# Explicit solutions to the Oppenheim conjecture for indefinite ternary
diagonal forms

Youssef Lazar<EMAIL_ADDRESS>
###### Abstract.

We give a new proof of the Oppenheim conjecture for indefinite ternary
diagonal forms of the type $x^{2}+y^{2}-\alpha z^{2}$ where $\alpha$ is an
irrational number. Our method is explicit in the sense that we are able to
construct a solution to the problem and we obtain an effective bound on the
solution. The method is geometrical and is based on continued fractions.

##  1\. Introduction

We are interested in the following diophantine problem, given any real number
$\varepsilon>0$ and a positive irrational number $\alpha$, is there exists a
nonzero vector $(x,y,z)\in\mathbb{Z}^{3}$ such that

$|x^{2}+y^{2}-\alpha z^{2}|\leq\varepsilon.$

This apparently simple question found a solution only in the mid-eighties
thanks to G.A. Margulis which proved that the answer is positive. In fact,
Margulis showed in [Mar89] a much more general statement which encompasses all
indefinite quadratic forms in $n\geq 3$ variables, provided they are not
proportional to a rational one. This result was conjectured by Oppenheim in
1929 [Opp29] and remained open in full generality until Margulis’
breakthrough. The Oppenheim conjecture reduces to the three dimensional case
which is strangely the most difficult case. The strategy of the proof used by
Margulis, was to solve a particular case of another conjecture due to M.S.
Raghunathan. The resolution of the Oppenheim conjecture is a consequence of
Margulis’ proof of the Raghunathan conjecture in the case $n=3$. Few time
later, Ratner’s proved the Raghunathan conjecture in full generality for any
connected Lie group [R90]. These results were the starting point of a
tremendous amount of activity around which is now called homogeneous dynamics.
This point of view shows to be very fruitful in order to treat various
unsolved problems especially in diophantine approximation. The litterature
about this conjecture and others related questions is abundent. The interested
reader may find most of the main contributions on this conjecture in Margulis’
survey [Mar03] which is by far the most complete.
A natural question is whether the Oppenheim conjecture could be proved with
another method, namely without using homogeneous dynamics. As far as we know
the answer is negative for $n=3$, unless for a very specific case due to
Watson which we will discuss later on. The most powerful method to solving
diophantine inequalities is the Circle method but it requires a large number
of variables compared to the degree of the polynomial involved. In the early
times of the conjecture, Davenport and Heilbronn succeeded to prove the
Oppenheim conjecture for irrational diagonal forms in five variables by using
a variant of the Circle method [DH46]. Their proof has the advantage to be
effective. The same result was proved earlier by Chowla for $n\geq 9$ using
lattice points counting in irrational ellipsoids. The barrier $n=4$ has been
breached by Oppenheim itself in its seminal paper [Opp29] using some old
results of Korkine and Zolotareff on representation of definite forms [KZ72].
In the late seventies, Iwaniec [Iw77] proved the Oppenheim conjecture for some
quaternary diagonal forms using sieve theory. A last attempt to prove the
conjecture was due to R.K. Baker and H.P. Schlickewei who proved the
conjecture in full generality for $n\geq 21$ [BK87]. For quadratic forms, i.e.
in degree 2, it seems that the circle method can only operate if $n\geq 5$.
Using the full power of analytic methods combined with geometry of numbers, an
effective version of the Oppenheim conjecture was proved very recently for
$n\geq 5$ by P. Buterus, F. Götze, T. Hille and G.A Margulis [BGHM]. This
results have been sharpened by P. Buterus, F. Götze, T. Hille in [BGH] for
diagonal forms extending Birch-Davenport method to dimensions at least five
combined with a result of Schlickewei. The latter proofs are quite involved
and very technical.

The three dimensional case.
It is noteworthy to mention the difficulty of the problem for $n=3$. The case
of forms $Q_{\alpha}(x,y,z)=x^{2}+y^{2}-\alpha z^{2}$ we are concerned with
shows a curious behaviour. Indeed it has been remarked by Eskin Margulis and
Mozes ([EMM98], Theorem 2.2.) that $Q_{\alpha}(\mathbb{Z}^{3})$ fails to be
equidistributed for a dense set of values of $\alpha$. This contrasts with the
analog in higher dimension, in the same paper it is proved that the set
$Q(\mathbb{Z}^{3})$ is equistributed in the real line given any form $Q$ of
signature $(p,q)\neq(2,1)$ or $(2,2)$ which satisfy the assumptions of the
Oppenheim conjecture.
For a very specific class of quadratics forms, Watson [Wat46] gave an explicit
proof of the Oppenheim conjecture by showing how to construct the solution and
therefore providing bounds for the solution.

Watson considered quadratic forms of the type $Q(x,y,z)=x^{2}-a\alpha
y^{2}-\alpha^{2}z^{2}$ where $a$ is a positive integer and $\alpha$ is an
irrational number with continued fraction representation
$[a;a,\ldots]=[a;\overline{a}]$. When $a\geqslant 2$ such numbers are
sometimes called silver means, in analogy with the case $a=1$ which is just
the golden ratio. The convergents of such numbers satisfies very a simple
reccurence relation, if $c_{n}=p_{n}/q_{n}$ is the $n^{th}$ convergent of
$\alpha$ then $q_{n}=p_{n-1}$. For each integer $n>0$, let us set

$x_{n}=q_{n+1}$, $y_{n}=q_{n}$ and $z_{n}=q_{n-1}$.

By means of easy manipulations Watson showed that

$\left|x_{n}^{2}-a\alpha
y_{n}^{2}-\alpha^{2}z_{n}^{2}\right|\leq\dfrac{\alpha+\overline{\alpha}}{q_{n}q_{n-1}B_{n}B_{n-1}}$

where $\overline{\alpha}$ is the algebraic conjugate of $\alpha$ and
$B_{n}=|\overline{\alpha}-p_{n}/q_{n}|$. Since $\alpha$ has bounded partial
quotients, in fact all equal to $a$, the $B_{n}$’s are bounded. Thus,

$\left|x_{n}^{2}-a\alpha
y_{n}^{2}-\alpha^{2}z_{n}^{2}\right|\ll_{n}\dfrac{1}{q_{n}q_{n-1}}.$

Let us choose an arbitrary $\varepsilon>0$, then if $n$ is taken large enough
in order to fullfill the inequality

$\dfrac{1}{q_{n}q_{n-1}}\leq\dfrac{1}{q_{n-1}^{2}}\leq\varepsilon.$

This ensures that $v_{n}=(q_{n+1},q_{n},q_{n-1})$ solves the Oppenheim
conjecture for $n$ as above. Note that this gives an asymptotic sequence of
solutions not only one solution.
A bound for the solution $v_{n}$ depends on the least integer $n_{1}$ such
that $z_{n_{1}}=q_{n_{1}-1}=\dfrac{1}{\sqrt{\varepsilon}}$. Thus for $n\geq
n_{1}$

$\|v_{n}\|_{\infty}=q_{n+1}\ll\dfrac{1}{\sqrt{\varepsilon}}.$ (1)

This result is quite exceptional among the bunch of results surrounding the
Oppenheim conjecture. In fact, it gives a computable solution and it is
effective in the sense that it gives a bound of the sequence
$F(N)=\displaystyle\min_{v\in\mathbb{Z}^{3},v\neq 0,\|v\|_{\infty}<N}|Q(v)|$.
As we have seen, by taking $N=\varepsilon^{-1/2}$, Watson’s result gives

$F(N)=\displaystyle\min_{v\in\mathbb{Z}^{3},v\neq 0,\|v\|_{\infty}<N}|Q(v)|\ll
N^{-2}.$

The problem of effectiveness in Margulis’ theorem amounts to finding optimal
bounds for $F(N)$. Although, Ratner’s theorems are not effective in general,
Lindenstrauss and Margulis [EL10] succeeded to overcome this issue by giving
upper bounds on $F(1/\varepsilon)$ of the form $e^{P(1/\varepsilon)}$ for some
polynomial $P$. Their deep result is valid for all indefinite forms in degree
three and is based on homogeneous dynamics. Shorty after Bourgain [Brg10] gave
optimal bounds for $F(N)$ for ternary diagonal forms. The works of Ghosh-
Gorodnik-Nevo [GGN20], Ghosh-Kelmer [GK18] and Athreya-Margulis [AM18] gave
closely related results for generic families of quadratic forms.
The bound provided in Watson’s result is outstanding, in the sense that, as
far as we know, this is the best known bound for an individual quadratic form.
Indeed, one of the output of Bourgain’s result predicts that, under Lindelöf
hypothesis for the Riemann Zeta function the best bound one can hope for a
generic form is $F(N)\ll N^{-1+o(1)}$. Watson’s peculiar example improves it
by a factor $N^{-1}$. Note a slight difference with Bourgain, indeed he
considered forms of the type
$Q(x,y,z)=x^{2}_{1}+\alpha_{2}x_{2}^{2}-\alpha_{3}x_{3}^{2}$ with
$\alpha_{2},\alpha_{3}>0$ whereas Watson’s example is of the form
$Q(x,y,z)=x^{2}_{1}-\alpha_{2}x_{2}^{2}-\alpha_{3}x_{3}^{2}$ with
$\alpha_{2},\alpha_{3}>0$. Be that as it may, Watson’s result is the best
result one can expect in solving diophantine inequalities of the form
$|Q(v)|\leq\varepsilon$.

Method | Variables(s) | Type | Quantative | Effective | Explicit
---|---|---|---|---|---
Homogeneous Dynamics | $n\geq 3$ | general | $\checkmark$ | $\checkmark$ | $\times$
Circle method | $n\geq 5$ | general | $\checkmark$ | $\checkmark$ | $\times$
Geometry of numbers |  $n=4$, $n\geq 9$ | diagonal | $\checkmark$ | $\checkmark$ | $\times$
Sieve Theory | $n=4$ | diagonal | $\times$ | $\times$ | $\times$
Continued Fractions | $n=3,4$ | diagonal | $\times$ | $\checkmark$ | $\checkmark$
Figure 1. Comparison of the different proofs of the Oppenheim conjecture.

#### The main results

The aim of the paper is to construct explicit solutions to the Oppenheim
conjecture for ternary forms of the type $Q_{\alpha}(x,y,z)=x^{2}+y^{2}-\alpha
z^{2}$ where $\alpha\notin\mathbb{Q}$. In turn, one is able to obtain
effective bounds on such solutions. The proof essentially relies on
diophantine properties of the irrational number $\beta=\sqrt{\alpha}$, more
precisely its measure of irrationality. The measure of irrationality of an
real number $\beta$ is defined as the least positive real number $\mu$ such
that for

$\mu(\beta)=\inf\\{\omega\ :\
\dfrac{C}{q^{\omega+\sigma}}<\left|\beta-\dfrac{p}{q}\right|$ for all rational
$p/q$ ($q>0$), every $\sigma>0$ and for some constant $C>0\\}$.

A deep theorem due to Roth states that $\mu(\beta)=2$ whenever $\beta$ is an
algebraic number. The converse is not true, indeed $\theta(e)=2$ whereas the
constant $e\approx 2.718$ is a transcendental number. There exists
transcendental numbers $x$ for which $\mu(x)=\infty$, these are termed
Liouville numbers.

Let us fix an arbitrary small parameter $\sigma>0$. For any irrational number
$\beta$ which is not a Liouville number, we define the following quantity

$\theta_{\sigma}(\beta):=\mu(\beta)-1+\sigma.$

In any case, one has $\theta>1$ and the following diophantine condition holds
for any irrational $\beta$

$\inf_{q\geq 1}q^{\theta}\langle q\beta\rangle>0$ (2)

where $\langle x\rangle$ denotes the distance of a real number $x$ to the
nearest integer. We are going to give an explicit proof of the Oppenheim
conjecture for quadratics forms of the type
$Q_{\alpha}(x,y,z)=x^{2}+y^{2}-\alpha z^{2}$ where $\alpha$ is an irrational
number. The convergents of $\beta$ are simply denoted
$\textbf{c}_{n}=\textbf{p}_{n}/\mathbf{q}_{n}$ and $\theta$ stands for
$\theta_{\sigma}(\beta)$.

######  Theorem 1.1.

Given any real number $\varepsilon>0$ and a positive irrational number
$\alpha$. There exists a nonzero vector $v=(x,y,z)\in\mathbb{Z}^{3}$ such that

$|Q_{\alpha}(v)|\leq\varepsilon.$

Moreover, if $\beta$ is not a Liouville number then the solution satisfies

$\|v\|_{\infty}\ll\mathbf{q}_{2n_{1}}^{2/(\theta+1)}$

where $\mathbf{q}_{2n_{1}}$ is the denomimator of the convergent of order
$2n_{1}$ of $\beta$ with

$n_{1}(\varepsilon)=2+\lfloor\dfrac{\theta+1}{\theta-1}\left|\ln\left(\varepsilon\right)\right|/\ln
2\rfloor.$

We can extend the class of forms for which the conjecture is valid by
considering classes of forms equivalent to the type $Q_{\alpha}$ as given in
Theorem 1.1. Given a subgroup $N$ of $\mathrm{GL}(3,\mathbb{R})$, we say that
two quadratic forms $Q_{1}$ and $Q_{2}$ are $N$-equivalent if there exists a
$g\in N$ such that $Q_{1}(x)=Q_{2}(gx)$ for every $x\in\mathbb{R}^{3}$. Any
indefinite ternary form $Q$ is $\mathrm{SL}(3,\mathbb{R})$-equivalent to
$Q_{0}(x,y,z)=x^{2}+y^{2}-z^{2}$. Let us denote by $H$ the subgroup of
$\mathrm{GL}(3,\mathbb{R})$ defined by

$H=\left\\{\left[\begin{array}[]{c|c}A&0\\\ \hline\cr
0&h_{33}\end{array}\right]:A\in\mathrm{SL}(3,\mathbb{Q}),h_{33}\notin\mathbb{Q}\right\\}.$

From Theorem 1.1 we derive the following result.

######  Corollary 1.2.

  1. (1)

Suppose that $Q$ is an indefinite quadratic form which is
$\mathrm{SL}(3,\mathbb{\mathbb{Q}})$-equivalent to a $Q_{\alpha}$ with
$\alpha\notin\mathbb{Q}$. Then the Oppenheim conjecture holds for $Q$.

  2. (2)

Suppose that $Q$ is an indefinite quadratic form which is $H$-equivalent to
$Q_{0}$. Then the Oppenheim conjecture holds for $Q$. In particular if
$Q_{1}=f(x,y)-\beta^{2}z^{2}$ where $f(x,y)$ is a rational binary form and
$\beta\notin\mathbb{Q}$ then the conjecture holds for $Q_{1}$.

Remarks. (1) A great advantage of our method is that we know how to construct
the solution. As a byproduct we obtain an effective bound on the solution. The
quality of the bound depends on the value of $\theta$ and the growth of the
denominators $(\mathbf{q}_{n}(\beta))_{n\geqslant 1}$ of the convergents of
$\beta$.

(2) The idea of the proof is geometrical and relies on the following
observations. The line of equation $x=\beta z$ is a generatrix for the cone
$\\{Q_{\alpha}=0\\}$ restricted to the plane $y=0$. For every $\varepsilon>0$,
this line is inside the region $\\{-\varepsilon\leq
Q_{\alpha}\leq\varepsilon\\}$ and because $\beta$ is irrational, this line
cannot contain a nontrivial lattice point. Nevertheless, Dirichlet’s
approximation theorem tells us that there exists lattice points lying
arbitrarily near the line at any level of precision. Given any
$\varepsilon>0$, one expects that such lattice point lies in the region
$\\{|Q_{\alpha}|\leq\varepsilon\\}$. Unsurprinsingly we show in section §2
that Dirichlet’s theorem is not enough to prove the Oppenheim conjecture for
$Q_{\alpha}$. To overcome this problem we introduce a sequence of rational
lines which are nearly parralel to the line passing through a lattice point
$u_{n}=(x_{n},0,z_{n})$ given by Dirichlet’s given a certain order of
approximation $\mathbf{q}_{2n}^{-1}$, i.e.

$\mathrm{dist}(u_{n},\mathcal{L}_{\beta})\ll\dfrac{1}{\mathbf{q}_{2n}^{1-\eta}}$

where $1\leq z_{n}\leq\mathbf{q}_{2n}^{1-\eta}$ with
$\eta=\dfrac{\theta-1}{\theta+1}$. Given any $n$, we define the line
$\mathcal{L}_{\beta}^{n}$ by setting

$(\mathcal{L}_{\beta}^{n}):~{}u_{n}+\mathbb{R}(c_{2n}(\beta),\dfrac{1}{\mathbf{q}_{2n}(\beta)},1).$

A parametrization of this line for the downward direction is given by

$(\mathcal{L}_{\beta}^{n})^{+}~{}:~{}v_{n}(t)=(x_{n}-tc_{2n}(\beta),-\dfrac{t}{\mathbf{q}_{2n}(\beta)},z_{n}-t)~{}~{}(t\geq
0).$

The proposition 2.1 is going to show that for $n$ large enough the
parametrization $v_{n}(t)$ of the intersection
${\mathcal{L}_{\beta}^{n}}^{+}\cap\\{|Q_{\alpha}|\leq\varepsilon\\}$ is
supported by two disjoints intevals $I_{n}^{1}$ and $I_{n}^{2}$. Thus, in
order to have a lattice point in $\\{|Q_{\alpha}|\leq\varepsilon\\}$ it
suffices to find a multiple of $\mathbf{q}_{2n}$, say $t_{n}$, in the union of
$I_{n}^{1}$ and $I_{n}^{2}$. In this case, one can clear denominators and the
solution is given by
$v_{n}(t_{n})\in\mathbb{Z}^{3}\cap{\mathcal{L}_{\beta}^{n}}^{+}\cap\\{|Q_{\alpha}|\leq\varepsilon\\}$.
The key lemma 3.1 says that this is possible if $n$ is greater or equal than
some integer $n_{1}(\varepsilon)$ which is explicitely computable.
3) When $\beta=\sqrt{\alpha}$ is a Liouville number, we can easily prove that
the Oppenheim conjecture is satisfied in dimension $n=2$ for the form
$q(x,z)=x^{2}-\beta^{2}z^{2}$. Since we have $Q_{\alpha}(x,0,z)=q(x,z)$, then
the Oppenheim conjecture is satisfied for $Q_{\alpha}$.
4) In the case when $\beta$ is not a Liouville number, it is always possible
to find a real number $\theta>1$ large enough such that for every integer
$q\geqslant 1$,

$q^{\theta}\langle q\beta\rangle>0.$

The irrationality measure $\mu(\beta)$ is introduced only with the aim of
obtaining optimal bounds and to quantify the growth of the sequence
$\mathbf{q}_{n+1}(\beta)/\mathbf{q}_{n}(\beta)$. For instance the main theorem
gives an explicit integral solution for our favorite example of irrational
indefinite form $Q(x,y,z)=x^{2}+y^{2}-\sqrt{2}z^{2}$.

5) The corollary 1.2 shows that we can find a solution to the Oppenheim
problem for quadratic forms of the type

$Q(x,y,z)=ax^{2}+bxy+cy^{2}-\alpha z^{2}$

where $a,b,c\in\mathbb{Q}$ and $\alpha\notin\mathbb{Q}$. This is the best we
can do, and it would be interesting to find explicit solutions for general
indefinite irrational forms. For the general case, one would be led to use the
mutidimensional version of the Dirichlet’s approximation theorem. Using the
same kind of strategy applied to a product of linear forms instead of a
quadratic form, the author was able to derive a set of sufficent conditions
for the Littlewood conjecture to hold.

##  2\. Sequences of Rational Lines intersecting
$\\{|Q_{\alpha}|\leq\varepsilon\\}$

We focus our attention on forms of the type

$Q_{\alpha}(x,y,z)=x^{2}+y^{2}-\alpha z^{2}$

where $\alpha\in\mathbb{R}_{+}$. We assume that $\alpha$ is irrational and
therefore the form $Q_{\alpha}$ is an indefnite quadratic form which is not
proportional to a form which rational coefficients. The output of Margulis’s
result tells us that for every $\varepsilon>0$, there must exist a nonzero
lattice vector $v\in\mathbb{Z}^{3}$ such that

$0\leq|Q_{\alpha}(v)|\leq\varepsilon.$ (3)

We are going to reprove this result by constructing an explicit solution to
this problem, i.e. to find a nonzero vector in
$\mathcal{A}(\varepsilon)\cap\mathbb{Z}^{3}$ where he domain
$\mathcal{A}(\varepsilon)$ is delimited by the level sets
$\\{Q_{\alpha}=-\varepsilon\\}$ and $\\{Q=\varepsilon\\}$.
A parametization of the cone $\\{Q_{\alpha}=0\\}$ is as follows,

$\left\\{\begin{array}[]{cc}x(t,\theta)&=\sqrt{\alpha}~{}t\cos\theta\\\
y(t,\theta)&=\sqrt{\alpha}~{}t\sin\theta\\\
z(t,\theta)&=t.\end{array}\right.(0\leqslant\theta<2\pi).$ (4)

This parametrization shows that the cone $\\{Q_{\alpha}=0\\}$ is generated by
a continous family of lines given by
$\mathcal{L}_{\alpha}(\theta)=\mathbb{R}(\sqrt{\alpha}~{}\cos\theta,\sqrt{\alpha}~{}\sin\theta,1)$
where the angle $\theta$ varies in $[0,2\pi)$. The line corresponding to the
intersection of the $xz$-plane with the cone $\\{Q_{\alpha}=0\\}$ is exactly
given by $\mathcal{L}_{\alpha}(0)=\mathbb{R}(\sqrt{\alpha},0,1)$, we denote it
by $\mathcal{L}_{\beta}$ where $\beta=\sqrt{\alpha}$. An equation of this line
in the $xz$-plane is just $x=\beta z$. Since $\beta^{2}=\alpha$ is irrational,
$\beta$ itself is irrational too. Thus given any positive integer $N>1$ we
obtain from Dirichlet’s Theorem that there exists
$(p_{0},q_{0})\in\mathbb{N}^{2}$ with $1\leqslant q_{0}\leqslant N$ such that

$|p_{0}-\beta q_{0}|\leqslant\frac{1}{N}.$ (5)

This tells us that we can always find a lattice vector $(p_{0},0,q_{0})$
arbitrarily near the line of equation $x=\beta z$ in the $xy$-plane provided
$\beta$ is irrational.

###  2.1.  Irrationality Measures

We follow the notations of [H90], section 3.
For each real number, let $\langle x\rangle$ denote the distance of $x$ to the
closest integer. Dirichlet’s theorem says that $\inf_{q\geq 1}q\langle
q\beta\rangle<1$, the question is to know in which extend one can improve this
approximation. We can assign to $\beta$ a number called the irrationality
measure of $\beta$ which is defined as follows,

$\mu(\beta):=\inf\\{\omega\in\mathbb{R}_{+}\ :\ \inf_{q\geq
1}q^{\omega-1+\sigma}\langle q\beta\rangle>0$ for every real $\sigma>0\\}$.

In other words,

$\mu(\beta)=\inf\\{\omega\ :\
\dfrac{C}{q^{\omega+\sigma}}<\left|\beta-\dfrac{p}{q}\right|$ for all rational
$p/q$ ($q>0$), every $\sigma>0$ and for some constant $C>0\\}$

An alternative definition of $\mu$
Suppose $\beta$ has an infinite continued fraction expansion
$\beta=[b_{0};b_{1},b_{2},\ldots]$, the $n^{th}$ convergent of $\beta$ is the
rational number $\mathbf{c}_{n}(\beta)=[b_{0};b_{1},\ldots,b_{n}]$ which has
reduced expression $\dfrac{\mathbf{p}_{n}(\beta)}{\mathbf{q}_{n}(\beta)}$.
Then the measure of irrationality of $\beta$ is related to the growth of the
denominators of $\mathbf{c}_{n}(\beta)$ through the following relation which
can be taken as an alternative defintion of $\mu$,

$\mu(\beta)=1+\limsup_{n}\dfrac{\ln\mathbf{q}_{n+1}(\beta)}{\ln\mathbf{q}_{n}(\beta)}.$

Provided the existence of the limit, one has the following asymptotic
behaviour

$q_{n+1}\asymp q_{n}^{\mu-1}.$

If we denote by $\lambda_{n}$ the ratio $q_{n+1}/q_{n}$, the last asymptotic
estimate could be read as follows

$\lambda_{n}\asymp q_{n}^{\mu-2}.$ (6)

Its lowest value for an irrational number is $\mu(\beta)=2$ and it is reached
for any algebraic number of degree $d\geq 2$. This fact is a highly non
trivial theorem due to Roth [Roth]. In the other extreme side, the value
$\mu(\beta)=\infty$ correspond to the case when $\beta$ is the Liouville
number. In general, it is extremely difficult to compute this measure in
practice.
A nice consequence of Roth’s theorem is that $x$ is a transcendental number as
soon as $\mu(x)>2$. Unfortunately, this criterion is not enough to
characterize transcendental numbers because the converse of Roth’s result is
not true. Indeed, Adams’ proved that $\theta(e)=2$ showing that a
transcendental number could reach the same bound (see e.g. [Dav78]). More
precisely, it can be proved that for all rational numbers $p/q$ ($q\geq 2$)

$|qe-p|>c_{1}\dfrac{\log\log q}{q\log q}.$

Since the continued fraction expansion of $e$ is given
$e=[2;1,2,\overline{1,1,2n}]^{n\geqslant 2}$, its partial quotients are
unbounded. This implies that $e$ is not a badly approximable number, thus
$\inf_{q\geq 1}q\langle qe\rangle=0$. But for every $\sigma>0$, it is not
difficult to see that for every $q\geqslant 2$

$\dfrac{\log\log q}{\log q}>\dfrac{1}{q^{\sigma}}.$

This shows that for every $\sigma>0$ and rationals $p/q$

$|qe-p|>\dfrac{c}{q^{1+\sigma}}$

for some constant $c$. The latter amounts to say that $\mu(e)=2$, and it shows
that some transcendental numbers are not well-approximated by rationals and
behave like algebraic numbers in view of Roth’s theorem.

The exponent theta associated to $\beta$
By definition suppose that $\beta$ is not a Liouville number i.e.
$\mu(\beta)<\infty$. Then for every $\sigma>0$ there exists $C>0$ such that
for any $p,q$ integers with $q\geq 1$,

$\dfrac{C}{{q}^{\mu(\beta)+\sigma}}<\left|\beta-\dfrac{p}{q}\right|$ (7)

or also,

$\dfrac{C}{{q}^{\mu(\beta)-1+\sigma}}<\left|q\beta-p\right|.$ (8)

Let us fix a real parameter $\sigma>0$ and introduce the following useful
quantity associated with any irrational number $\beta$

$\theta(\beta):=\mu(\beta)+\sigma-1.$

This exponent gives a lower bound for the approximation of the irrational
number $\beta$ by rational numbers provided it is not a Liouville number. It
particular, since $\beta$ is not a Liouville number one has that
$C=\inf_{q\geqslant 1}q^{\theta}\|q\beta\|$ is positive and therefore for
every $p$ and $q$ integers, $q\geq 1$

$\dfrac{C}{{q}^{\theta+1}}<\left|\beta-\dfrac{p}{q}\right|.$ (9)

###  2.2. Dirichlet versus Oppenheim

Dirichlet’s theorem does not give a very precise estimate about how close is
the lattice point $u_{0}=(p,0,q)$, obtained in (5), to the line
$\mathbb{R}(\beta,0,1)$. Let us explain why $u_{0}=(p_{0},0,q_{0})$ falls out
$\mathcal{A}(\varepsilon)$ for any choice of $N$. Otherwise the conjecture
would be proved for $Q_{\alpha}$ and $u_{0}$ would be the solution. It is not
difficult to quantify by how much Dirichlet’s fails to prove the Oppenheim
conjecture for $Q_{\alpha}$.

In particular, combining (9) with Dirichlet approximation (5) we have

$\dfrac{C}{q_{0}^{\theta+1}}<\left|\beta-\dfrac{p_{0}}{q_{0}}\right|\leqslant\dfrac{1}{q_{0}N}.$

Taking the inverse if necessary, we can assume that $u_{0}$ is in the first
octant with $q_{0}\leqslant N$ one infers that

$\dfrac{C}{q_{0}^{\theta}}<p_{0}-\beta q_{0}\leqslant\dfrac{1}{N}.$ (10)

and the latter inequality gives in addition sharp bounds for $q_{0}$

$(CN)^{1/\theta}<q_{0}\leqslant N.$ (11)

From (10) we get

$\beta q_{0}+\dfrac{C}{q_{0}^{\theta}}<p_{0}<\beta q_{0}+\frac{1}{N}.$ (12)

Using (11) one obtains

$\beta(CN)^{1/\theta}+\dfrac{C}{N^{\theta}}<p_{0}\leq\beta N+\frac{1}{N}.$
(13)

Therefore

$2\beta(CN)^{1/\theta}+\dfrac{C}{N^{\theta}}<p_{0}+\beta q_{0}\leq 2\beta
N+\frac{1}{N}.$ (14)

We finally obtain the following bounds for
$Q_{\alpha}(u_{0})=p_{0}^{2}-\beta^{2}q_{0}^{2}$

$\frac{C}{N^{\theta}}\left(2\beta(CN)^{1/\theta}+\dfrac{C}{N^{\theta}}\right)<p_{0}^{2}-\beta^{2}q_{0}^{2}\leq\frac{1}{N}\left(2\beta
N+\dfrac{1}{N}\right).$ (15)

In particular we can do than the inequality
$Q_{\alpha}(u_{0})<2\beta+\dfrac{1}{N^{2}}$. Thus Dirichlet’s theorem is
unable to provide a solution to the Oppenheim conjecture whatever the choice
of $N$.

###  2.3. Error in the approximation by the convergents

We have a precise of the rate of error of this approximation, set
$\mathbf{e}_{n}(\beta):=\beta-c_{n}(\beta)$, so we have (see e.g. Exercise
3.1.5. [EW])

$\dfrac{1}{2\mathbf{q}_{n+1}(\beta)^{2}}\leq|\mathbf{e}_{n}(\beta)|\leq\dfrac{1}{\mathbf{q}_{n}(\beta)\mathbf{q}_{n+1}(\beta)}<\dfrac{1}{\mathbf{q}_{n}(\beta)^{2}}.$
(16)

The sequence $\mathbf{q}_{n}(\beta)$ is increasing and the rate of convergence
is determined by the diophantine properties of $\beta$, in particular it tends
to infinity with at least exponential rate since
$2^{(n-2)/2}\leqslant\mathbf{q}_{n}(\beta)$ ([Kh], Theorem 12).

The convergents $c_{n}(\beta)$ tends to $\beta$ by oscillating so that the
sign of $\mathbf{e}_{n}(\beta)$ is alternating. From now on, we choose even
indices which implies that the error terms assume only positive values. We
infer that,

$\dfrac{1}{2\mathbf{q}_{2n+1}(\beta)^{2}}\leq\mathbf{e}_{2n}(\beta)<\dfrac{1}{\mathbf{q}_{2n}(\beta)^{2}}.$
(17)

###  2.4. Rational lines of approximation.

We introduce an object which is at the core of our strategy. It is a sequence
of rational lines which will cross $\mathcal{A}(\varepsilon)$ in a sufficently
large time in order to contain a lattice point. We have two degrees of freedom
given by the integral parameters $n$ and $N$. We are going to reduce to merely
one parameter, namely $n$. To do this, let us first fix the real parameter

$\eta:=\dfrac{\theta-1}{\theta+1}$

where

$\theta(\beta)=\mu(\beta)+\sigma-1.$

In all cases, $\theta>1$, and therefore

$0<\eta<1.$

Let us choose $N$ to be a sequence $\left(N_{n}\right)$ satisfying the growth
condition

$N_{n}=\mathbf{q}_{2n}^{1-\eta}.$ (18)

For each nonnegative integer $n$, Dirichlet’s theorem tells us that there
exists a two-dimensional lattice vector $(x_{n},z_{n})$ with $1\leqslant
z_{n}\leqslant N_{n}$ such that

$\left|\beta z_{n}-x_{n}\right|\leq\dfrac{1}{N_{n}}.$ (19)

Moreover, using $\theta$ there exists a constant $C$ such that

$\dfrac{C}{N_{n}^{\theta}}\leq\dfrac{C}{z_{n}^{\theta}}<\left|\beta
z_{n}-x_{n}\right|\leq\dfrac{1}{N_{n}}.$ (20)

This gives the crucial bound on the denominators,

$(CN_{n})^{1/\theta}<z_{n}\leq N_{n}$ (21)

For each $n$, from (19) we form the three-dimensional integral vector
$u_{n}:=(x_{n},0,z_{n})\in\mathbb{Z}^{3}$ which is close to the axis $x=\beta
z$. As we have seen earlier Dirichlet’s approximation theorem is not enough in
order to ensure that $u_{n}$ is in $\mathcal{A}(\varepsilon)$. However, we
have at our disposal a sequence of lattice points $(u_{n})_{n}$ near
$\mathcal{A}(\varepsilon)$ from which we built a sequence of affine lines
$\mathcal{L}_{\beta}^{n}$ by setting

$(\mathcal{L}_{\beta}^{n}):~{}u_{n}+\mathbb{R}(c_{2n}(\beta),\dfrac{1}{\mathbf{q}_{2n}(\beta)},1).$

The lines $(\mathcal{L}_{\beta}^{n})$ are good candidates for containing
lattice points in $\mathcal{A}(\varepsilon)$. Indeed the first interesting
feature is that this lines pass through lattice points, namely the $u_{n}$’s,
and such lines are directed by rational vectors so that they can contain
latiice points. Another crucial feature is geometrical, the fact that the
lines are nearly parralel to the generatrix of the cone, namely the line
$\mathbb{R}(\beta,0,1)$. This+ leads us to expect that
$(\mathcal{L}_{\beta}^{n})$ spends a sufficent amount of time in
$\mathcal{A}(\varepsilon)$ for $n$ large enough.

We will rather focus on the downward half-line parametrized as follows

$(\mathcal{L}_{\beta}^{n})^{+}~{}:~{}v_{n}(t)=(x_{n}-tc_{2n}(\beta),-\dfrac{t}{\mathbf{q}_{2n}(\beta)},z_{n}-t)~{}~{}(t\geq
0).$

We are interested in the intersection of this half-line with the domain
$\mathcal{A}(\varepsilon)$. Note that for each increment of the index $n$, the
line will never remains in a same plane, in that two successive lines
$(\mathcal{L}_{\beta}^{n})^{+}$ and $(\mathcal{L}_{\beta}^{n+1})^{+}$ will
never be coplanar. A geometric observation allows us to guess that this line
$(\mathcal{L}_{\beta}^{n})^{+}$ will cut the boundary of
$\mathcal{A}(\varepsilon)$, namely $\\{Q_{\alpha}=\pm\varepsilon\\}$, in at
most four points. This will be made explicit in our computations. Our first
task is to estimate the time spent by $(\mathcal{L}_{\beta}^{n})^{+}$ in
$\mathcal{A}(\varepsilon)$. The answer is given in the following proposition.

######  Proposition 2.1.

Let $I^{n}$ be the set of times at which the half-line
$(\mathcal{L}_{\beta}^{n})^{+}=\\{v_{n}(t)\mid t\geq 0)\\}$ intersects
$\mathcal{A}(\varepsilon)$. Then, for $n$ large enough, $I$ is the union of
two intervals $I_{1}^{n}$ and $I_{2}^{n}$.

Proof. Let us fix $\varepsilon>0$. For each positive integer $n$, the half-
line $(\mathcal{L}_{\beta}^{n})^{+}$ lies in $\mathcal{A}(\varepsilon)$ if and
only if for every $t\geq 0$

$v_{n}(t)=(x_{n}-tc_{2n}(\beta),-\dfrac{t}{\mathbf{q}_{2n}(\beta)},z_{n}-t)~{}\in\mathcal{A}(\varepsilon).$

This amounts to say that the time variable $t$ is constrained to satisfy the
inequalities

$-\varepsilon\leq(x_{n}-tc_{2n}(\beta))^{2}+(t\mathbf{q}_{2n}(\beta)^{-1})^{2}-\alpha(z_{n}-t)^{2}\leq\varepsilon.$

Let us define the quadratic polynomial in the real variable $t$ (the time)

$f_{n}(t):=(x_{n}-tc_{2n}(\beta))^{2}+(t\mathbf{q}_{2n}(\beta)^{-1})^{2}-\alpha(z_{n}-t)^{2}.$

Ordering the terms we get

$f_{n}(t)=\\{c_{2n}(\beta)^{2}+\mathbf{q}_{2n}(\beta)^{-2}-\beta^{2}\\}t^{2}-2\\{c_{2n}(\beta)x_{n}-y_{n}\beta^{2}\\}t+\\{x_{n}^{2}-\beta^{2}z_{n}^{2}\\}.$

Which is important to us is the intersection points of the graph of $f_{n}(t)$
with the two lines corresponding to $\pm\varepsilon$. Thus we are reduced to
solve the two following equations ( remember $\beta^{2}=\alpha$ ) provided
such solutions exists

$f_{n}(t)\pm\varepsilon=\\{c_{2n}(\beta)^{2}+\mathbf{q}_{2n}(\beta)^{-2}-\beta^{2}\\}t^{2}-2\\{c_{2n}(\beta)x_{n}-y_{n}\beta^{2}\\}t+\\{x_{n}^{2}-\beta^{2}z_{n}^{2}\pm\varepsilon\\}=0.$

Set $A_{n}=c_{2n}(\beta)^{2}+\mathbf{q}_{2n}(\beta)^{-2}-\beta^{2}$,
$B_{n}=-2\\{c_{2n}(\beta)x_{n}-z_{n}\beta^{2}\\}$ and
$C_{n}^{\pm}=x_{n}^{2}-\beta^{2}z_{n}^{2}\pm\varepsilon$.

Thus one has to solve the (two) equations

$A_{n}t^{2}+B_{n}t+C_{n}^{\pm}=0.$

We need to estimate the discriminants
$\Delta_{n}^{\pm}(\varepsilon)=B^{2}_{n}-4A_{n}C_{n}^{\pm}$ and in fact we
only need to focus on the roots and their relative distance not on their
ordering nor their signs.

Since we have a nice control of the error in the approximation by the
convergents, we replace $c_{2n}(\beta)$ by $\beta-\mathbf{e}_{2n}(\beta)$. The
coefficients are therefore given by,

$\left\\{\begin{array}[]{ccc}A_{n}&=&-2\beta\mathbf{e}_{2n}(\beta)+\mathbf{e}_{2n}(\beta)^{2}+\mathbf{q}_{2n}(\beta)^{-2}.\\\
B_{n}&=&-2(\beta
x_{n}-x_{n}\mathbf{e}_{2n}(\beta)-z_{n}\beta^{2})=-2(\beta(x_{n}-\beta
z_{n})-x_{n}\mathbf{e}_{2n}(\beta))\\\ C_{n}^{\pm}&=&(x_{n}-\beta
z_{n})(x_{n}+\beta z_{n})\pm\varepsilon.\end{array}\right.$ (22)

Let us set $\delta_{n}=x_{n}-\beta z_{n}$, $\overline{\delta_{n}}=x_{n}+\beta
z_{n}$, note that $Q_{\alpha}(u_{n})=\delta_{n}\overline{\delta_{n}}$, so that
$C_{n}^{\pm}=Q_{\alpha}(u_{n})\pm\varepsilon$ and
$B_{n}=-2(\beta\delta_{n}-x_{n}\mathbf{e}_{2n}(\beta))$. The Dirichlet lattice
point $u_{n}=(x_{n},0,z_{n})$ is exterior to $\\{-\varepsilon\leq
Q_{\alpha}\leq\varepsilon\\}$, changing $u_{n}$ to $-u_{n}$ if necessary we
can assume that $Q_{\alpha}(u_{n})>\varepsilon$. Thus,
$x_{n}^{2}>\beta^{2}z_{n}^{2}\pm\varepsilon$ and in particular
$C_{n}^{\pm}>0$.

We deduce from (20) and (21) the following bounds for $z_{n}$,$\delta_{n}$ and
$x_{n}$,

$(C\mathbf{q}_{2n}^{1-\eta})^{1/\theta}=(CN_{n})^{1/\theta}<z_{n}\leqslant
N_{n}=\mathbf{q}_{2n}^{1-\eta}.$ (23)

$\dfrac{C}{\mathbf{q}_{2n}^{(1-\eta)\theta}}=\dfrac{C}{N_{n}^{\theta}}\leq\dfrac{C}{z_{n}^{\theta}}<\delta_{n}\leq\dfrac{1}{N_{n}}=\dfrac{1}{\mathbf{q}_{2n}^{1-\eta}}.$
(24)

Since $x_{n}=\beta z_{n}+\delta_{n}$, using (23) and (24) we get

$\beta
C^{1/\theta}\mathbf{q}_{2n}^{(1-\eta)/\theta}+\dfrac{C}{\mathbf{q}_{2n}^{(1-\eta)\theta}}<\beta
z_{n}+\dfrac{C}{N_{n}^{\theta}}<x_{n}\leq\beta
z_{n}+\dfrac{1}{N_{n}}\leq\beta\mathbf{q}_{2n}^{1-\eta}+\dfrac{1}{\mathbf{q}_{2n}^{1-\eta}}.$
(25)

We define the following quantities,

$U_{n}:=-\dfrac{B_{n}}{A_{n}}$ and $V_{n}:=\dfrac{A_{n}}{B_{n}^{2}}$

thus
$\Delta_{n}^{\pm}(\varepsilon)=B_{n}^{2}\left(1-4V_{n}(Q_{\alpha}(u_{n})\pm\varepsilon)\right).$
We are going to show that $V_{n}$ tends to zero, this will prove that
$\Delta_{n}^{\pm}(\varepsilon)$ are both positive for $n$ large enough.

For any positive $n$, the inequalities in (17) gives

$\dfrac{1}{2\mathbf{q}_{2n+1}(\beta)^{2}}\leq\mathbf{e}_{2n}(\beta)<\dfrac{1}{\mathbf{q}_{2n}(\beta)^{2}}.$
(26)

Concerning $B_{n}=2\\{x_{n}\mathbf{e}_{2n}(\beta)-\beta\delta_{n}\\}$, with
(24), (25) and (26) one has that

$\dfrac{1}{2\mathbf{q}_{2n+1}^{2}}\left(\dfrac{C}{N_{n}^{\theta}}+\beta(CN_{n})^{1/\theta}\right)-\dfrac{2\beta}{N_{n}}<x_{n}\mathbf{e}_{2n}-2\delta_{n}\beta\leq\dfrac{1}{\mathbf{q}_{2n}^{2}}\left(\beta
N_{n}+\dfrac{1}{N_{n}}\right)-\dfrac{2\beta}{N_{n}^{\theta}}.$ (27)

We can rearrange the terms in order to get the following bounds for $B_{n}$

$\dfrac{1}{\mathbf{q}_{2n}}\left(\dfrac{C}{\lambda_{2n}^{2}\mathbf{q}_{2n}N_{n}^{\theta}}+\dfrac{\beta(CN_{n})^{1/\theta}}{\lambda_{2n}^{2}\mathbf{q}_{2n}}-\dfrac{2\beta\mathbf{q}_{2n}}{N_{n}}\right)\leq
B_{n}\leq\dfrac{2}{\mathbf{q}_{2n}^{2}}\left(\beta
N_{n}+\dfrac{1}{N_{n}}\right)-\dfrac{4\beta}{N_{n}^{\theta}}.$ (28)

Replacing $N_{n}$ by $\mathbf{q}_{2n}^{1-\eta}$ in (28) we obtain a lower
bound for $B_{n}$

$\dfrac{1}{\mathbf{q}_{2n}}\left(\dfrac{C}{\lambda_{2n}^{2}\mathbf{q}_{2n}^{1+(1-\eta)\theta}}+\beta
C^{1/\theta}\dfrac{\mathbf{q}_{2n}^{(1-\eta)/\theta-1}}{\lambda_{2n}^{2}}-2\beta\mathbf{q}_{2n}^{\eta}\right)\leq
B_{n}.$ (29)

$y$$y=\varepsilon$$y=-\varepsilon$$y=f_{n}(t)$ Figure 2.  The domain
$-\varepsilon\leq Q_{\alpha}(v_{n}(t))=f_{n}(t)\leq\varepsilon$ is supported
by two intervals.

We claim that $V_{n}$ tends to zero as $n$ goes to infinity. Indeed, one has

$V_{n}=\dfrac{\mathbf{e}_{2n}^{2}-2\beta\mathbf{e}_{2n}+1/\mathbf{q}_{2n}^{2}}{\left(x_{n}\mathbf{e}_{2n}-2\beta\delta_{n}\right)^{2}}.$

For the numerator of $V_{n}$ we have the bound

$A_{n}=\mathbf{e}_{2n}^{2}-2\beta\mathbf{e}_{2n}+\mathbf{q}_{2n}^{-2}\leq\dfrac{1}{\mathbf{q}_{2n}^{4}}-\dfrac{\beta}{\mathbf{q}_{2n+1}^{2}}+\dfrac{1}{\mathbf{q}_{2n}^{2}}=\dfrac{1}{\mathbf{q}_{2n}^{2}}\left(1-\dfrac{\beta}{\lambda_{2n}^{2}}+\dfrac{1}{\mathbf{q}_{2n}^{2}}\right).$
(30)

$x$$z$$z=(CN_{n})^{1/\theta}=(C\mathbf{q}_{2n}^{1-\eta})^{1/\theta}$$z=N_{n}=\mathbf{q}_{2n}^{1-\eta}$$z=\beta^{-1}x$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(\mathcal{L}^{n}_{\beta}):u_{n}+\mathbb{R}(c_{2n},\dfrac{1}{\mathbf{q}_{2n}},1)$$\bullet$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q_{\alpha}=-\varepsilon$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q_{\alpha}=\varepsilon$$u_{n}=(x_{n},0,z_{n})$
Figure 3. In blue the level sets $Q_{\alpha}=\pm\varepsilon$ and in grey the
generatrix $x=\beta z$ of the cone $Q_{\alpha}=0$ projected on the $xz$-plane.
The red line $(\mathcal{L}^{n}_{\beta})$ cuts
$\\{Q_{\alpha}=\pm\varepsilon\\}$ in 4 points as $n$ gets large. The dotted
lines represents the bounds for $z_{n}$.

Thus, using (29) we get

$0<|V_{n}|\ll\dfrac{\left(1-\dfrac{\beta}{\lambda_{2n}^{2}}+\dfrac{1}{\mathbf{q}_{2n}^{2}}\right)}{\left(\dfrac{C}{\lambda_{2n}^{2}\mathbf{q}_{2n}^{1+(1-\eta)\theta}}+\beta
C^{1/\theta}\dfrac{\mathbf{q}_{2n}^{(1-\eta)/\theta-1}}{\lambda_{2n}^{2}}-2\beta\mathbf{q}_{2n}^{\eta}\right)^{2}}.$
(31)

Taking under consideration the fact that
$\lambda_{2n}^{2}\asymp\mathbf{q}_{2n}^{2(\mu-2)}$ which follows from (6) we
obtain that

$\lim_{n}V_{n}=0.$

Using (15) we infer that

$|Q_{\alpha}(u_{n})|\leq
2C\beta+\dfrac{1}{N_{n}}=2C\beta+\dfrac{1}{\mathbf{q}_{2n}^{1-\eta}}.$ (32)

Thus the term

$1-4V_{n}(Q_{\alpha}(u_{n})\pm\varepsilon)$

can be made positive and less than 1 provided $n$ is taken large enough. Hence
the discriminants are always positive when $n$ becomes larger than some
positive integer $n_{0}=n_{0}(\varepsilon)$ depending on $\varepsilon$. In
this range the roots are given by

$t_{1,2,3,4}(n,\varepsilon)=\dfrac{1}{2}U_{n}\left(1\pm\sqrt{1-4V_{n}(Q_{\alpha}(u_{n})\pm\varepsilon)}\right).$

In more details, these correspond to the hitting times,
$t_{1}<t_{2}<t_{3}<t_{4}$ given by

$((\mathcal{L}_{\beta}^{n})^{+}\cap\\{Q_{\alpha}=-\varepsilon\\})\left\\{\begin{array}[]{ccc}t_{1}(n)&=&\dfrac{1}{2}U_{n}\left(1-\sqrt{1-4V_{n}(Q_{\alpha}(u_{n})+\varepsilon)}\right)\\\
\\\
t_{4}(n)&=&\dfrac{1}{2}U_{n}\left(1+\sqrt{1-4V_{n}(Q_{\alpha}(u_{n})+\varepsilon)}\right)\\\
\end{array}\right.$

and

$((\mathcal{L}_{\beta}^{n})^{+}\cap\\{Q_{\alpha}=\varepsilon\\})\left\\{\begin{array}[]{ccc}t_{2}(n)&=&\dfrac{1}{2}U_{n}\left(1-\sqrt{1-4V_{n}(Q_{\alpha}(u_{n})-\varepsilon)}\right)\\\
\\\
t_{3}(n)&=&\dfrac{1}{2}U_{n}\left(1+\sqrt{1-4V_{n}(Q_{\alpha}(u_{n})-\varepsilon)}\right).\\\
\end{array}\right.$

In particular the values of $t$ for which
$v_{n}(t)\in(\mathcal{L}_{\beta}^{n})^{+}\cap\mathcal{A}(\varepsilon)$ is the
union the two disjoint intervals $I_{1}^{n}(\varepsilon)=[t_{1},t_{2}]$ and
$I_{2}^{n}(\varepsilon)=[t_{3},t_{4}]$ when $n\geq n_{0}$. The proposition 2.1
is proved.

$\square$

Remark. The signs and the hitting times are not important for our purposes.
The two intervals of the proposition 2.1, $I_{1}^{n}$ and $I_{2}^{n}$ plays a
symmetric role and their size is the same. One interval should comprise
negative times while the other consists of positive ones. Assume for instance,
changing the order if necessary, that the roots are sorted such that
$t_{1}<t_{2}<t_{3}<t_{4}$ for $n\geqslant n_{0}$. Then in view of the previous
proposition, for $n\geqslant n_{0}$ the intersection of the half-line
$(\mathcal{L}_{\beta}^{n})^{+}$ with the two level sets
$\\{Q_{\alpha}=\pm\varepsilon\\}$ behaves as follows, see figure 3

$\left\\{\begin{array}[]{ccc}Q_{\alpha}(v_{n}(t))<-\varepsilon&\mathrm{if}&0<t<t_{1}\
\ \ (\mathrm{out})\\\ -\varepsilon\leq
Q_{\alpha}(v_{n}(t))\leqslant\varepsilon&\mathrm{if}&t_{1}\leqslant t<t_{2}\ \
\ (\mathrm{in})\\\ \varepsilon<Q_{\alpha}(v_{n}(t))&\mathrm{if}&t_{2}\leqslant
t<t_{3}\ \ \ (\mathrm{out})\\\
-\varepsilon<Q_{\alpha}(v_{n}(t))\leq\varepsilon&\mathrm{if}&t_{3}\leqslant
t<t_{2}\ \ \ (\mathrm{in})\\\
Q_{\alpha}(v_{n}(t))<-\varepsilon&\mathrm{if}&t_{4}<t\ \ \ \ \ \
(\mathrm{out}).\end{array}\right.$

##  3\. A Solution to the Oppenheim conjecture for $Q_{\alpha}$

Let $\varepsilon>0$ be an arbitrary small real number, we are interested to
finding $n$ and $t_{n}$ such that $v_{n}(t_{n})$ is a nonzero vector is in
$\mathbb{Z}^{3}\cap\mathcal{A}(\varepsilon)$ i.e.

$0<|Q_{\alpha}(v_{n}(t_{n}))|\leq\varepsilon.$

In order that $v_{n}(t_{n})$ provides the required lattice point, we
necessarily need $t_{n}$ to be a multiple of $\mathbf{q}_{2n}$ so that we can
clear the denominators. By symmetry, we only need to focus on one interval,
say $I_{1}^{n}(\varepsilon)$. The following combinatorial argument shows that
it is always possible to do so for large enough values of $n$.

######  Lemma 3.1.

There exists a positive integer $n_{1}(\varepsilon)$ such that the interval
$I_{1}^{n}(\varepsilon)=[t_{1},t_{2}]$ contains a multiple of
$\mathbf{q}_{2n}$ whenever $n\geq n_{1}$.

Proof. Let us set for each positive integer $n$, the following counting
function

$M_{n}:=\mathrm{Card}\left([t_{1},t_{2}]\cap\mathbb{Z}\mathbf{q}_{2n}\right).$

$M_{n}$ is the number of multiples of $\mathbf{q}_{2n}$ in
$I_{1}^{n}=[t_{1},t_{2}]$. In particular we have that
$M_{n}=\lfloor\dfrac{l(I_{1}^{n})}{\mathbf{q}_{2n}}\rfloor$ where
$l(I_{1}^{n})$ is the length of the interval $I_{1}^{n}$. The aim is to show
that this quantity is $\geq 1$ when $n$ is larger that a certain threshold
$n_{1}$. As $n$ gets large, we have that

$t_{1}\asymp
2U_{n}V_{n}(Q_{\alpha}(u_{n})-\varepsilon)=-\frac{2(Q_{\alpha}(u_{n})-\varepsilon)}{B_{n}}$
(33)

and

$t_{2}\asymp
2U_{n}V_{n}(Q_{\alpha}(u_{n})+\varepsilon)=-\frac{2(Q_{\alpha}(u_{n})+\varepsilon)}{B_{n}}.$
(34)

The length of the interval $I_{n}^{1}$ is asymptotically given by

$l(I_{1}^{n})=|t_{2}-t_{1}|\asymp\dfrac{4\varepsilon}{|B_{n}|}.$

Thus,

$M_{n}\asymp\dfrac{4\varepsilon}{\mathbf{q}_{2n}|B_{n}|}.$

Concerning the denominator,

$\mathbf{q}_{2n}|B_{n}|\leq\dfrac{1}{\mathbf{q}_{2n}}\left(\beta
N_{n}+\dfrac{1}{N_{n}}\right)-\dfrac{2\beta}{N_{n}^{\theta}}\mathbf{q}_{2n}$

$\ \ \ \ \
\leq\beta\dfrac{N_{n}}{\mathbf{q}_{2n}}+\dfrac{1}{\mathbf{q}_{2n}N_{n}}-2\beta\dfrac{\mathbf{q}_{2n}}{N_{n}^{\theta}}$

$\ \ \ \ \ \
\leq\dfrac{\beta}{\mathbf{q}_{2n}^{\eta}}+\dfrac{1}{\mathbf{q}_{2n}^{2-\eta}}-\dfrac{2\beta}{\mathbf{q}_{2n}^{(1-\eta)\theta-1}}.$

Then the choice111This is the only moment we need that $\theta>1$ of $\eta$
gives that $(1-\eta)\theta-1=\eta$, then

$\mathbf{q}_{2n}|B_{n}|\leq\dfrac{\beta}{\mathbf{q}_{2n}^{\eta}}+\dfrac{1}{\mathbf{q}_{2n}^{2-\eta}}-\dfrac{2\beta}{\mathbf{q}_{2n}^{\eta}}.$

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\leq\dfrac{\beta}{\mathbf{q}_{2n}^{\eta}}\left|1-\dfrac{1}{\beta\mathbf{q}_{2n}^{2(1-\eta)}}\right|.$

Thus we have the upper estimate

$\mathbf{q}_{2n}|B_{n}|\ll\dfrac{1}{\mathbf{q}_{2n}^{\eta}}.$

Taking the inverse,

$\varepsilon\mathbf{q}_{2n}^{\eta}\ll\dfrac{4\varepsilon}{\mathbf{q}_{2n}|B_{n}|}.$
(35)

Finally we infer the following crucial bound

$\varepsilon\mathbf{q}_{2n}^{\eta}\ll M_{n}$ (36)

In particular, since $(\mathbf{q}_{2n}^{\eta})_{n}$ diverges there exists
$n_{1}(\varepsilon)$ such that for all $n\geq n_{1}(\varepsilon)$

$1<\varepsilon\mathbf{q}_{2n}^{\eta}.$ (37)

Hence (36) shows that $M_{n}\geq 1$ for $n\geq n_{1}$, meaning that the
interval of times $I_{1}^{n}(\varepsilon)=[t_{1},t_{2}]$ contains at least one
multiple of $\mathbf{q}_{2n}$ for $n\geq n_{1}$. Let us estimate the integer
$n_{1}$ which depends on the choice of $\varepsilon$ and $\eta$ , and which
can be seen formally as

$\varphi_{\beta}(\varepsilon)=\min\\{n\geq n_{0}\ \ |\ \ M_{n}\geq 2\\}.$

Here $n_{0}=n_{0}(\varepsilon)$ is the least integer which ensures that
$I_{1}^{n}(\varepsilon)\neq\emptyset$ coming from Proposition 2.1 whereas
$n_{1}(\varepsilon)$ is the least integer such that $I_{1}^{n}(\varepsilon)$
contains a multiple of $\mathbf{q}_{2n}$. In particular,
$n_{0}(\varepsilon)<n_{1}(\varepsilon)$. The number $n_{1}$ is not going to be
optimal, i.e. it will be an upper estimate for $\varphi_{\beta}(\varepsilon)$.

Since $2^{n-1}\leq\mathbf{q}_{2n}$, a sufficient condition in order the
inequality $\varepsilon\mathbf{q}_{2n}^{\eta}>1$ to hold is

$\varepsilon 2^{\eta(n-1)}>1.$

Applying logarithms, we get

$n>1+\eta^{-1}\left|\ln\left(\dfrac{1}{\varepsilon}\right)\right|/\ln 2.$

Thus a good choice for $n_{1}$ is

$n_{1}(\varepsilon):=2+\lfloor\eta^{-1}\left|\ln\left(\varepsilon\right)\right|/\ln
2\rfloor.$

This finishes the proof of the Lemma.

$\square$

### Proof of Theorem 1.1.

Let $\varepsilon>0$ be fixed.

Case 1 Assume $\beta$ is a Liouville number and let $n$ be a positive large
enough integer so that

$2^{-n}\beta+2^{-2(n+2)}\leq\varepsilon.$

Since $\mu(\beta)=\infty$ and given $n$ as above we can always find a rational
number $p/q$ such that

$\left|q\beta-p\right|<\dfrac{1}{q^{n+2}}.$ (38)

From this, we deduce that $q\beta-1/q^{n+2}<p<q\beta-1/q^{n+2}$, thus

$2q\beta-1/q^{n+2}<p+\beta q<2q\beta+1/q^{n+2}.$ (39)

Thus,

$|Q_{\alpha}(p,0,q)|=|p^{2}-\beta^{2}q^{2}|=(p+\beta q)|p-\beta
q|<\dfrac{1}{q^{n+2}}(2q\beta+1/q^{n+2}).$ (40)

Since $q\geq 2$,

$|Q_{\alpha}(p,0,q)|<\dfrac{1}{2^{n+2}}(4\beta+1/2^{n+2}).$

The choice of $n$ implies that $v=(p,0,q)$ is a nonzero integral solution of

$|Q_{\alpha}(p.0.q)|<\varepsilon.$

In other words, the Oppenheim conjecture holds for $Q_{\alpha}$ in this case.

Case 2 Assume $\beta$ is a not a Liouville number,

The lemma (3.1) shows that there exists an explicit integer
$n_{1}(\varepsilon)>0$ such that $I_{1}^{n}(\varepsilon)$ contains a multiple
of $\mathbf{q}_{2n_{1}}$ say $a_{n_{1}}\mathbf{q}_{2n_{1}}\in[t_{1},t_{2}]$
where $a_{n_{1}}$ is a nonzero integer. The proposition (2.1) implies that
$v_{n_{1}}(a_{n_{1}}\mathbf{q}_{2n_{1}})\in\mathcal{A}(\varepsilon)$.
Moreover,

$v_{n_{1}}(a_{n_{1}}\mathbf{q}_{2n_{1}})=\left(x_{n_{1}}-a_{n_{1}}p_{2n_{1}},-a_{n_{1}},z_{n_{1}}-a_{n_{1}}\mathbf{q}_{2n_{1}}\right)\in\mathbb{Z}.$

Thus, we have a nonzero integral vector
$v_{1}:=v_{n_{1}}(a_{n_{1}}\mathbf{q}_{2n_{1}})$ in
$\mathcal{A}(\varepsilon)$, that is,

$|Q_{\alpha}\left(v_{1}\right)|\leqslant\varepsilon.$

This proves the first assertion of the theorem. We give an estimate the size
of the solution, set

$\|v_{1}\|_{\infty}=\max\\{|x_{n_{1}}-a_{n_{1}}p_{2n_{1}}|,|z_{n_{1}}-a_{n_{1}}p_{2n_{1}}|,|a_{n_{1}}|\\}.$

A crude bound is given by

$\|v_{1}\|_{\infty}\leq|x_{n_{1}}|+|a_{n_{1}}|p_{2n_{1}}.$

We know from (25) that

$|x_{n_{1}}|\lesssim\beta\mathbf{q}_{2n_{1}}^{1-\eta}.$

Also by construction we have
$a_{n_{1}}\in[\dfrac{t_{1}}{\mathbf{q}_{2n_{1}}},\dfrac{t_{2}}{\mathbf{q}_{2n_{1}}}]$,
thus in view of (32), (33) and (34) one has

$|a_{n_{1}}|\leq\dfrac{|Q_{\alpha}(u_{n})|+\varepsilon}{\mathbf{q}_{2n_{1}}|B_{n_{1}}|}\leq\dfrac{2C\beta+\mathbf{q}_{2n}^{-1+\eta}+\varepsilon}{\mathbf{q}_{2n_{1}}|B_{2n_{1}}|}.$

Thus,

$\|v_{1}\|_{\infty}\lesssim\mathbf{q}_{2n_{1}}^{1-\eta}+\dfrac{2C\beta+\varepsilon}{\mathbf{q}_{2n_{1}}|B_{n_{1}}|}\mathbf{p}_{2n_{1}}+\dfrac{c_{2n_{1}}}{\mathbf{q}_{2n_{1}}^{1-\eta}|B_{n_{1}}|}$

or equivalently

$\|v_{1}\|_{\infty}\lesssim\mathbf{q}_{2n_{1}}^{1-\eta}+\dfrac{2C\beta+\varepsilon}{|B_{n_{1}}|}\mathbf{c}_{2n_{1}}+\dfrac{c_{2n_{1}}}{\mathbf{q}_{2n_{1}}^{1-\eta}|B_{n_{1}}|}.$

Using (29)

$\dfrac{1}{\mid
B_{n_{1}}|}\leq\dfrac{1}{|C{\lambda_{2n_{1}}^{-2}\mathbf{q}_{2n_{1}}^{-2-(1-\eta)\theta}+\beta
C^{1/\theta}\lambda_{2n_{1}}^{-2}\mathbf{q}_{2n_{1}}^{(1-\eta)/\theta-2}-2\beta\mathbf{q}_{2n_{1}}^{\eta-1}\mid}}.$
(41)

We have the relation $1-\eta=2/(\theta(\theta+1))$ and the irrationality
measure $\mu$ which comes into play using (6), thus

$\lambda_{2n}^{-2}\mathbf{q}_{2n}^{-2-(1-\eta)\theta}\asymp\mathbf{q}_{2n}^{-2(\mu-2)-2-(1-\eta)\theta}=\mathbf{q}_{2n}^{-2\mu+2-2/(\theta+1)}=\mathbf{q}_{2n}^{-2(\mu-1+1/(\theta+1))}$

and

$\lambda_{2n}^{-2}\mathbf{q}_{2n}^{(1-\eta)/\theta-2}\asymp\mathbf{q}_{2n}^{-2(\mu-2)+(1-\eta)/\theta-2}=\mathbf{q}_{2n}^{-2\mu+2+2/(\theta^{2}(\theta+1))}=\mathbf{q}_{2n}^{-2(\mu-1-1/(\theta^{2}(\theta+1))}.$

Therefore,

$\dfrac{1}{\mid
B_{n_{1}}|}\ll\dfrac{1}{|C\mathbf{q}_{2n}^{-2(\mu-1+1/(\theta+1))}+\beta
C^{1/\theta}\mathbf{q}_{2n}^{-2(\mu-1-1/(\theta^{2}(\theta+1))}-2\beta\mathbf{q}_{2n_{1}}^{-2/(\theta(\theta+1))}\mid}.$

Let us set $\kappa=\dfrac{2}{\theta(\theta+1)}=1-\eta$,

$\dfrac{1}{\mid
B_{n_{1}}|}\ll\dfrac{1}{|C{\mathbf{q}_{2n_{1}}^{-2(\mu-1)-\kappa}+\beta
C^{1/\theta}\mathbf{q}_{2n_{1}}^{-2(\mu-1)+\kappa/\theta}-2\beta\mathbf{q}_{2n_{1}}^{-\kappa}\mid}}.$

$\ \ \ \ \ \ \ \ \
\ll\dfrac{\mathbf{q}_{2n_{1}}^{\kappa}}{|1-C{\mathbf{q}_{2n_{1}}^{-2(\mu-1)}/2\beta-C^{1/\theta}\mathbf{q}_{2n_{1}}^{-2(\mu-1)+\kappa/\theta+\kappa}/2\mid}}.$

Thus, since $2(\mu-1)\geq 2>\kappa/\theta+\kappa$, one has

$\dfrac{1}{|B_{n_{1}}|}\ll\mathbf{q}_{2n_{1}}^{\kappa}=\mathbf{q}_{2n_{1}}^{1-\eta}.$

Hence we get,

$\|v_{1}\|_{\infty}\ll\mathbf{q}_{2n_{1}}^{1-\eta}+O(1).$

In short,

$\|v_{1}\|_{\infty}\ll\mathbf{q}_{2n_{1}}^{1-\eta}.$

By definition we have $\eta=1-2/(\theta+1)$ so that the last inequality reads

$\|v_{1}\|_{\infty}\ll\mathbf{q}_{2n_{1}}^{2/(\theta+1)}$

where

$n_{1}(\varepsilon):=2+\lfloor\eta^{-1}\left|\ln\left(\varepsilon\right)\right|/\ln
2\rfloor.$

This finishes the proof of Theorem 1.1.

$\square$

### Proof of Corollary 1.2.

$(1)$ By assumption there exists $\gamma\in\mathrm{SL}(3,\mathbb{Q})$, such
that $Q(x)=Q_{\alpha}(\gamma x)$. Let us consider an arbitrary real
$\varepsilon>0$. Let
$a=\mathrm{lcm}\\{\mathrm{den}((\gamma^{-1})_{i,j}),1\leqslant i,j\leqslant
3\\}$ be the least common multiple of the denominator of the coefficients of
$\gamma^{-1}$. Thus $a\gamma^{-1}$ is an integral matrix. Theorem 1.1 gives
the existence of a nonzero integral vector $v\in\mathbb{Z}^{3}$ such that
$|Q_{\alpha}(v)|\leq\varepsilon/a^{2}$. Since
$a\gamma^{-1}\in\mathrm{SL}(3,\mathbb{Z})$, $v_{1}=a\gamma^{-1}v$ is a nonzero
integral vector such that

$|Q(v_{1})|=|Q(a\gamma^{-1}v)|=a^{2}|Q(\gamma^{-1}v)|=a^{2}|Q_{\alpha}(v)|\leq\varepsilon.$

Hence $Q$ satisfies the Oppenheim conjecture.

$(2)$ Let $h\in H$ such that $Q(x)=Q_{0}(hx)$, where for some
$A\in\mathrm{SL}(3,\mathbb{Q})$ and $h_{33}\notin\mathbb{Q}$ one has

$h=\left[\begin{array}[]{c|c}A&0\\\ \hline\cr 0&h_{33}\end{array}\right].$

The matrix $h$ can factorized as follows

$h=\left[\begin{array}[]{c|c}I_{2}&0\\\ \hline\cr
0&h_{33}\end{array}\right]\left[\begin{array}[]{c|c}A&0\\\ \hline\cr
0&1\end{array}\right].$

Set $\alpha=h_{33}^{2}$ and $\gamma=\left[\begin{array}[]{c|c}A&0\\\ \hline\cr
0&1\end{array}\right]\in\mathrm{SL}(3,\mathbb{Q})$, thus

$Q(x)=Q_{0}(hx)=Q_{\alpha}(\gamma x).$

The form $Q$ is $\mathrm{SL}(3,\mathbb{Q})$-equivalent to the form
$Q_{\alpha}$. Then the assertion $(1)$ of the corollary allows us to show that
$Q$ fullfills the conjecture.

$\square$

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# Application of a plasma-based energy booster
in operational free-electron lasers

S Schröder1,2 J Osterhoff1,2 S Wesch1 C B Schroeder2 1 Deutsches Elektronen-
Synchrotron DESY, 22603 Hamburg, Germany  2 Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA<EMAIL_ADDRESS>
###### Abstract

Plasma-based accelerators are a promising approach for reducing the size and
cost of future particle accelerators, making them a viable technology for
constructing and upgrading X-ray free-electron lasers (FELs). Adding an energy
booster stage to the linear accelerator of an operational X-ray FEL is
recognised as a realistic near-term application of plasma accelerators, with a
significant impact on the scientific reach of these facilities. Here, we
discuss potential use cases of such a plasma-based energy booster and apply
particle-in-cell simulations to estimate its ability to enhance the
performance of existing X-ray FEL facilities.

##  1 Motivation

Free-electron lasers (FELs)[1, 2] stand as unparalleled tools for exploring
matter at molecular to atomic scales, providing coherent and ultra-short
femtosecond photon pulses. FELs with wavelength extending into the hard X-ray
regime ($\mathcal{O}($0.1\text{\,}\mathrm{\SIUnitSymbolAngstrom}$)$) excel as
photon sources in pulse brightness, average and peak power, enabling major
scientific breakthroughs in structural biology[3, 4], atomic physics[5], and
material science[6], among other fields. Based on slightly different
technological concepts and designs, each of today’s FEL is a unique facility
(see Figure 1) attracting users with corresponding profiles, yet sharing the
common constant pursuit of expanding scientific reach. The performance of an
FEL is primarily determined by its radiation wavelength $\lambda_{r}$ and
brilliance $B$. The radiation wavelength determines the spatial resolution of
an FEL:

$\lambda_{r}\approx\frac{\lambda_{u}}{2\gamma^{2}}\left(1+\frac{K^{2}}{2}\right),$
(1)

where $\gamma$ is the relativistic gamma factor, $\lambda_{u}$ and $K$ are the
undulator period and undulator strength, respectively. The quality of the
radiation is directly related to the quality of the electron beam and is
expressed through the brilliance:

$B=\left[\frac{\text{ \\#
photons}}{\text{s}\cdot\text{mrad}^{2}\cdot\text{mm}^{2}\cdot 0.1\%\text{
BW}}\right]\propto\frac{\langle f\rangle
I}{\varepsilon_{x}\varepsilon_{y}\delta},$ (2)

where the transverse electron beam emittances $\varepsilon_{x/y}$ defines the
cross-section and the divergence of the photon pulse, the energy spread
$\delta$ determines the radiation bandwidth (BW), the beam current $I$ defines
the radiation intensity, and the average repetition rate $\langle f\rangle$
determines the resulting average power of the X-ray beam. The $1/\gamma^{2}$
scaling of the radiation wavelength but also the pulse pattern (i.e. the pulse
repetition rate and the length of the pulse train) depending on the
capabilities of the underlying linear electron accelerator (linac) make the
linac an integral part of FEL system upgrade considerations. Today’s FEL
linacs, based on radiofrequency (RF) acceleration technologies, account for
30–50% of the considerable size of FELs, ranging from hundreds of metres up to
several kilometres. A substantial boost in final electron beam energy—embedded
into the layout of the existing facility—thus requires compact, high-gradient
acceleration. In the following, we discuss the usage of a single plasma
accelerator stage based on the external injection beam-driven plasma wakefield
acceleration scheme[7, 8, 9].

| Undulator | Electron bunch | Linac
---|---|---|---
$\lambda_{u}$ (mm) | K | E (GeV) | $\varepsilon_{slice}$ (µm) | length (m)
EuXFEL (SASE1&2) | 35 | 1.65–3.9 | 8–17.5 | 0.1 | 2000
EuXFEL (SASE3) | 68 | 4–9 | 8–17.5 | 0.1
FLASH1 | 27.3 | 1.17–1.23 | 0.38–1.35 | 0.5 | 135
FLASH2 | 31.4 | 1.5–2.8 | 0.38–1.35 | 0.5
LCLS-II (1) | 39 | 1.24–5.43 | 4 | 0.2 | 930
LCLS-II (2) | 26 | 0.44–2.44 | 4 | 0.2
LCLS | 30 | 3.5–3.6 | 4 | 0.1 | 650
SwissFEL (Athos) | 40 | 1–3.05 | 2.1–3.4 | 0.1 | 290
SwissFEL (Aramis) | 15 | 1–1.04 | 2.1–5.8 | 0.1 | 440

Figure 1: FEL facility specifications (a) Achievable radiation wavelength
range according to Equation 1. FLASH, SwissFEL, LCLS-II and EuXFEL comprise
two beamlines equipped with different undulator types and/or operated at
different energies (SwissFEL) covering different wavelength ranges. (b) Pulse-
train timing pattern of facilities. (c) Undulator and electron bunch
specifications of facilities [10, 11, 12, 13].

Plasma wakefields sustain electric fields on the order of gigavolt-per-meter
(GV/m)[14], enabling the acceleration of electrons with orders of magnitude
higher acceleration gradients than today’s RF technology (MeV/m). Amplified
undulator radiation originating from an electron beam accelerated in a laser-
driven plasma wakefield [15, 16] recently represented a major milestone in the
field of plasma accelerators. While the laser-driven scheme promises the
realisation of truly compact linacs, the high-power lasers currently available
for wakefield excitation are orders of magnitude away from supporting the
megahertz repetition rate of today’s cutting-edge FELs and thus meet the user
profiles only for low-power operation. Beam-driven wakefield excitation, on
the other hand, is an interesting way also to facilitate high-power operation.
Based on continuous development and the establishment of novel methodologies
for operating a beam-driven plasma wakefield accelerator[17, 18, 19, 20, 21,
22, 23], significant progress has been made over the past decade towards
demonstrating high-efficiency acceleration [24], beam quality preservation[25,
26], and first amplified undulator radiation [27]. Achieving high average
power remains a critical milestone to confirm applicability as an energy
booster in FELs at full capacity, with the acceleration of bunch trains at
megahertz inter-bunch repetition rate while maintaining beam
characteristics[28] already presented.

Figure 2: Exemplary plasma booster scheme. The linear accelerator of an FEL is
extended by a plasma accelerator stage. Two consecutive bunches are
accelerated in the linac. Tightly focused into a plasma channel, the first
bunch drives a cavity-like wake structure with pronounced charge separation of
the mobile plasma electrons and the heavier, immobile plasma ions. The
resulting longitudinal and transverse electromagnetic fields accelerate and
focus the trailing bunch. The energy-depleted driving bunch is extracted from
the beamline; the accelerated trailing bunch is further delivered to the
undulator section for X-ray generation. Depending on the intended purpose, the
plasma stage can be integrated into individual beamlines, allowing the
operation of the beamlines at independent beam energies (as shown here), or
alternatively, it could serve as a common energy booster after the linac.

##  2 Plasma booster scheme

In the scheme of an external injection beam-driven plasma wakefield electron
energy booster stage for existing FELs (see Figure 2), both the wake-driving
and the trailing to-be-accelerated electron bunch are provided by the existing
FEL linac. With an FEL linac typically consisting of a photo-cathode followed
by RF-based acceleration sections interleaved with magnetic chicanes for
longitudinal bunch compression, the low-emittance electron bunches exhibit a
typical kilo-Ampère peak current with femtosecond duration—making them ideally
suited for resonantly driving strong wakefields in a plasma. The plasma
accelerator stage then acts as an energy transformer, wherein the wake-driving
bunch transfers its energy to the trailing bunch, with the plasma acting as
the mediating medium. After the acceleration process in the plasma, the
energy-depleted bunch that has driven the wakefield is extracted from the beam
transfer line, while the accelerated trailing bunch is transferred to the
undulator section for radiation generation. Such an energy booster can fulfil
various targeted linac upgrade objectives, which can be categorised as
follows:

  1. 1.

Reaching shorter X-Ray wavelength. According to Equation 1, an energy boost
generally enables shorter photon wavelengths. Whilst every FEL would, in
principle, benefit from a shorter wavelength range, i.e., a higher spatial
resolution, not every facility is equally suited to achieve this through an
electron energy boost. High-quality radiation generation requires low electron
beam emittance $\varepsilon_{n}<(\gamma\lambda_{r})/(4\pi)\propto 1/\gamma$
(dotted lines in Figure 1a) and energy spread $\sigma_{\delta}<\rho\propto
1/\gamma$ with the Pierce parameter $\rho$. Fulfilling both requirements
becomes increasingly challenging or even unfeasible with an energy boost. As
it stands today, a plasma booster could, in particular, help FLASH cover the
water window fully (see Figure 1a)–the most relevant wavelength range for
biological studies, or help LCLS-II to realise the already planned electron
energy doubling in a more cost-effective plasma-based variant. In contrast,
the wavelength range of LCLS and SwissFEL with today’s configuration is
already limited by beam emittance and energy spread, respectively.

  2. 2.

Parallel operation of beamlines at independent beam energies. Due to the
shared linear accelerator in FELs comprising more than one beamline, the beam
energy and, consequently, the achievable wavelength range in different
experimental areas are interdependent. Implementing a plasma booster in the
individual beamlines would allow their operation at different energies. For
instance, the soft X-ray EuXFEL beamline could operate in the water window,
and the hard X-ray beamline still reaches the lowest achievable wavelength
simultaneously. With a plasma booster, FELs that are typically overbooked gain
operational flexibility; costly downtime of individual beamlines due to
incompatibilities between user experiments will be reduced.

  3. 3.

Increasing pulse train length. The timing pattern of the photon pulse train
(see Figure 1b) dictates the observable dynamics. In a simplified picture, the
pulse width establishes the temporal resolution, the temporal gap between
pulses defines the sampling rate, and the length of the pulse train determines
the maximum observable duration of a dynamics. Thus, there is a pursuit of
ever shorter pulses, higher pulse-to-pulse repetition rates, and long pulse
trains. However, this has technological limits, namely the power acceptance in
the accelerator modules. Linear accelerators based on superconducting radio-
frequency (SRF) accelerator technology, operated at the maximum achievable
acceleration gradient, require a low duty cycle for cooling, leading to a
characteristic macro-/micro-pulse structure (FLASH, EuXFEL) (see Figure 1b).
Continuous pulse operation using SRF technology (LCLS-II) requires a reduction
in the acceleration gradient and, consequently, a larger facility or operation
at lower electron energy. Employing a plasma booster would allow an increase
of the duty cycle in SRF modules by reducing the acceleration gradient without
impacting the final beam energy. A favourable photon pulse train pattern could
be achieved without compromising the attainable wavelength range.

Beyond these FEL upgrade applications, electron bunches with
$\mathcal{O}($10\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$)$ energies are
imperative for future strong-QED[29, 30] and photo-nuclear[31] experiments. A
plasma booster stage could significantly increase the scientific reach of
those studies conducted in FELs, while the lower beam quality and average
power requirements serve the booster as a stepping stone to a full-capacity
plasma booster for actual FEL operation. Likewise, injector linacs of
synchrotrons and storage rings could employ a plasma booster.

Despite keen interest from the plasma accelerator as well as the FEL
community, there is currently no estimate for the achievable charge and energy
of a plasma booster system in existing free-electron lasers. The development
of such a plasma-based energy booster hinges on its intended application and
even more on the available infrastructure. In contrast to a detailed design
study tailored to the individual FEL facilities, the aspiration of this study
is, firstly, to provide an overview of the wealth of possible applications of
a booster stage and, secondly, to introduce quantitative figures into the
discussion—following the philosophy of keeping the existing facilities
unchanged as much as possible, exploiting the idealised capabilities of a
plasma accelerator stage.

##  3 Methodology

The achievable acceleration is fundamentally limited by the transferable
energy of the wake-driving bunch, $E_{kin}Q$, with the kinetic energy
$E_{kin}$ and the charge $Q$. To estimate a physical upper limit, we fully
exploit the currently available electron bunch at the facilities for wakefield
excitation and assume that the trailing bunch can be generated and accelerated
in the same linac111This assumption on two-bunch generation is non-trivial.
Testing its feasibility requires simulations that consider collective effects
(space charge, coherent synchrotron radiation), and complementing proof-of-
concept experiments.. The plasma response is simulated with single-step
particle-in-cell (PIC) simulations, based on which the achievable acceleration
gradient and bunch charge are estimated. The methodology of the simulation
study is illustrated in Figure 3 and follows these steps:

|  E |  Q |  $I_{p}$ |  $\sigma_{z}$ |  $\varepsilon_{x/y}$
---|---|---|---|---|---
|  (GeV) |  (nC) |  (kA) |  (µm) |  (µm)
FLASH |  1 |  1 |  2 |  60 |  0.5
EuXFEL |  17.5 |  1 |  6 |  30 |  0.1
SwissFEL |  3.4 |  0.2 |  0.8 |  12 |  0.1
LCLS-II |  4 |  0.1 |  0.8 |  10 |  0.1

Figure 3: Methodology of simulation study. (a) Two consecutive bunches, the
wakefield-driving and to-be-accelerated trailing bunch, are tightly focused
into a plasma. The first bunch drives a nonlinear wakefield in which the
trailing bunch is accelerated. The excited wakefield (dotted grey line) can be
flattened over the extent of the trailing bunch (solid blue line) by means of
beam loading when precisely tailoring the current profile of the trailing
bunch. In order to change the acceleration gradient, the position of the
trailing bunch is scanned over the accelerating wake phases (see bottom
panel), which also changes the loadable bunch charge. (b) Beam parameters of
the wakefield-driving bunch.

  1. 1.

Drive-bunch generation with today’s available electron bunch parameters.
Lowest bunch emittance, nominal correlated energy spread, maximum available
bunch charge, maximum achievable bunch energy, and a Gaussian current profile
with 80% of the maximum achievable peak current are assumed for the wake-
driving bunch (see Figure 3c). Maximum charge and energy allow driving and
sustaining the acceleration over the longest distance with maximum power and
the highest beam rigidity. The limitation of the peak current $I_{p}$ to 80%
makes the assumption of a Gaussian current profile more realistic and
mitigates coherent synchrotron radiation induced instabilities at maximum
compression.

  2. 2.

Longitudinal and transverse matching. For resonant excitation of the
wakefield, the wake-driving bunch’s length must match the resulting
wakefield’s decelerating phase. To the peak current at its maximum, the plasma
density is chosen according to the bunch length in the range of
$k_{p}\sigma_{z}=[0.6,1.8]$, where $k_{p}^{-1}=c/\omega_{p}$ is the plasma
skin depth with $c$ the speed of light and $\omega_{p}=\sqrt{4\pi
n_{p}e^{2}/m_{e}}$ (in CGS units) the plasma frequency with the elementary
electron charge $e$ and electron mass $m_{e}$. Higher plasma density will
generally allow higher gradients to be achieved, but require smaller bunch
dimensions and reduce the maximum charge load. Both the wake-driving and
trailing bunch are transversely matched to the focusing fields in the plasma,
allowing for in-principle transverse beam-quality preservation.

  3. 3.

Current profile shaping of the trailing bunch for field flattening. Injecting
charge into the plasma wakefield changes its shape by means of beam
loading[32]. This effect fundamentally limits the charge that can be injected
into a plasma accelerator. By precisely shaping the current profile of the
injected charge, beam loading can be used to flatten the field over the wake
phase extent of the to-be-accelerated bunch (see Figure 3a) such that the
bunch is uniformly accelerated and does not exhibit energy spread during
acceleration[32, 33]—a necessity for scalable acceleration over long plasma
distances and loss-free bunch transport.

  4. 4.

Scanning the acceleration gradient via the trailing bunch position. The
acceleration gradient is determined by the wakefield phase at which the
trailing bunch is injected, which at the same time determines the maximum
loadable charge under the assumption of field flattening. The link between
achievable gradient and loadable charge is gained by scanning the position of
the trailing bunch and tailoring the current profile at each injection
position (compare top and bottom panels in 3a).

The simulations are conducted with the 3D quasi-static PIC code HiPACE++[34]
using the built-in SALAME algorithm for current-profile shaping[35]. Both
bunches were modelled with $10^{6}$ numerical particles, with the macro-
particles of the drive bunch having constant weight. The plasma was sampled
with 4 particles per cell. A simulation box of size $20\times 20\times 16$
$k_{p}^{-3}$ (in $x\times y\times\zeta$) was resolved by a grid of $1024\times
1024\times 600$ cells.

##  4 Results and discussion

Based on the discussion about the most potent applicability of a plasma
booster (Section 2), simulations are performed for FLASH, SwissFEL, LCLS-II,
and EuXFEL. The expected trend of a lower loadable charge, $Q_{acc}$, at
higher acceleration gradients, $G$, is found across all facilities (see Figure
4a); the rate varies. The rapid reduction in bunch charge with increased
acceleration gradient for FLASH can be attributed to the comparatively long
bunch length and the correspondingly lower plasma densities. The loadable
charge depending on the acceleration gradient is empirically estimated (solid
lines) to extrapolate the achievable beam energy in finite plasma channel
lengths from 1 cm up to 2 m (see Figure 4b). For beam quality preservation,
the driven wakefield must not change over the acceleration length. The maximum
acceleration length is given by the depletion length,
$L_{dep}=E_{drive}/E_{z,max}$, at which the part of the drive-bunch that
experiences the maximum decelerating field ($E_{z,max}$) has transferred its
energy, $E_{drive}$, to the plasma. At the EuXFEL, the acceleration could be
sustained over meter-long plasma channels owing to the high initial beam
energy, whereas at LCLS-II and FLASH the acceleration is limited to 50 cm
plasma channel length and at SwissFEL to 20 cm. In the case of no charge
reduction of the accelerated beam, an energy gain of 0.5 GeV, 1.2 GeV, 2.5
GeV, 7 GeV to a final beam energy of 1.85 GeV, 3.3 GeV, 6.5 GeV, 24.5 GeV can
be achieved for FLASH, SwissFEL, LCLS-II, and EuxFEL, respectively. If the
final bunch charge can be reduced to 50%, an increased energy gain of 1 GeV, 2
GeV, 4 GeV, 14 GeV could be achieved for FLASH, SwissFEL, LCLS-II, and EuxFEL,
respectively—representing energy-doubling for FLASH and LCLS-II in only 50 cm.
Operating EuXFEL at reduced initial energy of 10 GeV, an energy gain of 5 GeV
and 11 GeV can be achieved for equal and 50% reduced bunch charge,
respectively.

This study is based on fundamental assumptions that need to be revisited in
the context of these results. First, the available bunches at the facilities
were used unchanged for driving the wakefield. Second, a perfect flattening of
the field was required. When assuming the drive-bunch current profile to be
also tailored, the transformer ratio, $E_{z,acc}/E_{z,max}$, [36, 37] and
depletion length increases, i.e., the energy transfer efficiency enhances[38].
The results presented here can therefore be considered conservative estimates,
and further optimisation is possible. Depending on the scheme, perfect field
flattening for energy-spread preservation may not be necessary. For
attosecond-pulse schemes[39], an increased correlated energy spread may be
desirable for post-plasma compression. Or, the typically negative chirp of the
initial beam can be reduced via overloading the wakefield. However, owing to
the inherently large changes in wakefield strength over the small scale of
wake structure, the current profile cannot deviate significantly from the here
optimised cases if an acceleration on the meter scale is to be sustained. The
values presented here are therefore also relevant for advanced schemes.

The here shown trade-off between bunch charge and final beam energy is used as
a figure of merit to infer on the trade-off between radiation brilliance and
wavelength. Using the bunch charge as the main indicator for the impairment of
brilliance is not entirely correct, as the current profile is ultimately
decisive. The low-current tail of the trailing bunch may not contribute to the
radiation generation in the undulators. Given the consistently triangular
shape and the limited extent of the wake structure, however, the average
current of the bunch scales with its bunch charge and is therefore a valid
measure of the qualitative effect on the resulting brilliance: lower charge
results in lower brilliance. Furthermore, as discussed above, FELs have
stringent requirements not only on the energy spread for the generation of
high-quality amplified undulator radiation, but also on the beam emittance.
Although emittance preservation is theoretically possible in the blowout
regime, it is technically increasingly difficult to achieve over a long plasma
channel where any transverse misalignment or source of instability grows.
Given the requirements of both parameters, energy spread and emittance,
scaling with $1/\gamma$, the ultimately achievable FEL radiation with the
accelerated bunches discussed here must be thoroughly quantified with
simulations and tolerance studies must be performed that consider the
transverse evolution of the bunches in the plasma and model the resulting
radiation process in the undulators.

Figure 4: Simulation results. (a) Coupling between maximum loadable charge and
achievable acceleration gradient under the assumption of optimal beam loading.
(b) Resulting bunch charge and energy for variable plasma channel lengths. Two
horizontal lines across each facility phase space highlight scenarios where
the injected bunch has the same and half the charge as the initial bunch at
the facilities. For the EuXFEL two options are shown with either an initial
beam energy of 17 GeV or 10 GeV (lighter color).

##  5 Conclusion and Outlook

Beam-driven plasma wakefield accelerators have made significant strides in
attaining electron beam characteristics relevant to X-ray FELs in recent
years. Consequently, exploring the quantitative reach of a plasma-based
accelerator stage as an upgrade option for existing free-electron lasers is
pertinent.

In the context of an all-encompassing upgrade of an FEL, equal attention must
be paid to the wavelength and brilliance of the radiation. A plasma booster
that upgrades an FEL to its full capacity must support the operation with its
diverse user groups with individual requirements for both performance
parameters and, therefore, should not compromise one against the other.
However, individual users may benefit from a significantly extended wavelength
range at lower brilliance and vice versa. The simulation results here
highlight a wide range of highly pertinent possibilities for upgrading FELs
through a plasma booster, which can greatly increase the phase space of the
available bunch charge and energy and, thus, the flexibility of the facility.
The study here focused on the limitations that arise through beam loading. To
gain a complete picture of the ultimate limitations that also comprise effects
such as head erosion, ion motion, and hosing, full-3D PIC simulations along
the full acceleration length with evolving bunches are required. Beyond pure
theoretical limitations, to also understand practical limitations, an extended
tolerance study based on start-to-end simulations is required including
collective effects and the FEL process. It is worth noting, that the gain
length scales linearly with beam energy, reaching similar saturation with
higher beam energy will thus also require an upgrade of the undulator beamline
and requires further studies.

Despite the ongoing successes in external injection beam-driven plasma
wakefield acceleration in recent years, implementing such an energy booster
stage still demands significant R&D efforts and encounters technical hurdles.
First and foremost, enabling high average power operation is imperative. The
precise two-bunch generation already represents a vast area of exploration and
is one of the most crucial technological developments to be made towards the
realisation of a booster stage. Last, but not least, the routine operation of
these micrometre-scale plasma accelerator structures demands precision in beam
delivery for both in- and out-coupling, surpassing the capabilities of today’s
FELs. Novel approaches employing computer-aided, autonomous precision control
and electron beam diagnostics will be indispensable and may revolutionise FEL
operation itself. The discussed booster stage is not only relevant for
upgrading existing FELs but is also an attractive avenue for constructing
additional FELs worldwide and preparing FELs for industrial applications. A
specialised service centre to support drug development through individualised
machines for the targeted development of advanced materials could be a
possible industrial application. Advanced operational controls will hereby
play a pivotal role in reducing operational costs—especially personnel
expenses, which currently constitute about 50% of the operational costs, with
a need for highly specialised experts available only at a few major research
laboratories around the world.

The successful integration of a plasma-based electron energy booster into the
operation of an FEL will represent a milestone in plasma wakefield
acceleration and demonstrate the maturity of this technology on the path to
future high-energy physics applications.

C.B.S, J.O., and S.S. are supported by the Director, Office of Science, Office
of High Energy Physics, of the U.S. Department of Energy under Contract No.
DE-AC02-05CH11231. S.S. was supported by the Helmholtz Innovation and Transfer
Center via the Field Study Fellowship.

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