# Length-Preserving Directions and Some Diophantine Equations

Juan Tolosa

###### Abstract

We study directions along which the norms of vectors are preserved under a
linear map. In particular, we find families of matrices for which these
directions are determined by integer vectors. We consider the two-dimensional
case in detail, and also discuss the extension to three-dimensional vector
spaces.

##  1  Introduction.

In the nice Webinar talk “Eigenpairs in Maple” of June 25, 2015 [3], Dr.
Robert Lopez discussed how to use Maple to find eigenvalues and eigenvectors
(eigenpairs) of a matrix $A$. An eigenvector of $A$ is a (nonzero) vector
whose direction is preserved under multiplication by $A$. By the end of the
talk, Dr. Lopez, as an aside, asked the question, what about preserving the
magnitude of the vector, rather than its direction? In other words, what about
(nonzero) vectors ${\bf v}$ such that $\|{\bf v}\|=\|{A{\bf v}}\|$, where
$\|{\bf v}\|$ is the usual Euclidean norm? He provided the example of

$A=\begin{pmatrix}4&3\cr-2&-3\end{pmatrix};$

regarded as a map from ${\mathbb{R}}^{2}$ to itself; it preserves the norms,
but not the directions, of the vectors with integer coordinates ${\bf
v}_{1}=\langle 1,-1\rangle$ and ${\bf v}_{2}=\langle 17,-19\rangle$. He
clarified he had found such matrix by using Maple and “for-loops” to find
matrices $A$ for which the equation $\|{\bf v}\|=\|{A{\bf v}}\|$ would have
integer solutions.

In this article we explore this intriguing idea, obtain several families of
such “nice” $2\times 2$ matrices, then consider a few $3\times 3$ examples,
and discuss a couple of related quadratic Diophantine equations.

To avoid repetition, when considering an equation involving a vector ${\bf
v}$, by a solution ${\bf v}$ we will consistently mean a nontrivial solution
${\bf v}\neq{\bf 0}$.

##  2 General considerations.

A given $n\times n$-matrix $A$ with real entries generates a related linear
map ${\bf v}\mapsto A{\bf v}$ of ${\mathbb{R}}^{n}$ into itself; we will use
the same letter $A$ to represent this map.

We seek nonzero vectors ${\bf v}$ whose norm is preserved under this linear
map; in other words, we seek (nonzero) solutions of the equation

$\|{\bf v}\|=\|{A{\bf v}}\|.$ (1)

First of all, observe that, since $\|{\bf\lambda{\bf
v}}\|=|\lambda|\cdot\|{\bf{\bf v}}\|$, then the entire line generated by any
nonzero solution of (1) will consist of vectors whose norm is preserved by
$A$; we will call these lines the norm-preserving lines.

Next, if $A$ has eigenvalue $1$, or $-1$, then the corresponding eigenspace
will consist entirely of solutions of (1) as well.

The interesting case, though, is when there are nonzero solutions of (1) that
are not eigenvectors. This may happen even if $A$ has an eigenvalue $\pm 1$.
For example, the matrix

$A=\begin{pmatrix}1&-8\cr 0&3\end{pmatrix}$

has eigenvalue $1$ with eigenline determined by ${\bf v}=\langle 1,0\rangle$,
but also has another, noninvariant, norm-preserving line, determined by ${\bf
w}=\langle 9,2\rangle$. Along this line, the map $A$ acts like a rotation.

At the other end of the spectrum we have the case of an orthogonal matrix $A$,
for which every line through the origin is norm-preserving. In the $2\times 2$
case these are typically rotations, for which these lines are noninvariant;
however, all lines are rotated by the same angle under $A$. In the general
case this does not happen: each norm-preserving line is usually rotated by a
different angle.

##  3 The $2\times 2$ case.

Let us discuss nonzero solutions of equation (1) for the case of a $2\times 2$
real-valued matrix $A$. We will first obtain conditions for existence of such
solutions, and next, find families of norm-preserving lines determined by
integer vectors.

###  3.1 Existence of a solution.

Let us study solutions of equation (1) for a general real-valued $2\times
2$-matrix

$A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$

Equation (1) is equivalent to $\|{\bf v}\|^{2}=\|{A{\bf v}}\|^{2}$, or $({\bf
v},{\bf v})=(A{\bf v},A{\bf v})$, where $({\bf v},{\bf w})$ is the usual
Euclidean inner product. The right-hand side becomes

$\|{A{\bf v}}\|^{2}=(A{\bf v},A{\bf v})=({\bf v},A^{t}A{\bf v})=({\bf v},B{\bf
v}),$

where $A^{t}$ is the transpose of $A$, and $B=A^{t}A$. Thus, equation (1) is
equivalent to

$({\bf v},(B-I){\bf v})=0,$ (2)

Where $I$ is the identity matrix.

Further, we have

$B=A^{t}\,A=\begin{pmatrix}a^{2}+c^{2}&ab+cd\\\
ab+cd&b^{2}+d^{2}\end{pmatrix}=\begin{pmatrix}m&p\\\ p&n\end{pmatrix},$

where

$\displaystyle m$ $\displaystyle=$ $\displaystyle a^{2}+c^{2},$ (3)
$\displaystyle n$ $\displaystyle=$ $\displaystyle b^{2}+d^{2},$ (4)
$\displaystyle p$ $\displaystyle=$ $\displaystyle ab+cd,$ (5)

so that if we denote ${\bf v}=\langle x,y\rangle$, then the quadratic form at
the left-hand side of (2) is

$\Phi(x,y)=(m-1)x^{2}+2pxy+(n-1)y^{2};$ (6)

with this notation, (1) or, equivalently, (2), is in turn equivalent to
$\Phi(x,y)=0$. We now prove the following result.

######  Theorem 1.

Norm-preserving lines exist if and only if

$a^{2}+b^{2}+c^{2}+d^{2}\geq 1+\det(A)^{2}.$ (7)

We will provide two proofs of this fact, one analytic, and one geometric.

###### First Proof.

Norm-preserving lines are determined by ${\bf v}=\langle x,y\rangle$, where
$(x,y)$ is a nontrivial solution of $\Phi(x,y)=0$, where $\Phi(x,y)$ is given
by (6). If $n=1$, that is, if $b^{2}+d^{2}=1$, this equation becomes

$(m-1)x^{2}+2pxy=x\left[(m-1)x+2py\right]=0,$

which has the nontrivial solution ${\bf v}=\langle 0,1\rangle$. Also, in this
case condition (7) is satisfied, since

$\displaystyle a^{2}+b^{2}+c^{2}+d^{2}-1-\det(A)^{2}$
$\displaystyle=a^{2}+c^{2}-\det(A)^{2}=(a^{2}+c^{2})-(ad-bc)^{2},$

which is equal to $(ab+cd)^{2}=p^{2}$ under the assumption that
$n=b^{2}+d^{2}=1$. This follows from the identity

$(a^{2}+c^{2})-(ad-bc)^{2}-(ab+cd)^{2}=[1-(b^{2}+d^{2})](a^{2}+c^{2}).$

Thus, in this case condition (7) holds.

If $n\neq 1$, a nontrivial solution of

$(m-1)x^{2}+2pxy+(n-1)y^{2}=0$

must satisfy $x\neq 0$. Dividing by $x^{2}$ and solving for $\frac{y}{x}$, we
obtain

${y\over x}={-p\pm\sqrt{p^{2}-(m-1)(n-1)}\over n-1}.$ (8)

A solution will exist if and only if the discriminant $p^{2}-(m-1)(n-1)$ is
non-negative. Further,

$p^{2}-(m-1)(n-1)=(p^{2}-mn)+m+n-1,$

and

$mn-p^{2}=\det(B)=\det(A)^{2}=(ad-bc)^{2},$

which can also be checked directly using (3)–(5). Thus, there is a norm-
preserving line if and only if

$m+n-1-\det(A)^{2}=(a^{2}+c^{2})+(b^{2}+d^{2})-1-\det(A)^{2}\geq 0,$

which coincides with condition (7). ∎

###### Second Proof.

The eigenvalues $\lambda_{1}$ and $\lambda_{2}$ of the symmetric matrix
$B=A^{t}A$ are (real and) nonnegative; assume
$0\leq\lambda_{1}\leq\lambda_{2}$. By the extreme properties of eigenvalues
(see, for example, [1] or [4]), we have

$\displaystyle\lambda_{1}$ $\displaystyle=$ $\displaystyle\min_{\|{\bf
v}\|=1}\|{A{\bf v}}\|^{2}=\min_{\|{\bf v}\|=1}({\bf v},B{\bf v})$
$\displaystyle\leq$ $\displaystyle\max_{\|{\bf v}\|=1}\|{A{\bf
v}}\|^{2}=\max_{\|{\bf v}\|=1}({\bf v},B{\bf v})=\lambda_{2}.$

Therefore, there will exist norm-preserving lines ${\bf v}$ such that $\|{\bf
v}\|=\|{A{\bf v}}\|$ if and only if

$\lambda_{1}\leq 1\leq\lambda_{2}.$

When the eigenvalues are strictly positive, this condition guarantees the
intersection of the ellipse $({\bf v},B{\bf v})=1$, for which the half-axes
are

${1\over\sqrt{\lambda_{1}}}\qquad\hbox{ and
}\qquad{1\over\sqrt{\lambda_{2}}},$

with the unit circle $\|{\bf v}\|=1$ (see Figure 1).

Figure 1: Illustration of R. Lopez’s example.

The eigenvalues of $B$ are found from the characteristic equation

$\lambda^{2}-t\lambda+\Delta=0,$ (9)

where

$t={\rm trace}\,{B}\qquad\hbox{ and }\qquad\Delta=\det B.$

In terms of $B$, condition (7) reads as

$t\geq 1+\Delta.$

Notice that $\Delta=\det(B)\geq 0$, so actually (7) implies $t\geq
1+\Delta\geq 1>0$. Solving (9) we find

$\lambda={t\over 2}\pm{1\over 2}\sqrt{t^{2}-4\Delta}.$

Let us assume now that $t\geq 1+\Delta$. Then

$t^{2}-4\Delta\geq(1+\Delta)^{2}-4\Delta=(1-\Delta)^{2}.$

so that $\sqrt{t^{2}-4\Delta}\geq|1-\Delta|$. For the largest eigenvalue
$\lambda_{1}$ we have

$\lambda_{1}\geq{1\over 2}(1+\Delta)+{1\over 2}|1-\Delta|.$

Considering both cases $1\geq\Delta$ and $\Delta\geq 1$, we conclude that

$\lambda_{1}\geq\max\\{\Delta,1\\}.$ (10)

In particular, we have $\lambda_{1}\geq 1$, as desired. Now let $\lambda_{2}$
be the smallest eigenvalue. Since $\lambda_{1}\lambda_{2}=\Delta$, we have

$\lambda_{2}={\Delta\over\lambda_{1}},$

and condition (10) now implies that indeed $\lambda_{2}\leq 1$. Thus,
condition (7) guarantees the existence of norm-preserving lines. The converse
is straightforward. ∎

###  3.2 Families of matrices with integer solutions.

We want to find matrices

$A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$

with integer or rational entries, for which the norm-preserving lines are
determined by vectors ${\bf v}$ with integer coordinates; we will call these
integer solution lines.111Of course, if, say, a norm-preserving line is
determined by ${\bf v}=\langle 1,2\rangle$, then it is also determined by
${\bf v}=\langle\sqrt{2},2\sqrt{2}\rangle$. The idea is that there exists a
determining vector with integer coordinates. In the $2\times 2$ case, this
means we are interested in solution lines with rational slopes. Recall that

$\|{A{\bf v}}\|^{2}=(A{\bf v},A{\bf v})=({\bf v},B{\bf v}),$

where

$B=A^{t}\,A=\begin{pmatrix}a^{2}+c^{2}&ab+cd\\\
ab+cd&b^{2}+d^{2}\end{pmatrix}=\begin{pmatrix}m&p\\\ p&n\end{pmatrix},$

so that the solutions of $\|{A{\bf v}}\|^{2}=\|{\bf{\bf v}}\|^{2}$ are given
by ${\bf v}=\langle x,y\rangle$, where $(x,y)$ is a (nontrivial) solution of

$(m-1)x^{2}+2pxy+(n-1)y^{2}=0.$ (11)

If either $m=1$ or $n=1$, it is not hard to get families of solutions. For
example, for the two-parameter family

$A=\begin{pmatrix}\frac{3}{5}&b\cr\frac{4}{5}&d\end{pmatrix},$

we get norm-preserving lines determined by ${\bf v}_{1}=\langle 1,0\rangle$
and

${\bf v}_{2}=\langle 5(1-b^{2}-d^{2}),2(3b+4d)\rangle.$

For the general case, let us assume that the entries of $A$ are integers, and
seek integer solutions of the Diophantine equation (11). Assuming $n\neq 1$
and solving for $y/x$, as we did in (8), we conclude that there are integer
solution lines if and only if the discriminant $p^{2}-(m-1)(n-1)$ is a perfect
square, say, $k^{2}$. This leads to the new Diophantine equation

$p^{2}-(m-1)(n-1)=k^{2},$

which can be rewritten as

$(m-1)(n-1)=p^{2}-k^{2}=(p+k)(p-k),$ (12)

where $m,n,p$ are given by (3)–(5).

We will now find two-parameter families of solutions. To this end, let us set

$\displaystyle m-1$ $\displaystyle=$ $\displaystyle p+k,$ $\displaystyle n-1$
$\displaystyle=$ $\displaystyle p-k,$

or

$\displaystyle(a^{2}+c^{2})-1$ $\displaystyle=$ $\displaystyle p+k,$ (13)
$\displaystyle(b^{2}+d^{2})-1$ $\displaystyle=$ $\displaystyle p-k,$ (14)

to which we will include the equation

$2(ab+cd)=2p,$ (15)

stemming from the definition (5) of $p$. Adding (13) and (14), and subtracting
(15), we get

$(a-b)^{2}+(c-d)^{2}=2,$ (16)

from which we conclude that

$|a-b|=1\qquad\hbox{ and }\qquad|c-d|=1.$ (17)

Next, subtracting (14) from (13), we obtain

$a^{2}-b^{2}+c^{2}-d^{2}=2k,$

or

$(a-b)(a+b)+(c-d)(c+d)=2k$ (18)

Considering all the possibilities in (17), we obtain the following four two-
parameter families of matrices

$\begin{pmatrix}a&a\pm 1\cr c&c\pm 1\end{pmatrix}.$

For each of them, we can use (18) to find the value of $k$. Incidentally,
their transposes,

$\begin{pmatrix}a&c\cr a\pm 1&c\pm 1\end{pmatrix},$

also have integer solution lines.

The example of Robert Lopez corresponds to the case $a-b=1$ and $c-d=1$, which
leads to

$A=\begin{pmatrix}a&a-1\cr c&c-1\end{pmatrix};$ (19)

he chose the particular values $a=4$ and $c=-2$. For the general solution of
type (19), we find from (18) that $k=a+c-1$. Substituting this value into (8),
which now looks like

${y\over x}={-p\pm k\over n-1},$

we conclude that the norm-preserving lines are determined by the vectors

${\bf v}_{1}=\langle 1,-1\rangle$

and

${\bf v}_{2}=\langle(a-1)^{2}+(c-1)^{2}-1,1-a^{2}-c^{2}\rangle.$

The remaining cases can be discussed similarly.

Figure 2: Norm-preserving directions shown.

Figure 2 illustrates the case $a=2$, $c=-3$; the two length-preserving lines
are shown. The direction vectors are ${\bf v}_{1}=\langle 1,-1\rangle$ and
${\bf v}_{2}=\langle 4,-3\rangle$. As expected, the solution lines pass
through the intersections of the ellipse $\|{A{\bf v}}\|=1$ with the unit
circle.

The corresponding picture for the matrix chosen by Robert Lopez was shown in
Figure 1; the solutions lines are not depicted, since they are rather close to
each other.

If the ellipse is tangent to the unit circle, as for example when $a=3$,
$c=-2$, we get only one integer solution line.

##  4 The $3\times 3$ case.

The $3\times 3$ case is considerably more complicated, as well as more
interesting.

If $A$ is a $3\times 3$ real-valued matrix, also regarded as a linear map from
${\mathbb{R}}^{3}$ to itself, then in general $\|{A{\bf v}}\|^{2}=1$ is an
ellipsoid, in terms of the coordinates of ${\bf v}=\langle x,y,z\rangle$. As
in the $2\times 2$ case, we have

$\|{A{\bf v}}\|^{2}=(A{\bf v},A{\bf v})=({\bf v},B{\bf v}),$

where $B=A^{t}A$ is a symmetric matrix with nonnegative eigenvalues. The
equation for norm-preserving vectors, $\|{A{\bf v}}\|=\|{\bf v}\|$, or $({\bf
v},B{\bf v})=({\bf v},{\bf v})$, is equivalent to the cone

$({\bf v},(B-I){\bf v})=0,$ (20)

where $I$ is the identity matrix.

As in the $2\times 2$ case, if we denote the eigenvalues of $B$ by
$0\leq\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}$, then there is a solution of
(20) if and only if

$\lambda_{1}\leq 1\leq\lambda_{3},$ (21)

which guarantees a nonempty intersection of the ellipsoid (or degenerate
ellipsoid) $\|{B{\bf v}}\|^{2}=1$ with the unit sphere $\|{\bf v}\|^{2}=1$.

If condition (21) is satisfied, then the cone determined by (20) will pass
through this intersection of the ellipsoid and the unit sphere. As an
illustration, for the matrix in Example 1, Figure 3 (a) shows the ellipsoid
$\|{A{\bf v}}\|^{2}=1$ and the unit sphere, and Figure 3 (b) has the added
solution cone $\|{A{\bf v}}\|=\|{\bf v}\|$; compare with Figure 2 for the
$2\times 2$ case.

(a)  (b)

Figure 3: Illustration for Example 1: (a) ellipsoid and sphere; (b) same, plus
solution cone.

For the $3\times 3$ case, however, it is considerably harder to find an
expression for condition (21) directly in terms of the entries of $A$.
Instead, we will limit ourselves to discuss several examples that exhibit the
various possible outcomes regarding the existence of integer solution lines.

Before giving examples, let us add another comment: unlike the $2\times 2$
case, in the $3\times 3$ case one cannot hope that in general all the lines in
the cone (20) for a given matrix $A$ with integer or rational coefficients
will turn out to be integer solution lines. Examples 2 and 3 show that,
however, we can still get infinitely many such lines.

###  4.1 Example 1: no integer solution lines.

Let us consider the symmetric matrix

$A=\begin{pmatrix}1&1&\frac{1}{2}\cr
1&\frac{1}{2}&1\cr\frac{1}{2}&1&1\end{pmatrix}$ (22)

The form $\|{A{\bf v}}\|^{2}=(A{\bf v},A{\bf v})=({\bf v},B{\bf v})$ will in
this case have matrix

$B=A^{t}A=A^{2}=\begin{pmatrix}\frac{9}{4}&2&2\cr 2&\frac{9}{4}&2\cr
2&2&\frac{9}{4}\end{pmatrix}$

so that equation (20) will be

$\frac{5}{4}x^{2}+4xy+4xz+\frac{5}{4}y^{2}+4yz+\frac{5}{4}z^{2}=0.$ (23)

This equation has infinitely many real-valued solutions, which constitute the
solution cone shown in Figure 3 (b). We want to show, however, that there are
no integer solution lines, that is, no nontrivial vectors ${\bf v}$ with
integer coordinates such that $\|{A{\bf v}}\|=\|{\bf v}\|$.

Indeed, (23) is a quadratic equation in $z$, which we can solve:

$z=-{8\over 5}(x+y)\pm{\sqrt{39x^{2}+48xy+39y^{2}}\over 5}.$

Therefore, there will be integer solution lines if and only if the
discriminant $39x^{2}+48xy+39y^{2}$ is a perfect square. We now prove this
does not happen.

######  Theorem 2.

The Diophantine equation

$39x^{2}+48xy+39y^{2}=u^{2}$ (24)

has no nontrivial solutions, that is, no nonzero integer solutions.

###### Proof.

Let $x=2^{a}v$ and $y=2^{b}w$, with $a,b$ nonnegative integers and $v,w$ odd.
We may assume that $a\leq b$; otherwise, we interchange $x$ and $y$. Then

$39x^{2}+48xy+39y^{2}=2^{2a}(39v^{2}+2^{b-a}48\,vw+2^{2(b-a)}39\,w^{2}),$

and this will be a square only if and only if the expression in parentheses is
a square. But if $b=a$ this expression is congruent to $2$ mod 4, while if
$b>a$ this expression is congruent to $3$ mod 4, so in either case this is
impossible. ∎

###  4.2 Example 2: a dense set of integer solution lines.

Consider the symmetric matrix

$A=\begin{pmatrix}1&2&2\cr 2&1&2\cr 2&2&1\end{pmatrix}.$ (25)

Here

$B=A^{t}A=A^{2}=\begin{pmatrix}9&8&8\cr 8&9&8\cr 8&8&9\end{pmatrix}$

so that equation (20) will be

$8x^{2}+16xy+16xz+8y^{2}+16yz+8z^{2}=0,$

or (after dividing by $8$)

$(x+y+z)^{2}=0.$ (26)

In our case, the cone degenerates into the plane $x+y+z=0$. We can pick an
integer basis, say ${\bf v}_{1}=\langle 1,0,-1\rangle$ and ${\bf
v}_{2}=\langle 0,1,-1\rangle$, and obtain every integer norm-preserving line
as generated by $\alpha{\bf v}_{1}+\beta{\bf v}_{2}$, with integer
coefficients $\alpha,\beta$ such that $\alpha^{2}+\beta^{2}>0$. This
constitutes a dense set of integer solution lines, among all possible
solutions in the plane $x+y+z=0$.

The ellipsoid $\|{A{\bf v}}\|^{2}=1$ lies inside the unit sphere, and is
tangent to it along the intersection of the sphere with the solution plane;
Figure 4 depicts the situation.

Figure 4: Solution cone is degenerate.

###  4.3 Example 3: infinitely many integer solution lines.

Finally, let us consider an example when there are still infinitely many
integer solution lines, yet we cannot guarantee they are dense in the cone of
all solution lines. Consider the matrix

$A=\begin{pmatrix}1&2&3\cr 2&1&1\cr 1&1&1\end{pmatrix}.$ (27)

One eigenvalue of $A$ is $-1$, with eigenvector ${\bf=}\langle-1,1,0\rangle$,
which therefore provides one integer solution line. Are there any other such
lines? The matrix $B=A^{t}A$ is

$B=\begin{pmatrix}6&5&6\cr 5&6&8\cr 6&8&11\end{pmatrix}$

and consequently equation (20) becomes

$5x^{2}+10xy+12xz+5y^{2}+16yz+10z^{2}=0.$

Solving for $x$ (which provides a slightly shorter answer than solving for
$z$) yields

$x=-y-{6\over 5}z\pm{\sqrt{-20yz-14z^{2}}\over 5}.$

We conclude that there will be integer solutions lines if and only if the
discriminant $-20yz-14z^{2}$ is a perfect square.222Notice, by the way, that
by setting $z=0$ we get back the eigenline determined by
$\langle-1,1,0\rangle$. This leads to the Diophantine equation

$-20yz-14z^{2}=u^{2}.$ (28)

Let us find all integer solutions of (28). As remarked previously, if $z=0$ we
get back the eigenline corresponding to $\lambda=-1$, generated by
$\langle-1,1,0\rangle$, so let us assume $z\neq 0$.

If $z$ is odd, then

$-20yz-14z^{2}\equiv-14\equiv 2\pmod{4},$

and hence cannot be a square. Therefore $z$ must be even, so $z=2a$ for some
integer $a$. Substituting into (28) we get

$8(-5ya-7a^{2})=u^{2},$

whence $u$ is divisible by $4$. Letting $u=4b$ and dividing by $8$, we obtain

$a(-5y-7a)=2b^{2}.$

Letting $a=v^{2}q$, with $q$ square-free, we obtain

$v^{2}q(-5y-7v^{2}q)=2b^{2}.$ (29)

Denoting for a moment $p=-5y-7v^{2}q$, the above equation reads

$v^{2}pq=2b^{2},$ (30)

where $q$ is square-free. This implies that $pq$ is even.

If $p$ is even, so $p=2t$ for some integer $t$, then (30) becomes
$v^{2}tq=b^{2}$, and it follows that $t=qr^{2}$ for some integer $r$, so that

$p=2qr^{2}.$ (31)

On the other hand, if $p$ is odd, then we must have $q=2s$, with $s$ odd;
hence (30) is now $v^{2}ps=b^{2}$, whence $p=sr^{2}$ for some integer $r$, and
therefore

$p=sr^{2}={q\over 2}r^{2}=2q\left({r\over 2}\right)^{2},$

and we get back expression (31) for $p$, if we allow $r$ to be an integer or a
half-integer.

We conclude that $-5y-7v^{2}q$ must be equal to $2qr^{2}$, for some integer or
half-integer $r$. Solving for $y$, we obtain

$y=-{q\over 5}(7v^{2}+2r^{2}),$ (32)

where $v,q$, and $2r$ are arbitrary integers, subject only to the condition
that $y$ be an integer. Moreover, since rational coordinate vectors also
determine integer solution lines, we can even drop this additional condition.

To find the value of the other variables in terms of $(v,q,r)$, we observe
first that

$z=2a=2v^{2}q,$ (33)

and that, from (29), we get $b=vqr,$ whence

$u=4vqr.$ (34)

(We have chosen only the “$+$” sign for $b$, since the “$\pm$” is recovered
when computing the $x$-value; see (35).) Next, since

$x=-y-{6\over 5}z\pm{1\over 5}u,$ (35)

we get

$x=\displaystyle{q\over 5}(2r^{2}-5v^{2}\pm 4vr).$ (36)

Equations (35), (32), and (33) provide the coordinates of all possible integer
solution lines. Observing, however, that $q$ is a common factor of all three
coordinates, and proportional vectors provide the same solutions, we may just
set $q=1$, replace $r$ by $r/2$, and conclude that all solution lines of (28)
are given by

$\begin{cases}x=\displaystyle{1\over 10}(r^{2}-10v^{2}\pm 4vr)\cr
y=-\displaystyle{1\over 10}(14v^{2}+r^{2})\cr z=2v^{2},\cr\end{cases}$

where $v$ and $r$ are arbitrary integers.

For example, choosing $(v,r)=(1,4)$ we get the vectors

$\left\langle{11\over 5},-3,2\right\rangle\sim\langle
11,-15,10\rangle\qquad\hbox{ and }\qquad\langle 1,3,-2\rangle;$

and choosing $(v,r)=(1,1)$ we obtain

$\left\langle-{1\over 2},-{3\over 2},2\right\rangle\sim\langle
1,3,-4\rangle\qquad\hbox{ and }\qquad\left\langle-{13\over 10},-{3\over
2},2\right\rangle\sim\langle 13,15,-20\rangle.$

###  4.4 A general method.

There is a method for obtaining infinitely many integer solution lines, which
can be applied to a general $3\times 3$ matrix with integer or rational
coefficients; the drawback is that one must know (at least) one nontrivial
solution. This is an idea by T. Piezas [5]. Namely, if we know one particular
integer solution $(y,z,u)=(m,n,p)$ of the Diophantine equation

$ay^{2}+byz+cz^{2}=du^{2},$

then a two-parameter family of solutions is given by

$\displaystyle y$ $\displaystyle=$ $\displaystyle(am+bn)s^{2}+2cnst-cmt^{2},$
$\displaystyle z$ $\displaystyle=$ $\displaystyle ans^{2}+2amst+(bm+cn)t^{2},$
$\displaystyle u$ $\displaystyle=$ $\displaystyle p(as^{2}+bst+ct^{2}),$

where $s$ and $t$ are arbitrary integers.

For example, for the matrix

$A=\begin{pmatrix}1&2&3\cr 3&4&5\cr 2&3&4\end{pmatrix},$

equation (20) for ${\bf v}=\langle x,y,z\rangle$ is

$13x^{2}+40xy+52xz+28y^{2}+76yz+49z^{2}=0,$

which solved with respect to $x$ produces

$x={-20y\over 13}-2z\pm{\sqrt{36y^{2}+52yz+39z^{2}}\over 13},$ (37)

so to get integer solutions we need to solve the Diophantine equation

$36y^{2}+52yz+39z^{2}=u^{2}.$ (38)

Setting $z=0$, it is not hard to guess the particular solution
$(y,z,u)=(m,n,p)=(1,0,6)$. The corresponding two-parameter solution family
looks like

$\displaystyle y$ $\displaystyle=$ $\displaystyle 36s^{2}-39t^{2}$ (39)
$\displaystyle z$ $\displaystyle=$ $\displaystyle 72st+52t^{2}$ (40)
$\displaystyle u$ $\displaystyle=$ $\displaystyle 216s^{2}+312st+234t^{2}.$
(41)

Each choice of $(s,t)$ produces two integer solution lines ${\bf v}=\langle
x,y,z\rangle$, using the two $x$-values provided by (37). For example, for
$(s,t)=(1,1)$ we get

$\left\langle-{2402\over
13},-3,124\right\rangle\sim\langle-2402,-39,1612\rangle,\qquad\hbox{ and
}\qquad\langle-302,-3,124\rangle;$

and $(s,t)=(1,2)$ yields

$\left\langle-{4976\over
13},-120,352\right\rangle\sim\langle-4976,-1560,4576\rangle,\qquad\hbox{ and
}\qquad\langle-656,-120,352\rangle.$

##  5 Questions for further study, applications.

  * •

Unlike the case of eigenpairs, the solution lines to equation (1) are very
much dependent on the chosen norm in ${\mathbb{R}}^{n}$. It would be
interesting to discuss similar solutions for other norms in Euclidean space.

  * •

We have only grazed the case of a $3\times 3$ matrix $A$. Can we find
solvability conditions of (1) in terms of the coefficients of $A$, as we did
in the $2\times 2$ case? Can one find nontrivial families of integer matrices
$A$ for which (1) has integer solutions?

  * •

Application to toral automorphisms. Many of the $2\times 2$ integer matrices
we studied, for example, the subfamily

$A=\begin{pmatrix}q+1&q\cr q&q-1\end{pmatrix},$ (42)

with $q$ integer, are symmetric and have determinant $-1$; therefore, they can
be regarded also as linear automorphisms of the (flat) 2-torus, which possess
very interesting dynamical properties; see, for example [2], p. 42. Nontrivial
such automorphisms have eigenvectors with irrational slopes; on the other
hand, the integer solution lines of (42) bisect the eigendirections. We can
therefore use integer arithmetic to compute iterates of vectors in the stable
and in the unstable manifolds of such automorphisms. For example, the matrix

$A=\begin{pmatrix}3&2\cr 2&1\end{pmatrix}$ (43)

has integer solution lines generated by ${\bf v}_{1}=\langle 1,-1\rangle$ and
${\bf v}_{2}=\langle-1,3\rangle$, and irrational eigenvalues

$\lambda_{1}=2+\sqrt{5}\qquad\hbox{ and }\qquad\lambda_{2}=2-\sqrt{5}.$

If we choose the equal-norm vectors

${\bf v}_{1}=\langle 1,-1\rangle\qquad\hbox{ and }\qquad{\bf
v}_{3}=\frac{1}{\sqrt{5}}{\bf v}_{2}=\frac{1}{\sqrt{5}}\langle-1,3\rangle,$

then ${\bf u}={\bf v}_{1}+{\bf v}_{3}$ will be along the unstable direction of
$A$, and ${\bf w}={\bf v}_{1}-{\bf v}_{3}$ will be along the stable direction.
Therefore, on the one hand

$A^{n}{\bf u}=\lambda_{1}^{n}{\bf u}=(1+\sqrt{5})^{n}{\bf u},$

and on the other hand

$A^{n}{\bf u}=A^{n}{\bf v}_{1}+\frac{1}{\sqrt{5}}A^{n}{\bf v}_{2}.$

For example,

$\displaystyle A^{10}{\bf u}=\begin{pmatrix}1346269&832040\cr
832040&514229\end{pmatrix}{\bf u}=$ $\displaystyle=\left\langle
514229+\frac{1}{\sqrt{5}}1149851,\,317811+\frac{710647}{\sqrt{5}}\right\rangle,$

provides a way to compute $(2+\sqrt{5})^{10}{\bf u}$ using only integer
arithmetic. A similar calculation can be used for iterates of vectors in the
stable direction.

##  6 Acknowledgments.

I am grateful to Dr. Robert Lopez for introducing the interesting idea of
vectors whose norms are preserved by a linear map. I also wish to thank the
Editorial Board member of the American Mathematical Monthly, for valuable
comments, corrections, and suggestions that were used to significantly improve
the overall quality of the paper. This applies, especially, to a shorter and
more conceptual proof of Theorem 2, and to a complete solution of the
Diophantine equation (28).

## References

  * [1] Gel’fand, I. M. (1989). Lectures on Linear Algebra. New York: Dover Publications.
  * [2] Katok A., Hasselblatt B. (1995). Introduction to the Modern Theory of Dynamical Systems. New York: Cambridge University Press.
  * [3] Lopez, R. (2015). Eigenpairs in Maple. https://www.maplesoft.com/webinars/recorded/featured.aspx?id=1181.
  * [4] Shilov, G. E. (1977). Linear Algebra. New York: Dover Publications.
  * [5] Weiseintein, E. W. (2019). Diophantine equation–2nd powers. MathWorld–A Wolfram Web Resource. mathworld.wolfram.com/DiophantineEquation2ndPowers.html.


1in1in0.8in1in

#  $(1+)$-complemented, $(1+)$-isomorphic copies of $L_{1}$
in dual Banach spaces

Dongyang Chen  School of Mathematical Sciences
Xiamen University, Xiamen, 361005, China<EMAIL_ADDRESS>,  Tomasz Kania
Mathematical Institute
Czech Academy of Sciences
Žitná 25
115 67 Praha 1, Czech Republic
and
Institute of Mathematics and Computer Science
Jagiellonian University
Łojasiewicza 6
30-348 Kraków, Poland<EMAIL_ADDRESS><EMAIL_ADDRESS>and
Yingbin Ruan  School of Mathematics and Statistics
Fujian Normal University, Fuzhou, 350007, China<EMAIL_ADDRESS>
###### Abstract.

The present paper contributes to the ongoing programme of quantification of
isomorphic Banach space theory focusing on the Hagler–Stegall characterisation
of dual spaces containing complemented copies of $L_{1}$. As a corollary, we
obtain the following quantitative version of the Hagler–Stegall theorem
asserting that for a Banach space $X$ the following statements are equivalent:

  * •

$X$ contains almost isometric copies of
$(\bigoplus_{n=1}^{\infty}\oplus\ell_{\infty}^{n})_{\ell_{1}}$,

  * •

for all $\varepsilon>0$, $X^{*}$ contains a $(1+\varepsilon)$-complemented,
$(1+\varepsilon)$-isomorphic copy of $L_{1}$,

  * •

for all $\varepsilon>0$, $X^{*}$ contains a $(1+\varepsilon)$-complemented,
$(1+\varepsilon)$-isomorphic copy of $C[0,1]^{*}$.

Moreover, if $X$ is separable, one may add the following assertion:

  * •

for all $\varepsilon>0$, there exists a $(1+\varepsilon)$-quotient map
$T\colon X\rightarrow C(\Delta)$ so that $T^{*}[C(\Delta)^{*}]$ is
$(1+\varepsilon)$-complemented in $X^{*}$,

where $\Delta$ is the Cantor set.

###### Key words and phrases:

Isomorphic copies of $L_{1}$; Complemented subspaces; Quotient maps; Banach
spaces

###### 2010 Mathematics Subject Classification:

46B15 (primary), 46C05 (secondary).

##  1\. Introduction

In 1968, Pełczyński [17] showed that if a Banach space $X$ contains an
isomorphic copy of $\ell_{1}$, then the dual space $X^{*}$ contains an
isomorphic copy of $L_{1}$ and proved that the converse holds as well subject
to a mild technical condition that was later removed by Hagler [7]. More
precisely, the result stated that the isomorphic containment of $\ell_{1}$ is
equivalent to the following assertions: $X^{*}$ contains a subspace isomorphic
to $L_{1}$, $X^{*}$ contains a subspace isomorphic to $C[0,1]^{*}$. When $X$
is separable, these are further equivalent to the assertions: $X^{*}$ contains
a subspace isomorphic to $l_{1}([0,1])$, and $C[0,1]$ is a quotient of $X$.

Shortly after, Hagler and Stegall [8] obtained a ‘complemented’ version of the
aforementioned Pełczyński’s classical work:

Theorem (Hagler–Stegall). Let $X$ be a Banach space. Then the following
assertions are equivalent:

  1. (1)

$X$ contains a subspace isomorphic to
$(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{n})_{\ell_{1}}$;

  2. (2)

$X^{*}$ contains a complemented subspace isomorphic to $L_{1}$;

  3. (3)

$X^{*}$ contains a complemented subspace isomorphic to $C[0,1]^{*}$;

  4. (4)

$X^{*}$ contains an infinite set $K$ such that $K$ is equivalent to the usual
basis of $\ell_{1}(\Gamma)$ for some $\Gamma$, $[K]$ is complemented in
$X^{*}$, and $K$ is dense in itself in the weak* topology on $X^{*}$;

If, in addition, $X$ is separable, then the assertions (1)–(4) are equivalent
to

  * (5)

There exists a surjective operator $T\colon X\rightarrow C[0,1]$ such that
$T^{*}[C[0,1]^{*}]$ is complemented in $X^{*}$.

The purpose of this note is to quantify the Hagler–Stegall Theorem in the
spirit of a large number of recent results on quantitative versions of various
theorems on and properties of Banach spaces, such as, quantitative versions of
Krein’s theorem [6], Eberlein–S̆mulyan and Gantmacher theorems [2], James’
compactness theorem [5], weak sequential continuity and the Schur property
[11, 12], the (reciprical) Dunford–Pettis property [13, 10], the Banach–Saks
property [3], etc. More broadly speaking, the present paper contributes to the
on-going programme of quantification of Banach space theory.

In the present paper, we quantify the Hagler–Stegall theorem by introducing
the following three quantities denoted by lower-case Greek letters and defined
as infima of certain sets (when the sets happen to be empty, we use the
convention that the corresponding value is $\infty$).

Hereinafter $X$ and $Y$ will stand for Banach spaces; $\mathscr{B}(X,Y)$ is
the space of (bounded, linear) operators from $X$ to $Y$. We then introduce
the following quantities.

  * •

$\alpha_{Y}(X)=\inf\\{\operatorname{d}(Y,Z)\colon Z\text{ is a subspace of X
}\\}$, where $\operatorname{d}(Y,Z)$ is the Banach–Mazur distance between $Y$
and $Z$.

The quantity $\alpha_{Y}(X)$, being directly related to the Banach–Mazur
distance, measures how well $Y$ is from being isomorphically embeddable into
$X$. Obviously, $\alpha_{Y}(X)=1$ if and only if $X$ contains almost isometric
copies of $Y$, that is, for every $\varepsilon>0$, $X$ contains a subspace
$(1+\varepsilon)$-isomorphic to $Y$.

  * •

$\beta_{Y}(X)=\inf\\{\|A\|\|B\|\colon
A\in\mathscr{B}(X,Y),B\in\mathscr{B}(Y,X)$, $AB=I_{Y}\\}$.

The quantity $\beta_{Y}(X)$ measures how well $Y$ is from being isomorphic to
a complemented subspace of $X$. It is easy to see that $\beta_{Y}(X)=1$ if and
only if for every $\varepsilon>0$, there exists a subspace $M$ of $X$ so that
$M$ is $(1+\varepsilon)$-isomorphic to $Y$ and $(1+\varepsilon)$-complemented
in $X$.

  * •

$\theta_{Y}(X)=\inf\\{\|A\|\|S\|\colon
A\in\mathscr{B}(X,Y),S\in\mathscr{B}(X^{*},Y^{*}),SA^{*}=I_{Y^{*}}\\}$.

The quantity $\theta_{Y}(X)$ measures how well $Y$ is isomorphic to a quotient
of $X$ and its dual $Y^{*}$ is isomorphic to a complemented subspace of
$X^{*}$. We see that $\theta_{Y}(X)=1$ if and only if, for every
$\varepsilon>0$, there exists a $(1+\varepsilon)$-quotient map $T\colon
X\rightarrow Y$ so that $T^{*}[Y^{*}]$ is $(1+\varepsilon)$-complemented in
$X^{*}$.

A straightforward argument shows that

(1.1) $\beta_{Y^{*}}(X^{*})\leqslant\theta_{Y}(X)\leqslant\beta_{Y}(X).$

By using the aforementioned three quantities, we quantify the Hagler–Stegall
theorem as follows.

######  Theorem A.

Let $X$ be a Banach space. Then

$\alpha_{(\oplus_{n=1}^{\infty}\ell_{\infty}^{n})_{l_{1}}}(X)=\beta_{C[0,1]^{*}}(X^{*})=\beta_{L_{1}}(X^{*}).$

If, in addition, $X$ is separable, then

$\theta_{C(\Delta)}(X)=\beta_{L_{1}}(X^{*}).$

The following $(1+\varepsilon)$-version of the Hagler–Stegall theorem follows
from Theorem A.

######  Corollary 1.1.

Let $X$ be a Banach space. Then the following assertions are equivalent:

  1. (1)

$X$ contains almost isometric copies of
$(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{n})_{l_{1}}$;

  2. (2)

$X^{*}$ contains a $(1+\varepsilon)$-complemented subspace that is
$(1+\varepsilon)$-isomorphic to $L_{1}$ for every $\varepsilon>0$;

  3. (3)

$X^{*}$ contains a $(1+\varepsilon)$-complemented subspace that is
$(1+\varepsilon)$-isomorphic to $C[0,1]^{*}$ for every $\varepsilon>0$.

If, in addition, $X$ is separable, then

  * (4)

For every $\varepsilon>0$, there exists a $(1+\varepsilon)$-quotient map
$T\colon X\rightarrow C(\Delta)$ so that $T^{*}[C(\Delta)^{*}]$ is
$(1+\varepsilon)$-complemented in $X^{*}$.

##  2\. Preliminaries

Our notation and terminology are standard and mostly in-line with [1, 16].
Throughout the paper, all Banach spaces can be considered either real or
complex. We work with real scalars but the results can be easily amended to
the complex too. By a _subspace_ we understand a closed, linear subspace and
by an _operator_ we understand a bounded, linear map. If $X$ is a Banach
space, we denote by $B_{X}$ the closed unit ball of $X$, by $I_{X}$ the
identity operator on $X$ and, for a subset $K\subseteq X$, $[K]$ for the
closed linear span of $K$. For $\lambda\geqslant 1$, we say that a surjective
operator $T\colon X\rightarrow Y$ is a $\lambda$-quotient map if
$\|T\|\operatorname{co}(T)\leqslant\lambda$. _Quotient maps_ are 1-quotient
maps according to the above terminology. A norm-one surjective operator
$T\colon X\rightarrow Y$ is a quotient map if and only if $T$ is a
$(1+)$-quotient map, that is, a $(1+\varepsilon)$-quotient map for every
$\varepsilon>0$.

The Banach–Mazur distance $\operatorname{d}(X,Y)$ between two isomorphic
Banach spaces $X$ and $Y$ is defined by $\inf\|T\|\|T^{-1}\|$, where the
infimum is taken over all isomorphisms $T$ from $X$ onto $Y$. As defined by
Lindenstrauss and Rosenthal [15], for $\lambda\geqslant 1$, a Banach space $X$
is said to be a $\mathcal{L}_{1,\lambda}$-space whenever for every finite-
dimensional subspace $E$ of $X$ there is a finite-dimensional subspace $F$ of
$X$ such that $F\supseteq E$ and $\operatorname{d}(F,l_{1}^{\dim
F})\leqslant\lambda$. We say that a Banach space $X$ is an
$\mathcal{L}_{1,\lambda+}$-space if it is an
$\mathcal{L}_{1,\lambda+\varepsilon}$-space for all $\varepsilon>0$.

Following the notation from [8], we denote

$\mathcal{F}=\\{(n,i)\colon n=0,1,\ldots,i=0,1,\ldots,2^{n}-1\\}$

and, for $(n,i),(m,j)\in\mathcal{F}$ we write $(n,i)\geqslant(m,j)$ whenever

  * •

$n\geqslant m$,

  * •

$2^{n-m}j\leqslant i\leqslant 2^{n-m}(j+1)-1$.

Let $\Delta=\\{0,1\\}^{\mathbb{N}}$ be the Cantor set endowed with the metric

$\operatorname{d}((a_{n})_{n=1}^{\infty},(b_{n})_{n=1}^{\infty})=\sum_{n=1}^{\infty}\frac{1}{2^{n}}|a_{n}-b_{n}|\quad\big{(}(a_{n})_{n},(b_{n})_{n}\in\Delta\big{)}.$

By Miljutin’s Theorem ([1, Lemma 4.4.7]), $C[0,1]$ is isomorphic (but not
isometric) to $C(\Delta)$. It is well-known that $C(\Delta)^{*}$ and
$C[0,1]^{*}$ are linearly isometric, though.

##  3\. Proof of Theorem A

The present section is devoted to the proof of Theorem A and is conveniently
split into more digestible parts.

###### Proof of Theorem A.

We split the proof into a number of steps.

Step 1.
$\beta_{C(\Delta)^{*}}(X^{*})\leqslant\alpha_{(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{n})_{\ell_{1}}}(X)$.

Since $Z=(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{2^{n}})_{\ell_{1}}$ embeds
isometrically into $(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{n})_{\ell_{1}}$,
it suffices to prove that
$\alpha_{Z}(X)\geqslant\beta_{C(\Delta)^{*}}(X^{*})$. For this, let us fix
$c>\alpha_{Z}(X)$. Then there exists a contractive operator $R\colon
Z\rightarrow X$ that is bounded below by $1/c$.

Let us consider a double-indexed family
$(\Delta_{n,i})_{n=0,i=0}^{\infty,2^{n}-1}$ of clopen subsets of the Cantor
set such that

  1. (1)

$\Delta_{0,0}=\Delta$, $\Delta_{n,i}=\Delta_{n+1,2i}\cup\Delta_{n+1,2i+1}$
($(n,i)\in\mathcal{F}$) and $\Delta_{n,i}\cap\Delta_{n,j}=\varnothing$ if
$i\neq j$;

  2. (2)

the diameter of $\Delta_{n,i}$ is $1/{2^{n}}$ ($0\leqslant i\leqslant
2^{n}-1$).

We set $g_{n,i}=\mathds{1}_{\Delta_{n,i}}$, which is a continuous function,
$[g_{n,i}]_{i=0}^{2^{n}-1}\subseteq[g_{n+1,i}]_{i=0}^{2^{n+1}-1}$,
$(g_{n,i})_{i=0}^{2^{n}-1}$ is isometrically equivalent to the unit vector
basis of $\ell_{\infty}^{2^{n}}$ ($n\in\mathbb{N}$) and
$\bigcup_{n=0}^{\infty}[g_{n,i}]_{i=0}^{2^{n}-1}$ is dense in $C(\Delta)$. We
may then define an operator $T\colon Z\rightarrow C(\Delta)$ by the assignment
$Te_{n,i}=g_{n,i}$. For each $n$, $T$ is an isometry when restricted to
$[e_{n,i}\colon 0\leqslant i\leqslant 2^{n}-1]$. Clearly, $\|T\|=1$.

_Claim_ 1\. If $W$ is a finite-dimensional Banach space and $S\colon
W\rightarrow C(\Delta)$ is an operator, then for every $\varepsilon>0$, there
exists an operator $\widehat{S}\colon W\rightarrow Z$ so that
$\|\widehat{S}\|\leqslant(1+\varepsilon)\|S\|$ and
$\|S-T\widehat{S}\|\leqslant\varepsilon$.

###### Proof of Claim 1.

Let us fix an Auerbach basis $(w_{k},w_{k}^{*})_{k=1}^{N}$ for $W$ ($\dim
W=N$). So if $w=\sum_{k=1}^{N}a_{k}w_{k}\in W$, then for each $1\leqslant
j\leqslant N$, we get

$|a_{j}|\leqslant|\langle
w^{*}_{j},\sum_{k=1}^{N}a_{k}w_{k}\rangle|\leqslant\|w^{*}_{j}\|\|w\|=\|w\|.$

It follows that $\sum_{k=1}^{N}|a_{k}|\leqslant N\|w\|.$ Let $\delta>0$ be
such that $\delta N\leqslant\varepsilon\|S\|$ and $\delta
N\leqslant\varepsilon$. Then, there exist a $n\in\mathbb{N}$ and
$(f_{k})_{k=1}^{N}$ in $[g_{n,i}]_{i=0}^{2^{n}-1}$ so that
$\|Sw_{k}-f_{k}\|<\delta$ ($k=1,2,\ldots,N$). Let us write
$f_{k}=\sum_{i=0}^{2^{n}-1}t_{k,i}g_{n,i}$ $(k=1,2,\ldots,N).$

Define an operator $\widehat{S}\colon W\rightarrow Z$ by

$\widehat{S}w_{k}=\sum_{i=0}^{2^{n}-1}t_{k,i}e_{n,i}.$

We _claim_ that $\|\widehat{S}\|\leqslant(1+\varepsilon)\|S\|$ and
$\|S-T\widehat{S}\|\leqslant\varepsilon$. Indeed, for
$w=\sum_{k=1}^{N}a_{k}w_{k}\in W$, we have

$\begin{array}[]{lcl}\|\widehat{S}w\|&=&\|\sum_{k=1}^{N}a_{k}\widehat{S}w_{k}\|=\|\sum_{k=1}^{N}a_{k}T\widehat{S}w_{k}\|\\\
&=&\|\sum_{k=1}^{N}a_{k}f_{k}\|\leqslant\|\sum_{k=1}^{N}a_{k}(f_{k}-Sw_{k})\|+\|\sum_{k=1}^{N}a_{k}Sw_{k}\|\\\
&\leqslant&\sum_{k=1}^{N}|a_{k}|\|f_{k}-Sw_{k}\|+\|S\|\|w\|\\\
&\leqslant&N\|w\|\delta+\|S\|\|w\|\\\
&\leqslant&(1+\varepsilon)\|S\|\|w\|.\end{array}$

Furthermore,

$\begin{array}[]{lcl}\|Sw-T\widehat{S}w\|&=&\|\sum_{k=1}^{N}a_{k}(Sw_{k}-\sum_{i=0}^{2^{n}-1}t_{k,i}g_{n,i})\|\\\
&=&\|\sum_{k=1}^{N}a_{k}(Sw_{k}-f_{k})\|\\\ &\leqslant&\delta N\|w\|\\\
&\leqslant&\varepsilon\|w\|.\end{array}$

∎

Let $\varepsilon>0$. Since $C(\Delta)$ has the metric approximation property,
there exists a net $(T_{\alpha})_{\alpha}$ of finite-rank operators on
$C(\Delta)$ such that

  * •

$\limsup_{\alpha}\|T_{\alpha}\|\leqslant 1+\varepsilon$,

  * •

$\dim T_{\alpha}(C(\Delta))\rightarrow\infty$,

  * •

$T_{\alpha}\rightarrow I_{C(\Delta)}$ strongly.

For each $\alpha$, we may apply Claim 1 to the inclusion map $I_{\alpha}\colon
T_{\alpha}[C(\Delta)]\rightarrow C(\Delta)$ in order to get an operator
$\widehat{I_{\alpha}}\colon T_{\alpha}[C(\Delta)]\rightarrow Z$ such that

  * •

$\|\widehat{I_{\alpha}}\|\leqslant 1+\varepsilon$,

  * •

$\|I_{\alpha}-T\widehat{I_{\alpha}}\|\leqslant(1+\dim
T_{\alpha}[C(\Delta)])^{-2}$.

Hence, for $f\in C(\Delta)$, we get

$\begin{array}[]{lcl}\|T\widehat{I_{\alpha}}T_{\alpha}f-f\|&\leqslant&\|T\widehat{I_{\alpha}}T_{\alpha}f-I_{\alpha}T_{\alpha}f\|+\|T_{\alpha}f-f\|\\\
&\leqslant&\|T\widehat{I_{\alpha}}-I_{\alpha}\|\|T_{\alpha}\|\|f\|+\|T_{\alpha}f-f\|\rightarrow
0.\end{array}$

Let $S$ be a
$\sigma(\mathscr{B}(Z^{*},C(\Delta)^{*}),Z^{*}\widehat{\otimes}_{\pi}C(\Delta))$-cluster
point of the net $((\widehat{I_{\alpha}}T_{\alpha})^{*})_{\alpha}$. We show
that $ST^{*}=I_{C(\Delta)^{*}}$. Indeed, we choose a subnet
$((\widehat{I_{\alpha^{\prime}}}T_{\alpha^{\prime}})^{*})_{\alpha^{\prime}}$
of $((\widehat{I_{\alpha}}T_{\alpha})^{*})_{\alpha}$ so that
$(\widehat{I_{\alpha^{\prime}}}T_{\alpha^{\prime}})^{*}\rightarrow S$ in
$\sigma(\mathscr{B}(Z^{*},C(\Delta)^{*}),Z^{*}\widehat{\otimes}_{\pi}C(\Delta))$-topology.
Then, for $f\in C(\Delta)$ and $\mu\in C(\Delta)^{*}$, we get
$\langle(\widehat{I_{\alpha^{\prime}}}T_{\alpha^{\prime}})^{*}T^{*}\mu,f\rangle\rightarrow\langle
ST^{*}\mu,f\rangle$. On the other hand, we have

$\langle(\widehat{I_{\alpha^{\prime}}}T_{\alpha^{\prime}})^{*}T^{*}\mu,f\rangle=\langle\mu,TI_{\alpha^{\prime}}T_{\alpha^{\prime}}f\rangle\rightarrow\langle\mu,f\rangle.$

Therefore, $\langle ST^{*}\mu,f\rangle=\langle\mu,f\rangle.$

_Claim_ 2\. There exists an operator $\widetilde{T}\colon
C(\Delta)^{*}\rightarrow X^{*}$ so that $R^{*}\widetilde{T}=T^{*}$ and
$\|\widetilde{T}\|\leqslant c(1+\varepsilon)$.

The proof of the claim is a variation of the Lindenstrauss’ compactness
argument (see [9, Proposition 1] and [14, Lemma 2]). Since some amendments are
required, we present the full reasoning.

###### Proof of Claim 2.

We use the fact that $C(\Delta)^{*}$ is isometric to $L_{1}(\mu)$ for some
infinite measure $\mu$, and as such, it is a $\mathcal{L}_{1,1+}$-space. Let
$\Lambda$ be the collection of all finite-dimensional subspaces of
$C(\Delta)^{*}$. Then, for each $\gamma\in\Lambda$ there exist
$E_{\gamma}\in\Lambda$ with $\gamma\subseteq E_{\gamma}$ together with an
isomorphism $U_{\gamma}\colon\ell_{1}^{\dim E_{\gamma}}\rightarrow E_{\gamma}$
so that $\|U_{\gamma}\|\|U_{\gamma}^{-1}\|\leqslant 1+\varepsilon$. Let
$S_{\gamma}\colon Z\rightarrow E_{\gamma}^{*}$ be an operator such that
$S_{\gamma}^{*}=T^{*}|_{E_{\gamma}}$ ($\gamma\in\Lambda$). By the
1-injectivity of $\ell_{\infty}^{\dim E_{\gamma}}$, there is an operator
$R_{\gamma}\colon X\rightarrow\ell_{\infty}^{\dim E_{\gamma}}$ so that
$R_{\gamma}R=U_{\gamma}^{*}S_{\gamma}$ and
$\|R_{\gamma}\|\leqslant\|U_{\gamma}^{*}S_{\gamma}\|\|R^{-1}\|\leqslant\|U_{\gamma}\|\|T\|\|R^{-1}\|$.
Let $T_{\gamma}=R_{\gamma}^{*}U_{\gamma}^{-1}\colon E_{\gamma}\rightarrow
X^{*}$. Then $R^{*}T_{\gamma}=T^{*}|_{E_{\gamma}}$ and
$\|T_{\gamma}\|\leqslant c(1+\varepsilon)\|T\|$. For each $\gamma$, we define
a non-linear, discontinuous function from $C(\Delta)^{*}$ to $X^{*}$ by

$\widetilde{T_{\gamma}}f=\left\\{\begin{array}[]{l@{\quad}l}T_{\gamma}f,&f\in
E_{\gamma}\\\ 0,&\text{otherwise.}\end{array}\right.$

Then $(\widetilde{T_{\gamma}})_{\gamma}$ is a net in the compact space

$\prod_{f\in C(\Delta)^{*}}c(1+\varepsilon)\|T\|\|f\|B_{X^{*}}.$

and as such, it has a cluster point $\widetilde{T}$. Standard arguments show
that $\widetilde{T}$ is linear, $R^{*}\widetilde{T}=T^{*}$ and
$\|\widetilde{T}\|\leqslant c(1+\varepsilon)\|T\|=c(1+\varepsilon)$.∎

Finally, we get $SR^{*}\widetilde{T}=ST^{*}=I_{C(\Delta)^{*}}$ and hence

$\beta_{C(\Delta)^{*}}(X^{*})\leqslant\|\widetilde{T}\|\|SR^{*}\|\leqslant
c(1+\varepsilon)^{3}.$

Letting $\varepsilon\rightarrow 0$, we get
$\beta_{C(\Delta)^{*}}(X^{*})\leqslant c$. As $c$ is arbitrary, we get Step 1.

Step 2. $\beta_{L_{1}}(X^{*})\leqslant\beta_{C[0,1]^{*}}(X^{*})$.

It is well known that $L_{1}$ is isometric to a $1$-complemented subspace of
$C[0,1]^{*}$ (see, _e.g._ , [1, p. 85]), which implies Step 2.

Step 3.
$\alpha_{(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{n})_{\ell_{1}}}(X)\leqslant\beta_{L_{1}}(X^{*})$.

Let $c>\beta_{L_{1}}(X^{*})$. Then there exist operators $A\colon
L_{1}\rightarrow X^{*},B\colon X^{*}\rightarrow L_{1}$ so that
$BA=I_{L_{1}},\|A\|=1$ and $\|B\|<c$. Let $0<\varepsilon<1$ and
$\varepsilon_{n}=\varepsilon/2^{2n+3}$ ($n=0,1,\ldots$).

By [8, Lemma 3], we get $(f_{n,i})_{(n,i)\in\mathcal{F}}$ in $L_{\infty}$ and
$(x_{n,i})_{(n,i)\in\mathcal{F}}$ in $X$ satisfying

  1. (1)

$\|f_{n,i}\|_{1}=1$ and $f_{n,i}\geqslant 0$ everywhere for all
$(n,i)\in\mathcal{F}$;

  2. (2)

For each $n$ and $i\neq j$, $f_{n,i}(t)$ and $f_{n,j}(t)$ cannot be both non-
zero for the same $t\in[0,1]$;

  3. (3)

$\langle
Af_{n,i},x_{m,j}\rangle=\left\\{\begin{array}[]{r@{\quad}l}1,&(n,i)\geqslant(m,j),\\\
0,&\text{otherwise;}\end{array}\right.$

  4. (4)

$\max_{0\leqslant i\leqslant
2^{n}-1}|t_{i}|\leqslant\|\sum_{i=0}^{2^{n}-1}t_{i}x_{n,i}\|\leqslant
c(1+\varepsilon_{n})\max_{0\leqslant i\leqslant 2^{n}-1}|t_{i}|$
($n=0,1,\cdots$; $t_{0},\ldots,t_{2^{n}-1}\in\mathbb{R}$).

We may now define recursively a sequence $(W_{n,i})_{(n,i)\in\mathcal{F}}$ of
non-empty weak*-closed subsets of $B_{X^{*}}$ as follows:

  * •

$W_{0,0}=\\{x^{*}\in B_{X^{*}}\colon|\langle
x^{*},x_{0,0}\rangle-1|\leqslant\varepsilon_{0}\\},$

  * •

$W_{1,0}=W_{0,0}\cap\\{x^{*}\in B_{X^{*}}\colon|\langle
x^{*},x_{1,0}\rangle-1|\leqslant\varepsilon_{1},|\langle
x^{*},x_{1,1}\rangle|\leqslant\varepsilon_{1}\\},$

  * •

$W_{1,1}=W_{0,0}\cap\\{x^{*}\in B_{X^{*}}\colon|\langle
x^{*},x_{1,1}\rangle-1|\leqslant\varepsilon_{1},|\langle
x^{*},x_{1,0}\rangle|\leqslant\varepsilon_{1}\\},$

  * •

$W_{2,0}=W_{1,0}\cap\\{x^{*}\in B_{X^{*}}\colon|\langle
x^{*},x_{2,0}\rangle-1|\leqslant\varepsilon_{2},|\langle
x^{*},x_{2,j}\rangle|\leqslant\varepsilon_{2},j=1,2,3\\},$

  * •

$W_{2,1}=W_{1,0}\cap\\{x^{*}\in B_{X^{*}}\colon|\langle
x^{*},x_{2,1}\rangle-1|\leqslant\varepsilon_{2},|\langle
x^{*},x_{2,j}\rangle|\leqslant\varepsilon_{2},j=0,2,3\\},$

  * •

$W_{2,2}=W_{1,1}\cap\\{x^{*}\in B_{X^{*}}\colon|\langle
x^{*},x_{2,2}\rangle-1|\leqslant\varepsilon_{2},|\langle
x^{*},x_{2,j}\rangle|\leqslant\varepsilon_{2},j=0,1,3\\},$

  * •

$W_{2,3}=W_{1,1}\cap\\{x^{*}\in B_{X^{*}}\colon|\langle
x^{*},x_{2,3}\rangle-1|\leqslant\varepsilon_{2},|\langle
x^{*},x_{2,j}\rangle|\leqslant\varepsilon_{2},j=0,1,2\\}$,

and so on. By (3), each $W_{n,i}$ is non-empty. By the choice of
$\varepsilon_{n}$, the sets $W_{n,i},W_{n,j}$ are disjoint as long as $i\neq
j$. Let

$K=\bigcap_{n=0}^{\infty}(\bigcup_{i=0}^{2^{n}-1}W_{n,i})\quad\text{and}\quad
K_{n,i}=W_{n,i}\cap K\;\;\big{(}(n,i)\in\mathcal{F}\big{)}.$

By (3), $Af_{n,i}\in W_{m,j}$ if $(n,i)\geqslant(m,j)$, which implies that
each $K_{n,i}$ is non-empty. By the construction of the sequence $(W_{n,i})$,
we see that $K_{0,0}=K,K_{n+1,2i}\cup K_{n+1,2i+1}=K_{n,i}$ and $K_{n,i}\cap
K_{n,j}=\varnothing$ if $i\neq j$.

Let us define an operator $T\colon X\rightarrow C(K)$ by $\langle
Tx,x^{*}\rangle=\langle x^{*},x\rangle$ ($x\in X,x^{*}\in K$). Then $|\langle
Tx_{n,i},x^{*}\rangle-1|\leqslant\varepsilon_{n}$ if $x^{*}\in K_{n,i}$, and
$|\langle Tx_{n,i},x^{*}\rangle|\leqslant\varepsilon_{n}$ if
$x^{*}\in\bigcup_{j\neq i}K_{n,j}$. Set $g_{n,i}=\mathds{1}_{K_{n,i}}$, which
is continuous as $K_{n,i}$ is clopen. Then
$\|Tx_{n,i}-g_{n,i}\|\leqslant\varepsilon_{n}$. Moreover,
$[g_{n,i}]_{i=0}^{2^{n}-1}\subseteq[g_{n+1,i}]_{i=0}^{2^{n+1}-1}$,
$(g_{n,i})_{i=0}^{2^{n}-1}$ is isometrically equivalent to the unit vector
basis of $\ell_{\infty}^{2^{n}}$ for all $n$, and

$[g_{n,i}\colon(n,i)\in\mathcal{F}]=\overline{\bigcup_{n=0}^{\infty}[g_{n,i}]_{i=0}^{2^{n}-1}}$

is isometric to $C(\Delta)$. Let $Z$ be a subspace of $C(\Delta)$ isometric to
$(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{n})_{\ell_{1}}$ and let
$(z_{n,j})_{n=1,j=0}^{\infty,n-1}$ be a basis of $Z$ isometrically equivalent
to the unit vector basis of
$(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{n})_{\ell_{1}}$. Fix $n\geqslant 1$.
Then there exist $m>n$ and unit vectors $h_{n,j}\in[g_{m,i}]_{i=0}^{2^{m}-1}$
so that $\|z_{n,j}-h_{n,j}\|\leqslant\varepsilon/2^{n+3}$
($j=0,1,\ldots,n-1$). We write $h_{n,j}=\sum_{i=0}^{2^{m}-1}a_{i,j}g_{m,i}$
and define $y_{n,j}=\sum_{i=0}^{2^{m}-1}a_{i,j}x_{m,i}\in X.$

_Claim 2a._ For all
$(t_{n,j})_{n=1,j=0}^{\infty,n-1}\in(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{n})_{\ell_{1}}$
we have

$(1-\frac{\varepsilon}{2})\sum_{n=1}^{\infty}\max_{0\leqslant j\leqslant
n-1}|t_{n,j}|\leqslant\|\sum_{n=1}^{\infty}\sum_{j=0}^{n-1}t_{n,j}y_{n,j}\|\leqslant
c(1+\varepsilon)^{2}\sum_{n=1}^{\infty}\max_{0\leqslant j\leqslant
n-1}|t_{n,j}|.$

Indeed, by (4) we get

$\displaystyle\|\sum_{j=0}^{n-1}t_{n,j}y_{n,j}\|$
$\displaystyle=\|\sum_{i=0}^{2^{m}-1}(\sum_{j=0}^{n-1}a_{i,j}t_{n,j})x_{m,i}\|$
$\displaystyle\leqslant c(1+\varepsilon_{m})\max_{0\leqslant i\leqslant
2^{m}-1}|\sum_{j=0}^{n-1}a_{i,j}t_{n,j}|$
$\displaystyle=c(1+\varepsilon_{m})\|\sum_{j=0}^{n-1}t_{n,j}h_{n,j}\|$
$\displaystyle\leqslant
c(1+\varepsilon_{m})\Big{(}\|\sum_{j=0}^{n-1}t_{n,j}z_{n,j}\|+\sum_{j=0}^{n-1}t_{n,j}(h_{n,j}-z_{n,j})\|\Big{)}$
$\displaystyle\leqslant c(1+\varepsilon_{m})\Big{(}\max_{0\leqslant j\leqslant
n-1}|t_{n,j}|+n\varepsilon/2^{n+3}\max_{0\leqslant j\leqslant
n-1}|t_{n,j}|\Big{)}$ $\displaystyle\leqslant
c(1+\varepsilon)^{2}\max_{0\leqslant j\leqslant n-1}|t_{n,j}|.$

Consequently,

$\|\sum_{n=1}^{\infty}\sum_{j=0}^{n-1}t_{n,j}y_{n,j}\|\leqslant\sum_{n=1}^{\infty}\|\sum_{j=0}^{n-1}t_{n,j}y_{n,j}\|\leqslant
c(1+\varepsilon)^{2}\sum_{n=1}^{\infty}\max_{0\leqslant j\leqslant
n-1}|t_{n,j}|.$

On the other hand, by the choice of $m$ and $h_{n,j}$, we arrive at

$\displaystyle\|Ty_{n,j}-z_{n,j}\|$
$\displaystyle\leqslant\|Ty_{n,j}-h_{n,j}\|+\|h_{n,j}-z_{n,j}\|$
$\displaystyle=\|\sum_{i=0}^{2^{m}-1}a_{i,j}(Tx_{m,i}-g_{m,i})\|+\varepsilon/2^{n+3}$
$\displaystyle\leqslant\varepsilon_{m}2^{m}\max_{0\leqslant i\leqslant
2^{m}-1}|a_{i,j}|+\varepsilon/2^{n+3}$
$\displaystyle\leqslant\varepsilon/2^{n+3}+\varepsilon/2^{n+3}=\varepsilon/2^{n+2}.$

This implies

$\displaystyle\|\sum_{n=1}^{\infty}\sum_{j=0}^{n-1}t_{n,j}y_{n,j}\|$
$\displaystyle\geqslant\|\sum_{n=1}^{\infty}\sum_{j=0}^{n-1}t_{n,j}Ty_{n,j}\|$
$\displaystyle\geqslant\|\sum_{n=1}^{\infty}\sum_{j=0}^{n-1}t_{n,j}z_{n,j}\|-\|\sum_{n=1}^{\infty}\sum_{j=0}^{n-1}t_{n,j}(Ty_{n,j}-z_{n,j})\|$
$\displaystyle\geqslant\sum_{n=1}^{\infty}\max_{0\leqslant j\leqslant
n-1}|t_{n,j}|-\sum_{n=1}^{\infty}n\max_{0\leqslant j\leqslant
n-1}|t_{n,j}|\frac{\varepsilon}{2^{n+2}}$
$\displaystyle\geqslant(1-\frac{\varepsilon}{2})\sum_{n=1}^{\infty}\max_{0\leqslant
j\leqslant n-1}|t_{n,j}|.$

Finally, by Claim, we get

$\alpha_{(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{n})_{\ell_{1}}}(X)\leqslant
c(1+\varepsilon)^{2}/(1-\frac{\varepsilon}{2}).$

Letting $\varepsilon\rightarrow 0$ yields
$\alpha_{(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{n})_{\ell_{1}}}(X)\leqslant
c$; since $c$ was arbitrary the proof of Step 3 is complete.

Step 4. $\beta_{L_{1}}(X^{*})\leqslant\theta_{C(\Delta)}(X)$.

This step follows from (1.1) together with Step 2. We are now ready to
establish the final step of the proof.

Step 5. Suppose that $X$ is separable. Then
$\theta_{C(\Delta)}(X)\leqslant\beta_{L_{1}}(X^{*})$.

Let $c>\beta_{L_{1}}(X^{*})$. Then there exist operators $A\colon
L_{1}\rightarrow X^{*},B\colon X^{*}\rightarrow L_{1}$ so that
$BA=I_{L_{1}},\|A\|=1$, and $\|B\|<c$.

Let $(f_{n,i})_{(n,i)\in\mathcal{F}}$ be a family of functions in
$L_{\infty}$, $(x_{n,i})_{(n,i)\in\mathcal{F}}$ in $X$, and
$(W_{n,i})_{(n,i)\in\mathcal{F}}$ associated to $\varepsilon_{n}=1/2^{2n+2}$
($n=0,1,\ldots$) as described in Step 3. Since $X$ is separable, we may assume
that the $\operatorname{d}$-diameter of $W_{n,i}\leqslant 2^{-n}$ for each
$i$, where $\operatorname{d}$ is a metric giving the relative
$\sigma(X^{*},X)$-topology on $B_{X^{*}}$. Let

$K=\bigcap_{n=0}^{\infty}(\bigcup_{i=0}^{2^{n}-1}W_{n,i})\quad\text{ and
}\quad K_{n,i}=W_{n,i}\cap K\;\;\big{(}(n,i)\in\mathcal{F}\big{)}.$

Then $K$ is a compact, totally disconnected metric space without isolated
points, hence homeomorphic to $\Delta$. Moreover, $K_{0,0}=K,K_{n+1,2i}\cup
K_{n+1,2i+1}=K_{n,i}$ and $K_{n,i}\cap K_{n,j}=\varnothing$ if $i\neq j$.
Hence $K=\bigcup_{i=0}^{2^{n}-1}K_{n,i}$ for all $n$. As seen in Step 3, the
operator $T\colon X\rightarrow C(K)$, defined by $\langle
Tx,x^{*}\rangle=\langle x^{*},x\rangle$ ($x\in X,x^{*}\in K$), satisfies
$\|Tx_{n,i}-g_{n,i}\|\leqslant\varepsilon_{n}$, where
$g_{n,i}=\mathds{1}_{K_{n,i}}\in C(K)$.

An argument analogous to Step 1 yields that, if $W$ is a finite-dimensional
Banach space and $S\colon W\rightarrow C(K)$ is an operator, then, for every
$\varepsilon>0$, there exists an operator $\widehat{S}\colon W\rightarrow X$
so that $\|\widehat{S}\|\leqslant c(1+\varepsilon)\|S\|$ and
$\|S-T\widehat{S}\|\leqslant\varepsilon$.

Fix $\varepsilon>0$. By an argument analogous to the one from Step 1, we get
an operator $S\colon X^{*}\rightarrow C(K)^{*}$ with $\|S\|\leqslant
c(1+\varepsilon)^{2}$ so that $ST^{*}=I_{C(K)^{*}}$. This means that

$\theta_{C(\Delta)}(X)=\theta_{C(K)}(X)\leqslant c(1+\varepsilon)^{2}.$

Letting $\varepsilon\rightarrow 0$, we arrive at
$\theta_{C(\Delta)}(X)\leqslant c.$ As $c$ is arbitrary, the proof is
complete.∎

### Acknowledgements

The authors would like to thank Prof. W. B. Johnson for helpful discussions
and comments. The first-named author was supported by the National Natural
Science Foundation of China (Grant No.11971403) and the Natural Science
Foundation of Fujian Province of China (Grant No. 2019J01024). The second-
named author acknowledges with thanks funding received from SONATA 15 No.
2019/35/D/ST1/01734.

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# Rational Ponzi Game in Algorithmic Stablecoin

Shange Fu1, Qin Wang2, Jiangshan Yu1, Shiping Chen2 1 Monash University,
Australia
2 CSIRO Data61, Australia

###### Abstract

Algorithmic stablecoins (AS) are one special type of stablecoins that are not
backed by any asset. They stand to revolutionize the way a sovereign fiat
operates. As implemented, AS are poorly stabilized in most cases; their prices
easily deviating from the target or even falling into a catastrophic collapse,
and are as a result often dismissed as a Ponzi scheme. However, what is the
essence of Ponzi? In this paper, we try to clarify such a deceptive concept
and reveal how AS work from a higher level. We find that Ponzi is basically a
financial protocol that pays existing investors with funds collected from new
ones. Running a Ponzi, however, does not necessarily imply that any
participant is in any sense losing out, as long as the game can be perpetually
rolled over. Economists call such realization as a rational Ponzi game. We
thereby propose a rational model in the context of AS and draw its holding
conditions. We apply the model to examine: whether or not the algorithmic
stablecoin is a rational Ponzi game. Accordingly, we discuss two types of
algorithmic stablecoins (Rebase & Seigniorage Shares) and dig into the
historical market performance of a number of impactful projects to demonstrate
the effectiveness of our model.

###### Index Terms:

Ponzi, Algorithmic Stablecoin, Rational Model

##  I Introduction

Ponzi is making a comeback that is used to describe the hyped algorithmic
stablecoins today. The most recent critique was directed at the one called
TerraUSD (UST), which crashed by almost 98% within 24 hours on May
12${}^{\text{th}}$, 2022 (cf.[1]), impacting a direct of $85B market value
evaporation and the collapse of entire crypto markets. This worst-hit reveals
a new type of bank run and has triggered repetitive rounds of criticism and
discussion about the stablecoin and its algorithmic stabilized approaches.
Stablecoins are created to inherit all great features that cryptocurrencies
hold (e.g., decentralization, automation, cross-border), without, however,
suffering from the same volatility, making them much more applicable as a
store of value, medium of exchange, and unit of account [2, 3, 4]. In an ideal
form, stablecoins are simply cryptocurrencies with stable value. Current
stablecoin implementations (Tab.I) aim to bridge external assets with the
cryptocurrency space by anchoring a peg (e.g., USD) in price.

Stablecoins are generally tagged as either collateralized or algorithmic
stablecoins based on whether they are backed by collateral or not. The
collateralized stablecoin, as the name indicates, requires collateral and thus
ensures the circulating token has a redemption value. Major collateral options
include fiat money, cryptocurrency, and other valuable assets. Currently,
stablecoins are primarily backed by fiat money (Tab.I). USDC [5], for example,
is a fully collateralized stablecoin pegged to USD. The USDC issuer claims
that they store an equivalent amount of US dollars of the supply of USDC in a
list of banks. It is reasonable to argue that stablecoins backed by fiat money
are of a high degree of centralization in essence. Using cryptocurrency as
collateral can instead circumvent this problem. However, considering its high
volatility, an over-collateralization is often required if relying on
cryptocurrency as collateral. One representative in this category is
stablecoin DAI [6], where a user needs to deposit Ether (the native token of
Ethereum blockchain) in excess of the amount of created DAIs. This way, even
if the underlying crypto depreciates, each unit of the stablecoin still has
enough space to be redeemed for at least the same value of its peg.

Algorithmic stablecoin, however, does NOT require any collateral. Its money
supply is elastic and managed by algorithms (i.e., coded in smart contracts),
and thus not limited in scale. As for stabilization, the par value of an
algorithmic stablecoin is basically preserved by expanding supply when the
price is too high and contracting supply when the price is too low. There are
two major designs in algorithmic stablecoins [7], namely, Rebase and
Seigniorage Shares. Rebase [8] is an algorithm that automatically adjusts the
money supply in response to the market demand on a routine basis. A rebase
protocol can add ($\mathsf{mint}$) or remove ($\mathsf{burn}$) coins from
circulation according to the stablecoin’s price deviation from the peg, and
this is usually achieved by managing token balance pro rata across all
wallets. Instead of rebasing one coin, seigniorage shares [9] normally has a
dual (or multiple) coin design: the price-stable coins and flexible investment
shares. The shares can algorithmically adjust the supply of price-stable coins
much like a central bank does with fiat currencies. Seigniorage shares model
has, to varying degrees, served as a foundation for TerraUSD [10], Basis Cash
[11], and Frax [12].

TABLE I: An Overview of Stablecoins (Ranked by Mrk Cap)

Name | Token | Centralised | Supply | Peg | Mrk Cap
---|---|---|---|---|---
Tether | USDT | Centralised | Collateral | USD | $157.4B
USD Coin | USDC | Centralised | Collateral | USD | $68.5B
BinanceUSD | BUSD | Centralised | Collateral | USD | $21.5B
Dai | DAI | Decentralised | Collateral | USD | $6.2B
Frax | FRAX | Decentralised | Fractional Alg. | USD | $1.4B
TerraUSD | UST-LUNA | Decentralised | Alg. (S.Share) | USD | $471.1M
Olympus | - | Decentralised | Collateral | - | $303.4M
Fei USD | FEI | Decentralised | Collateral B.C. | USD | $57.2M
Ampleforth | AMPL | Decentralised | Alg. (Rebase) | CPI USD | $45.6M
Empty Set | ESD | Decentralised | Alg. (S.Share) | USD | $370.5K
Basis Cash | BAC-BAB-BAS | Decentralised | Alg. (S.Share) | USD | $281.7K

  * Data: Fetched on $12^{\text{th}}$ Dec 2022. Abbreviation: Alg. = Algorithmic;

  * Mrk = Market; S.Share = Seigniorage Shares; B.C. = Bonding Curve111A bonding curve is a mathematical concept used to describe the relationship between the supply of a token and its price. In its simplest form, the more token that has been issued, the higher the token price is..

Contributions. However, there is no satisfactory algorithmic stablecoin that
has been healthily operated as of now. Non-collateral property and non-stable
performance undermine people’s already fragile trust, and algorithmic
stablecoins are thus often questioned as Ponzi schemes. This motivates our
observations towards this fancy but fuzzy concept of Ponzi. First, we review
existing in-the-wild algorithmic stablecoin projects and explore the design
primitives they share in common (Sec.I). We then accordingly establish a
rational Ponzi model in the context of algorithmic stablecoins (Sec.II) and
evaluate three representative projects, namely, Ampleforth (rebase), TerraUSD
(seigniorage share), and Basis Cash (seigniorage share), based on six-month
historical data from Mar 1${}^{\text{st}}$ to Oct 17${}^{\text{th}}$ 2022
(Sec.III). We also find several interesting insights (Sec.IV) that are driven
by our model. At last, we deliver two taking-home messages saying:

  * $\diamond$

It is more likely to run a rational Ponzi game in decentralized finance (DeFi)
protocols than in traditional finance. In traditional finance, default (a
failure to make repayments on a debt) is an intrinsically intractable problem
as human being’s financial behavior is non-controllable. DeFi, however,
empowered by the automatically operated smart contracts, can run functions
such as enforceable payment to ensure a Ponzi-style debt to roll over.

  * $\diamond$

Algorithmic stablecoin has a chance to achieve the defined rational Ponzi
model under certain conditions. Based on our investigations, the rebase
approach is relatively easy to implement as a rational Ponzi game as long as
there always exists an active trading market for the token always exitsts.
Whereas the seigniorage share method is of low probability to realize so. This
is due to its inability to cope with the joint fall effect caused by the
multi-coin design when a death spiral is triggered.

Recent Studies. A series of papers and reports [4, 13, 14, 15] establish and
extend the taxonomies in stablecoins based on the peg type, the collateral
type, and the collateral amount. They give an initial sight to demystify its
assemblies and components. Besides, with in-depth analysis, a number of
follow-up works has been proposed. Moin et al. [2] provide a systematical
classification of existing stablecoins in terms of their design elements.
Ariah et al. [3] investigate the economic structure of stablecoins and
establish a risk-based functional characterization. Pernice et al. [16]
research stablecoins from a monetary perspective. Salehi et al. [17] explore
the liquidation process in stablecoins, and put forward a research agenda for
alternatives to liquidation. Clements R. [18] studies the inherent fragility
in algorithmic stablecoins and claims that they are built to fail. Zhao et al.
[7] conduct an empirical analysis to explain the volatility of algorithmic
stablecoins.

##  II Modeling Rational Ponzi Game

Ponzi [19] has always been such a term that is instinctively resented by the
public. For malicious organizers, Ponzi schemes are used to lure new
investors, and leverage incoming money to pay earlier ones while keeping the
rest as profit. In general, Ponzi organizers normally will promise an
abnormally high return to investors within the short term. With little or no
legitimate earnings, Ponzi games require a constant cash flow of new funds to
survive. Once becoming difficult to recruit new participants, or whenever a
large number of existing investors cash out, the game would tend to collapse.
However, stemming from its origin, the substance of Ponzi is neutral yet
subtle. Economists [20, 21] discuss a perfect foresight version of the Ponzi
model, where games based on irrational lenders or imperfect information are
ruled out. They call one possibility, in which all debts can be forever rolled
over, i.e., financed by issuing new debt, a rational Ponzi game [22].
Following such narrative connotations, we develop a rational Ponzi model in
the context of algorithmic stablecoins.

Rational Ponzi Game. An entity that plans to issue debt and roll it over the
possibly infinite future is the borrowing side of a rational Ponzi game. Any
plan will impose a set of net cash inflows $I_{s}$ from the lending side. The
present value of borrower’s net indebtedness $D_{T}$ of at any time $T$ equals
the present value of the stream of prospective incomes generated by lenders
between $0$ and $T$:

$\Gamma(T)D_{T}=\sum_{s=1}^{T}\Gamma(s)I_{s}$ (1)

where $\Gamma(T)$ is the discount factor222In corporate finance, a discount
factor is a number that is derived from the discount rate. It is used to
discount future cash flows back to their present value at a specific discount
rate. Here, we assume $r_{t}$ are identical for all assets under a perfect
foresight setting. of the borrowing side, it is calculated by the interest
rate $r_{t}$ between periods $t-1$ and $t$, and $\Gamma(s)$ is the discount
factor applicable in period 0 to cash inflows received in the period $s$,
where $\Gamma(s)\equiv\prod_{j=1}^{s}\left(1+r_{j}\right)^{-1}.$ We now
provide the definition of the rational Ponzi game.

######  Definition 1 (Rational Ponzi Game).

A rational Ponzi game is a sequence of debt transactions satisfying the
following sense:

  * (i)

to the borrower, it is a positive net present indebtedness value such that
$\lim_{T\rightarrow\infty}$ $\Gamma(T)D_{T}>0$, if the limit of
$\Gamma(T)D_{T}$ exists;

  * (ii)

to all the participants, i.e., both the borrowing and lending sides, the game
enables a Pareto improvement333A Pareto improvement is an improvement to a
given situation, where a change in allocation of goods harms no one and
benefits at least one individual..

The first holding condition in Definition 1 implies that debts are
unnecessarily fully paid off by future cash inflows if in an infinite horizon
setting. The second condition imposes restrictions that no one is being worse
(debt can always be redeemed at least equal to the market value of assets
being deposited). Therefore, the two conditions guarantee a rational
realization. We provide a general operation process of the rational Ponzi
model as in Fig.1. The red plus sign and the green minus sign represent cash
inflow and cash outflow, respectively. The blue values in parentheses
following each participant represent their corresponding utility.

Figure 1: How a rational Ponzi game works.

Scope of application. One may consider that only governments are entitled to
run such rational Ponzi games. However, in a world where rational Ponzi games
can exist, any infinitely lived entity can issue debt and perpetually roll it
over. In principle, even a finitely lived entity can issue bonds with zero
coupons, i.e., fiat currency. All required is to make participants believe
that there will always exist followers being willing to purchase the debt. At
the same time, economists have also proven that the introduction of
perpetually rolled-over debt will never make the lending market worse off
relative to an economy in which no Ponzi game runs [22, 23, 24].

##  III Applied to Algorithmic Stablecoin

Typically, algorithmic stablecoins, based on their operating mechanisms, can
be considered a Ponzi game: an on-chain protocol launches a coin (equiv. debt)
claimed to be stable (investors can always get the equivalent fund back when
selling the coin), and one joins the game by purchasing the coin and can quit
the game if others are willing to buy his coins on the market. However, is
such an algorithmic stablecoin game can being rational under our model? In
this section, we investigate three representative instances, Ampleforth
(rebase), TerraUSD (seigniorage shares), and Basis Cash (seigniorage shares),
to answer this question. Supposing that all the lending protocols are deployed
on smart contracts on a public ledger so that all steps are perfectly
enforceable (equiv. no default risk). We define the notation $\mathcal{U}$ as
the investor’s utility; it is basically the product of the number of held
assets $Q$ and the market price of the token $P$, in symbols as

$\mathcal{U}=Q\times P.$ (2)

Then, we apply our model to three concrete instances by investigating their
six-month historical data (Mar – Oct, 2022). We analyze the utility of
investors based on extracted data covering price, quantity, and market cap.

Ampleforth (Rebase). Rebasing is a process applied by several algorithmic
stablecoins to maintain their peg to a certain value, such as the US dollar.
When a stablecoin rebases, it adjusts the supply of tokens in circulation in
response to changes in the value of the underlying asset or assets. It’s
important to note that rebasing can be a complex process, and different
algorithmic stablecoins may use different algorithms to determine when and how
to rebase. Ampleforth (AMPL) [25] is one representative protocol that applies
the design of rebase. AMPL is pegged to the CPI-adjusted USD, and its rebase
procedure occurs at 02:00 UTC on a daily basis. The quantity of AMPL tokens in
user accounts automatically expands or contracts based on everyday price.
According to Ampleforth whitepaper [26], the absolute supplyDelta of AMPL is
equal to
$\frac{(\mathsf{price}-\mathsf{target})*\mathsf{totalSupply}}{\mathsf{target}}$.
For example, at the time of rebase, if the price of AMPL is $1.1, 10% greater
than the standard 1 USD peg, a wallet with a balance of 100 AMPL will be
rebased as 110 accordingly. In contrast, when the AMPL price is less than 1
USD, the protocol will proportionally decrease the quantity of tokens in
wallets, aiming to bring the price up to its anchor.

Investor’s utility analysis. Considering an investor who participates in the
Ampleforth game at any time by purchasing one unit of AMPL, we analyze the
utility changes over time. At each rebase time (cf. Fig.4), the investor’s
utility (green line) stays the same444Except for the rebase time in a day, the
number of AMPL in wallets will not change. If the coin price rises and the
investor selects to sell, his or her utility can be even greater than his
original investment., meaning that the investor can always redeem (trade) the
identical value of his investment back. However, the price of AMPL (blue line)
is unstable, and the AMPL balance (brown line)555To better demonstrate the
nature of rebase, we calculate daily AMPL numbers applying a simplest setting
based on its 1st version whitepaper [26]. is in the investor’s wallet. In
terms of a rational Ponzi model, AMPL has the potential to satisfy all listed
conditions as long as there exists an lively traded market.

Since any investor can retain at least the same utility when participating in
the AMPL game, regardless of the token price, one may wonder why the rebase is
not being adopted on a larger scale. The most obvious reason is due to its
highly volatile price, resulting in holding AMPL being no different than
holding a regular cryptocurrency. At the same time, one’s balance varies
proportionally to the price every day, making the holder fall into passive and
anxious emotions. Besides, AMPL, as an ERC-20 token that works natively on
Ethereum blockchain, cannot perform its rebase implementation smoothly in a
centralized exchange or other blockchain platforms, significantly restricting
its use scenarios. Ampleforth mitigates this constraint with the release of
Wrapped-AMPL, a token that wraps AMPL similar to wrapped ETH. Wrapped-AMPL
facilitates ecosystem integrations on both centralized and decentralized
platforms.

(a) AMPL price

(b) AMPL amounts

(c) Investor’s utility

Figure 2: AMPL status over time.

(a) UST price

(b) UST amounts

(c) Investor’s utility

Figure 3: UST status over time.

(a) BAC price

(b) BAC amounts

(c) Investor’s utility

Figure 4: BAC status over time.

TerraUSD (Seiniorage Shares). Seniorage Shares is a monetary system design
that is well-established in traditional finance. The term “seigniorage” refers
to the profits that are generated by a central bank through the creation of
new money. In the case of “seigniorage” shares stablecoins, this profit is
generated by an algorithm that controls the supply of the stablecoin. The
seigniorage shares stablecoin model is designed to be self-regulating and can
utilize a dual coin system to stabilize the value of one token. The two coins
are known as the “shares” and the “price-stable coin”. The share is designed
to have a fluctuating value, much like a typical cryptocurrency. Its value is
determined by market demand and supply. The price-stable coin, on the other
hand, is pegged to a specific value. The price-stable coin is used to
stabilize the value of the shares, and it can be redeemed for the underlying
asset at any time. The two coins work together to provide a stable and
flexible monetary system. When the value of the shares starts to rise too
fast, the price-stable coin is sold on the market, creating more supply and
reducing the value of the shares. When the value of the shares starts to fall
too fast, the price-stable coin is redeemed for the underlying asset, reducing
the supply and increasing the value of the share. This dual coin design is
intended to provide the benefits of a cryptocurrency while also providing the
stability of a traditional currency. It allows for market-based pricing of the
shares while ensuring that the overall value of the currency remains
relatively stable. TerraUSD (UST) [10] is a seiniorage shares style stablecoin
pegging to the USD on top of Terra’s blockchain. LUNA is the native staking
token of the Terra blockchain, and it absorbs the volatility of UST as the
token of shares. Aibitrage can help making UST price stable: when the UST’s
price is $<1$ USD, arbitragers can send 1 UST to the system and receive 1
USD’s worth of LUNA; whereas the UST’s price becomes $>1$ USD, arbitragers
will send 1 USD’s worth of LUNA to the system and receive 1 UST.

Investor’s utility analysis. We also consider an investor who participates in
the UST game at any time by purchasing one unit of UST. For simplicity, we
assume that the investor will always hold UST without converting it to LUNA in
the mid-run before exiting the game. A chart of UST price (blue line), the
number of UST in the wallet (brown line), and the investor’s utility (green
line) over time are presented in Fig.4. The practice proves the
ineffectiveness of UST’s stabilization as the UST eventually collapsed.
Investors suffered a significant loss from their initial investments, making
UST unlikely to become a rational Ponzi game.

As a matter of fact, current dual-coin designs in the scope of seigniorage
shares are practically difficult in realizing a rational model. These
stablecoins are not substantially secured by anything, other than the
expectation that the price-stable coin will eventually appreciate in value.
Hence, a necessary condition for the seigniorage shares to work healthy is
that the shares, or the secondary coin, must have some values. LUNA, as the
secondary coin, should have, at least, trading values in this case, as people
do use Terra blockchain every day. However, in extreme scenarios, investors
may still lose confidence in it, and the massive sell-off has accelerated both
two types of coins into a death spiral faster.

Basis Cash (Seiniorage Shares). Seiniorage Shares systems in algorithmic
stablecoin are not necessarily limited to dual-coin designs. Basis Cash [11]
is also a seigniorage shares style protocol but with three tokens in the
system, namely Basis Cash (BAC), Basis Bonds (BAB), and Basis Shares (BAS).
The protocol is designed to expand and contract supply similar to the way
central banks trade fiscal debt to stabilize purchasing power. Basis Cash
(BAC), in this case, serves as the price-stable coin that maintains its peg to
the MakerDAO Multi-Collateral Dai token. In the previous examples, issuers
either use one coin such as AMPL to rebase the user accounts, or apply a
secondary token such as LUNA to absorb the volatility of the price-stable
token of UST. In the Basis Cash system, Basis Bonds (BAB) and Basis Shares
(BAS) are both non-stable tokens that are designed to help stabilize the value
of Basis Cash (BAC) by controlling its supply. BAB are debt securities that
can be minted and redeemed to incentivize changes in the BAC supply. Bonds are
always on sale to BAC holders. The price of a Basis Bond is determined by
market forces and can fluctuate based on demand. When the price of BAC is
below its peg, BAB can be purchased for less than the face value. The
bondholders can then redeem the bonds for the face value at a later date,
earning a profit. BAS are share tokens that loosely represent the value of the
Basis Cash network. When the price of BAC is above its peg, new Basis Shares
are minted and sold to users, increasing the supply of BAS. Conversely, when
the price of BAC is below its target price, existing BAS can be burned,
reducing the supply of BAS. This mechanism helps to stabilize the price of BAC
over time.

Investor’s utility analysis. Due to the three-token design in Basis Cash, the
complexity of the investor’s possible operations increases dramatically. For
simplicity, we also apply the same setup as the previous examples. An investor
can participate in the Basis cash game at any time by purchasing one unit of
BAC. We assume that the investor will always hold BAC without converting it to
BAB, and will not consider purchasing BAS in the mid-run before exiting the
game. Accordingly, a chart of price (blue line), the number of BAC in the
wallet (brown line), and the investor’s utility (green line) over time are
presented in Fig.4. In practice, BAC, BAB, and BAS have all failed to maintain
their coin prices and eventually went to almost zero in value. Similar to the
collapse of UST, BAC, as a more sophisticated seigniorage shares design, did
not achieve a Rational Ponzi model result.

One thing worth noting is that BAC and BAS tokens are distributed through
yield farming [11]. Yield farming, also referred to as liquidity mining, is a
way to generate additional rewards by depositing your cryptocurrency into a
pool. In Basis cash, providing liquidity to BAC-DAI and BAS-DAI pairs result
in additional BAS tokens being distributed. This allows the participants in
the game, to some extent, to gain tokens (utility) at no cost, and also can
act as an incentive mechanism against lack of liquidity and bond death
spirals. In general, seigniorage shares designs are without any rebase or
collateral risk, and there are theoretically infinite ways to split the
volatility of the price-stable coin across multiple tokens. This provides many
possibilities to the algorithmic stablecoin practitioners with different
expectations and risk preferences to design their customized monetary system.
However, seigniorage shares is inherently difficult to maintain the price
stability, as the cases of UST and BAC have already demonstrated.

##  IV Discussion

We now provide several additional thinking under the scope of this study in
this section.

Market cap vs price stability. Investors involved in stablecoin markets may
often have an intuitive feeling that the larger the market cap of the
stablecoin, the better stability the stablecoin can achieve with regard to its
peg. However, according to the price and market cap data from CoinGecko [1],
this conjecture may not be valid as imagined for the case algorithmic
stablecoins. The statement of ineffective application to the rebase design is
explicit, as its market capitalization will vary as the supply of the currency
is adjusted. For the seigniorage share design, this statement will be invalid
as well. Due to its relative complexity, there are a lot of possibilities
resulting in this situation, one of which is the loss of confidence. The
recent joint collapse of LUNA/UST was a loop in which the market lost
confidence and entered a death spiral, pushing them out of the market cap
top-10 within merely several hours. Thus, a larger market cap of an
algorithmic stablecoin does not necessarily guarantee a better stability.

Price oracle risk. Price oracle provides the price feed for on-chain DeFi
protocols [27], and it is crucial to the secure operation of an algorithmic
stablecoin. Algorithmic stablecoins may have special price-feeding needs
according to their different algorithms, which may require the establishment
of customized oracles. Constructing an internal oracle is one way, but this
may lead to an additional layer with potential centralization and incentivize
the stakeholder to update the price in his favor. Opting for an external price
oracle can mitigate this problem, but it is still vulnerable to external price
attacks include flash loan price manipulation attack, pump, and dump [28].
Selecting a trustworthy price oracle is not easy for an algorithmic
stablecoin, and a comprehensive ex-ante risk assessment is required.

Reconsider the necessity of a peg. Stablecoins are designed to be pegged to
some underlying assets, with the USD being a popular choice. Anchoring a
benchmark is an effective way to achieve stability in stablecoins. People may
thus generally assume a fact that the stablecoin is perfectly fungible for its
peg. However, this is not the full truth. Stablecoins can be, in fact, rather
considered as their own free-floating assets that closely track the value of a
peg. Besides, it is also possible that stablecoins could provide the desired
stability without pegging to an asset. In one possible future, once the
economy develops surrounding the stablecoin itself, and such stablecoins have
also been widely used as a medium of exchange, maintaining a perfect peg
becomes unnecessary.

Rational Ponzi in DeFi. At a high level, DeFi naturally has an advantage in
realizing a rational Ponzi game. In traditional finance, default has always
been a tough problem to deal with. Default is a situation in which a borrower
is unable to meet their financial obligations to a lender, such as failing to
make scheduled loan payments. In other words, default occurs when a borrower
cannot pay back their debt as agreed. However, DeFi protocols are built on top
of blockchain technology and are designed to be trustless, meaning they do not
rely on intermediaries such as banks or other financial institutions to manage
transactions. Instead, of leveraging automated smart contracts, DeFi has many
advanced and innovative financial tools such as automated market makers (AMM),
flash loans, and enforceable payment. This ensures all parties involved in a
game adhere to the terms of the agreement and thus provides more possibility
to run a rational Ponzi in DeFi successfully.

##  V Conclusion

This study applies O’Connell & Zeldes’s model (1988, cf.[22]) for rational
Ponzi games to the context of algorithmic stablecoins. Following its
connotation, we establish a rational Ponzi model as the measurement and
accordingly evaluate three mainstream algorithmic stablecoin projects
(Ampleforth, TerraUSD, and Basis Cash) as instances. Our investor utility
analysis indicates that the rebase design has the potential to realize a
rational Ponzi game. However, the seigniorage share approach seems hard to do
so. It is seldom possible to run a rational Ponzi in seigniorage share styled
algorithmic stablecoins, as once entering a death spiral, the selling pressure
in the case of a multi-coin design is intrinsically uncontrollable. But in
general, implementing rational models in DeFi is always easier than in
traditional finance. Beyond that, we also point out the hurdles existing in
current algorithmic stablecoin implementations and identify directions for
future endeavors.

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