1}{n}\sum_{1\leq k\leq n}\mu^{*k}*\delta_x$ converges weakly to a measure $\bar\nu_F\otimes m$ if and only if the $D$-orbit or the $D^{\pm}$- orbit (respectively in Case 2.2.a or Case 2.2.b) starting at $z \in S/\Lambda$ equidistributes to the measure $m$.

For $\Gamma_\mu< \PSL_2(\R)\simeq \PGL_2(\R)^o$, we take the lift $\mu_1$ on $\SL_2(\R)$ of $\mu$ on $\PSL_2(\R)$ with equal probability on two preimages of each element. Then we can apply Theorem <ref> to this new measure $\mu_1$. Here the element $\diag(-1,-1)$ (which is the non-trivial element of $M$ in the case of $H \simeq \SL_2(\R)$) maps to identity in $G$, which acts trivially. Then the same argument as in the proof of Theorem <ref> readily yields Theorem <ref>.

[1]
Aoun, R. and Sert, C. Random walks on hyperbolic spaces: Concentration inequalities and probabilistic Tits alternative. Probab. Theory Relat. Fields vol: 184, pages 323–365 (2022).

[2] Bénard T. and de Saxcé, N.
Random walks with bounded first moment on finite-volume spaces.
Geom. Func. Anal. 32, no. 4 (2022): 687–724.

[3] Benoist, Y. and Quint, J.-F.
Mesures stationnaires et fermés invariants des espaces homogènes.
Ann. Math. 174 (2011), no. 2, 1111–1162.

[4]
Benoist, Y. and Quint, J.-F.
Stationary measures and invariant subsets of homogeneous spaces (II).
J. Amer. Math. Soc. 26 (2013), no. 3, 659–734.

[5]
Benoist, Y. and Quint J.-F.
Random walks on finite volume homogeneous spaces.
Invent. Math. 187 (2012), no. 1, 37–59.

[6] Benoist, Y. and Quint, J.-F.
Stationary measures and invariant subsets of homogeneous spaces (III).
Ann. Math. 178 (2013), no. 3, 1017–1059.

[7] Benoist, Y. and Quint, J.-F.
Random walks on projective spaces.
Compos. Math. 150 (2014), no. 9, 1579–1606.

[8] Benoist, Y. and Quint, J-F.
Random walks on reductive groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Vol 62. Springer, Cham, 2016.

[9] Bougerol, Ph. B. and Lacroix, J.
Products of random matrices with applications to Schrödinger operators.
Vol. 8. Springer Science & Business Media, 2012.

[12]
Eskin, A. and Lindenstrauss E.
Zariski dense random walks on homogeneous spaces.
Preprint on webpage at <https://www.math.uchicago.edu/ eskin/RandomWalks/short_paper.pdf>.

[13] Eskin, A. and Lindenstrauss, E.
Random walks on locally homogeneous spaces. Preprint on webpage at <http://www.math.uchicago.edu/ eskin/RandomWalks/paper.pdf>.

[14] Eskin, A. and Mirzakhani, M.
Invariant and stationary measures for the action on moduli space.
Publ. Math. Inst. Hautes Études Sci. 127 (2018), 95–324.

[15]  Furstenberg, H.
A Poisson formula for semi-simple Lie groups.
Ann. Math. 77 (1963), 335–386.

[16]  Furstenberg, H.
Noncommuting random products.
Trans. Amer. Math. Soc. 108 (1963), 377–428.

[17] Furstenberg, H.
Boundary theory and stochastic processes on homogeneous spaces.
Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), pp. 193–229. Amer. Math. Soc., Providence, R.I., 1973.

[18]
Guivarc’h, Y. and  Raugi, A.
Actions of large semigroups and random walks on isometric extensions of boundaries.
Ann. Sci. École Norm. Sup. 40 (2007), no. 2, 209–249.

[20]  Goldsheid, I. and  Margulis, G.
Lyapunov Indices of a Product of Random Matrices.
Russian Math. Surveys 44 (1989), no. 5, 11–71

[21]  Guivarc’h, Y. and  Le Page, É.
Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions.
Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 2, 503–574.

[22] Guivarc’h, Y. and Raugi, A.
Frontière de Furstenberg, propriéetés de contraction et théeorèmes de convergence.
Z. Wahrsch. Verw. Gebiete 69 (1985), no. 2, 187–242.

[23] Li, J.
Fourier decay, Renewal theorem and Spectral gaps for random walks on split semisimple Lie groups.
To appear in Ann. Sci. École Norm. Sup.

[24]  Li, J. and Sahlsten, T.
Fourier transform of self-affine measures.
Adv. Math. 374 (2020), 107349, 35 pp.

[25] Knapp, A. W.
Lie groups beyond an introduction.
Progress in Mathematics, 140. Birkhäuser Boston, Inc., Boston, MA, 2002.

[26] Kostant, B. The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math., 81 (1959): 973–1032.

[27] Mackey, G. W.
Imprimitivity for representations of locally compact groups I.
Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 537–545.

[28] Mackey, G. W.
Unitary representations of group extensions. I.
Acta Math. 99 (1958), 265–311.

[29] Neil, C. and Ginzburg, V.
Representation theory and complex geometry.
Vol. 42. Boston: Birkhäuser, 1997.

[34] Prohaska, R., Sert, C. and Shi, R.
Expanding measures: Random walks and rigidity on homogeneous spaces.

[35]
Ratner, M.
On measure rigidity of unipotent subgroups of semisimple groups.
Acta Math. 165 (1990), no. 3–4, 229–309.

[36]
Ratner, M.
Raghunathan’s topological conjecture and distributions of unipotent flows.
Duke Math. J. 63 (1991), no. 1, 235–280.

[39]
Sargent, O. and Shapira, U.
Dynamics on the space of 2-lattices in 3-space.
Geom. Funct. Anal. 29 (2019), no. 3, 890–948.

[41] Varadarajan, V. S.
Geometry of quantum theory.
The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968.

[43] Zimmer, R. J.
Ergodic theory and semisimple groups.
Vol. 81. Springer Science $\&$ Business Media, 2013.