# On inscribed trapezoids and affinely $3$-regular maps

Florian Frick  Dept. Math. Sciences, Carnegie Mellon University, Pittsburgh,
PA 15213, USA
Inst. Math., Freie Universität Berlin, Arnimallee 2, 14195 Berlin, Germany
<EMAIL_ADDRESS>and  Michael Harrison  Institute for Advanced Study, 1 Einstein
Drive, Princeton, NJ 08540<EMAIL_ADDRESS>
###### Abstract.

We show that any embedding $\mathbb{R}^{d}\to\mathbb{R}^{2d+2^{\gamma(d)}-1}$
inscribes a trapezoid or maps three points to a line, where $2^{\gamma(d)}$ is
the smallest power of $2$ satisfying $2^{\gamma(d)}\geq\rho(d)$, and $\rho(d)$
denotes the Hurwitz–Radon function. The proof is elementary and includes a
novel application of nonsingular bilinear maps. As an application, we recover
recent results on the nonexistence of affinely $3$-regular maps, for
infinitely many dimensions $d$, without resorting to sophisticated algebraic
techniques.

FF was supported by NSF grant DMS-1855591, NSF CAREER grant DMS 2042428, and a
Sloan Research Fellowship. MH was supported by Mathematisches
Forschungsinstitut Oberwolfach with an Oberwolfach Leibniz Fellowship and by
the Institute for Advanced Study through the NSF Grant DMS-1926686.

##  1\. Introduction

The main results of this paper are as follows.

######  Theorem 1.

Let $d\geq 1$ be an integer and let $n\leq 2d+\rho(d)-1$. Then any embedding
$f\colon\mathbb{R}^{d}\to\mathbb{R}^{n}$ inscribes a trapezoid or maps three
distinct points to a line.

Here $\rho(d)$ denotes the Hurwitz–Radon function and is defined as follows:
decompose $d$ as the product of an odd number and $2^{4a+b}$ for $0\leq b\leq
3$, then $\rho(d)\coloneqq 2^{b}+8a$. In particular, Theorem 1 implies:

######  Corollary 2.

Any embedding $\mathbb{R}^{2}\to\mathbb{R}^{5}$,
$\mathbb{R}^{4}\to\mathbb{R}^{11}$, or $\mathbb{R}^{8}\to\mathbb{R}^{23}$
inscribes a trapezoid or maps three distinct points to a line.

With a small technical modification we obtain the following improved bound.

######  Theorem 3.

Let $d\geq 1$ be an integer and let $n\leq 2d+2^{\gamma(d)}-1$, where
$2^{\gamma(d)}$ is the smallest power of $2$ satisfying
$2^{\gamma(d)}\geq\rho(d)$. Then any embedding
$f\colon\mathbb{R}^{d}\to\mathbb{R}^{n}$ inscribes a trapezoid or maps three
distinct points to a line.

######  Corollary 4.

Any embedding $\mathbb{R}^{16}\to\mathbb{R}^{47}$ inscribes a trapezoid or
maps three distinct points to a line.

The idea of the proofs is simple: given an embedding
$f\colon\mathbb{R}^{d}\to\mathbb{R}^{n}$, we define a suitable test function
which takes the value zero if and only if $f$ inscribes a trapezoid or maps
three points to a line, and then we apply a Borsuk-Ulam type result to show
that the test function must hit zero in the stated dimensions. While this
general idea is ubiquitous in topology, we emphasize that our specific test
function utilizes the Hurwitz–Radon function in a novel way to capture the
geometry of the situation more adequately than the “obvious” test function;
see Section 4 for a detailed discussion. Additional context and known results
related to Theorems 1 and 3 may be found in Section 3.

The test function makes use of the following equivalent definition of the
Hurwitz–Radon function: $\rho(d)$ is the largest integer such that there
exists a nonsingular bilinear map
$B\colon\mathbb{R}^{\rho(d)}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$; here
nonsingularity of $B$ means that $B(x,y)=0$ if and only if $x=0$ or $y=0$.
Nonsingular bilinear maps generalize the multiplication in the classical
division algebras $\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$ and were
primarily studied in a series of articles by K.Y. Lam (e.g. [24, 25, 26, 27,
28, 29]) and by Berger and Friedland [3]. They have appeared prominently in
topology alongside the Hurwitz–Radon function; for example, a famous result of
Adams states that there exist $q-1$ linearly independent tangent vector fields
on the sphere $S^{d-1}$ if and only if $q\leq\rho(d)$ [1]. Nonsingular
bilinear maps can be used to construct immersions of projective spaces (see
e.g. [23]) and have recently appeared in studies of skew fibrations ([19, 21,
30, 31]), totally nonparallel immersions [20], and coupled embeddability [12].
Nevertheless, the idea to use nonsingular bilinear maps to improve the
effectiveness of a test function appears to be new.

##  2\. The test function and the proofs of the main results

Fix an integer $d\geq 1$, write $\rho=\rho(d)$, let
$B\colon\mathbb{R}^{\rho}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ be a
nonsingular bilinear map, and let $X=(0,1)\times F_{2}(\mathbb{R}^{d})$, where
$F_{2}(M)\coloneqq\left\\{(x,y)\in M\times M\mid x\neq y\right\\}$ is the
configuration space consisting of pairs of distinct points of a space $M$. Now
given an embedding $f\colon\mathbb{R}^{d}\to\mathbb{R}^{n}$, we define
$\Phi\colon F_{2}(X)\times S^{\rho-1}\to\mathbb{R}^{n}$ by

$\displaystyle\Phi((t_{1},x_{1},y_{1}),(t_{2},x_{2},y_{2}),z)=\ \ $
$\displaystyle
t_{1}[f(x_{1}+y_{1}+B(z,x_{1}-y_{1}))-f(x_{1}+y_{1}-B(z,x_{1}-y_{1}))]$
$\displaystyle-$ $\displaystyle
t_{2}[f(x_{2}+y_{2}+B(z,x_{2}-y_{2}))-f(x_{2}+y_{2}-B(z,x_{2}-y_{2}))],$

and we show that this test function detects degeneracy in the following sense:

######  Lemma 5.

If zero is in the image of $\Phi$, then the embedding $f$ inscribes a
trapezoid or maps three distinct points to a line.

###### Proof.

If $\Phi((t_{1},x_{1},y_{1}),(t_{2},x_{2},y_{2}),z)=0$, the four image points

$\displaystyle u_{i}=f(x_{i}+y_{i}+B(z,x_{i}-y_{i}))\ \mbox{ and }\
v_{i}=f(x_{i}+y_{i}-B(z,x_{i}-y_{i})),\ i=1,2,$

satisfy the equation $t_{1}(u_{1}-v_{1})=t_{2}(u_{2}-v_{2})$. Equivalently,

$\displaystyle\frac{t_{1}}{t_{1}+t_{2}}u_{1}+\frac{t_{2}}{t_{1}+t_{2}}v_{2}=\frac{t_{1}}{t_{1}+t_{2}}v_{1}+\frac{t_{2}}{t_{1}+t_{2}}u_{2}.$
(1)

That is, the diagonal from $u_{1}$ to $v_{2}$ and the diagonal from $v_{1}$ to
$u_{2}$ intersect, and the point of intersection splits both diagonals in a
$t_{1}$ to $t_{2}$ ratio. So $u_{1},u_{2},v_{1},v_{2}$ are image points of $f$
which form a (possibly degenerate) trapezoid. We check that at least three of
these four points are distinct.

To verify that $u_{i}\neq v_{i}$, note that $x_{i}\neq y_{i}$ by assumption,
and so $B(z,x_{i}-y_{i})\neq 0$ by nonsingularity, and then apply injectivity
of $f$.

Now assume for contradiction that $u_{1}=u_{2}$ and $v_{1}=v_{2}$. Since $f$
is injective, we obtain

$\displaystyle x_{1}+y_{1}+B(z,x_{1}-y_{1})$
$\displaystyle=x_{2}+y_{2}+B(z,x_{2}-y_{2}),\ \text{and}$ $\displaystyle
x_{1}+y_{1}-B(z,x_{1}-y_{1})$ $\displaystyle=x_{2}+y_{2}-B(z,x_{2}-y_{2}).$

Adding and substracting these equations, respectively, yields

$x_{1}+y_{1}=x_{2}+y_{2}\quad\text{and}\quad
B(z,x_{1}-y_{1})=B(z,x_{2}-y_{2}).$

By bilinearity and nonsingularity of $B$, the second equation gives
$x_{1}-y_{1}=x_{2}-y_{2}$. Together with the first equation, this yields
$x_{1}=x_{2}$ and $y_{1}=y_{2}$. Since
$(t_{1},x_{1},y_{1})\neq(t_{2},x_{2},y_{2})$ by assumption, we have $t_{1}\neq
t_{2}$, which together with the assumptions $u_{1}=u_{2}$ and $v_{1}=v_{2}$
and Equation (1) implies that $u_{1}=v_{1}$, a contradiction. Therefore,
$u_{1}\neq u_{2}$ or $v_{1}\neq v_{2}$.

Similarly, if $u_{1}=v_{2}$ and $u_{2}=v_{1}$, then together with the
assumption $t_{1}(u_{1}-v_{1})=t_{2}(u_{2}-v_{2})$, we obtain
$t_{1}(u_{1}-v_{1})=-t_{2}(u_{1}-v_{1})$, impossible since $u_{1}\neq v_{1}$
and $t_{i}>0$ by assumption. Thus, $u_{1}\neq v_{2}$ or $u_{2}\neq v_{1}$.

Therefore Equation (1) is a non-trivial affine combination that involves at
least three pairwise distinct points. In the degenerate situation, where
$\\{u_{1},u_{2},v_{1},v_{2}\\}$ is a set of three points, Equation (1) implies
that these points lie on a common line. ∎

In Section 4 we explain the geometric motivation which led to the definition
of $\Phi$.

Now to prove Theorem 1, we need only show that $\Phi$ must hit zero whenever
$n\leq 2d+\rho-1$. We make use of a classical result in equivariant topology.
Consider the $(\mathbb{Z}/2)^{2}$-action on $S^{m}\times S^{q}$ defined by
letting the first copy of $\mathbb{Z}/2$ act antipodally on $S^{m}$ and the
second copy of $\mathbb{Z}/2$ act antipodally on $S^{q}$. Similarly we define
a $(\mathbb{Z}/2)^{2}$-action on $\mathbb{R}^{m+q}$ by letting both generators
act by $x\mapsto-x$. A map $S^{m}\times S^{q}\to\mathbb{R}^{m+q}$ that
commutes with these actions is a _$(\mathbb{Z}/2)^{2}$ -map_.

######  Lemma 6 (Hopf [22]).

Let $m\geq 1$ and $q\geq 1$ be integers that do not share a one in any digit
of their binary expansions. Then any $(\mathbb{Z}/2)^{2}$-map $S^{m}\times
S^{q}\to\mathbb{R}^{m+q}$ has a zero.

Hopf’s proof was one of the earliest applications of cohomology theory. In
modern language, a $(\mathbb{Z}/2)^{2}$-map which avoids zero induces a map in
cohomology $H^{*}(\mathbb{R}P^{m+q-1};\mathbb{Z}/2)\to
H^{*}(\mathbb{R}P^{m}\times\mathbb{R}P^{q};\mathbb{Z}/2)$ which sends the
generator $\gamma_{m+q-1}$ to the sum of generators $\gamma_{m}+\gamma_{q}$.
Therefore $(\gamma_{m}+\gamma_{q})^{m+q}=0$, hence the binomial coefficient
$\scriptsize\Big{(}\begin{array}[]{c}m+q\\\ m\end{array}\Big{)}$ is even. By
Lucas’ theorem, this occurs if and only if $m+q$ has a zero in a digit of its
binary expansion in which $m$ has a one, which occurs if and only if $m$ and
$q$ share a one in some digit.

Now consider the $\mathbb{Z}/2$-action on $F_{2}(X)$ which swaps the two
points of $X$. Lemma 6 has the following simple consequence.

######  Corollary 7.

Let $m\geq 1$ and $q\geq 1$ be integers that do not share a one in any digit
of their binary expansions. If there exists a $\mathbb{Z}/2$-equivariant
embedding $S^{m}\to F_{2}(X)$, then (by restriction) any
$(\mathbb{Z}/2)^{2}$-equivariant map $F_{2}(X)\times S^{q}\to\mathbb{R}^{m+q}$
has a zero.

###### Proof of Theorem 1.

Let $n=2d+\rho-1$ and let $f\colon\mathbb{R}^{d}\to\mathbb{R}^{n}$ be an
embedding. By Lemma 5, we only need to show that $\Phi$ has a zero. There
exists an embedding $h\colon S^{2d}\to X$, since $X$ is a $(2d+1)$-dimensional
manifold, and this induces a $\mathbb{Z}/2$-equivariant embedding
$S^{2d}~{}\to~{}F_{2}(X)\colon z\mapsto(h(z),h(-z))$. Moreover, the map $\Phi$
is $(\mathbb{Z}/2)^{2}$-equivariant, so by Corollary 7, it suffices to check
that $2d$ and $\rho-1$ do not share any common ones in their binary
expansions.

To verify this, write $d$ as the product of an odd number with $2^{\ell}$,
where $\ell=4a+b$ and $b\in\\{0,1,2,3\\}$. Then
$\rho=2^{b}+8a=2^{\ell-4a}+8a<2^{\ell+1}$. The last $\ell+1$ digits in the
binary expansion of $2d$ (corresponding to $2^{\ell},2^{\ell-1},\dots,2^{0}$)
are zero, and only those digits may be non-zero for $\rho-1$. ∎

Next we present the proof of Theorem 3, which relies on a few small
modifications to the previous proof.

###### Proof of Theorem 3.

Let
$C\colon\mathbb{R}^{2^{\gamma(d)}-\rho+1}\times\mathbb{R}^{\rho}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$
be the trilinear map defined by $C(w,z,x)=B(w,B(z,x))$. This is well-defined
by restriction in the first factor, since $2^{\gamma(d)}-\rho+1\leq\rho$ by
assumption. Moreover, nonsingularity of $B$ yields nonsingularity of $C$,
where by nonsingularity of $C$ we mean that $C(w,z,x)=0$ if and only if one of
the factors is zero.

Using $C$ in place of $B$, we define the $(\mathbb{Z}/2)^{3}$-map $\Phi\colon
F_{2}(X)\times S^{2^{\gamma(d)-\rho}}\times S^{\rho-1}\to\mathbb{R}^{n}$ by

$\displaystyle\Phi((t_{1},x_{1},y_{1}),$
$\displaystyle(t_{2},x_{2},y_{2}),w,z)$ $\displaystyle=\ \ $ $\displaystyle
t_{1}[f(x_{1}+y_{1}+C(w,z,x_{1}-y_{1}))-f(x_{1}+y_{1}-C(w,z,x_{1}-y_{1}))]$
$\displaystyle-$ $\displaystyle
t_{2}[f(x_{2}+y_{2}+C(w,z,x_{2}-y_{2}))-f(x_{2}+y_{2}-C(w,z,x_{2}-y_{2}))].$

The proof of Lemma 5 is otherwise unchanged, and so we only need to check that
$\Phi$ has a zero when $n\leq 2d+2^{\gamma(d)}-1$.

For this, we use the obvious generalization of Hopf’s lemma (with nearly-
identical proof): any $(\mathbb{Z}/2)^{3}$-map $S^{m_{1}}\times
S^{m_{2}}\times S^{m_{3}}\to\mathbb{R}^{m_{1}+m_{2}+m_{3}}$ has a zero
provided that no two of the $m_{i}$ share a one in any digit of their binary
expansions. This applies to the integers $m_{1}=2^{\gamma(d)}-\rho$,
$m_{2}=\rho-1$, and $m_{3}=2d$. Indeed, the integers $m_{1}$ and $m_{2}$ share
no ones since they sum to $2^{\gamma(d)}-1$, and the argument that $m_{1}$ and
$m_{3}$ share no ones is identical to that for $m_{2}$ and $m_{3}$, which was
given in the proof of Theorem 1. ∎

##  3\. Context and history: $k$-regular embeddings

The main theorem and corollary are best understood in the context of _regular
maps_ , first defined and studied by Borsuk in 1957 [6].

######  Definition 8.

A continuous map $f\colon\mathbb{R}^{d}\to\mathbb{R}^{n}$ is called _$k$
-regular_ if for any $k$ pairwise disjoint points
$x_{1},\dots,x_{k}\in\mathbb{R}^{d}$ the points $f(x_{1}),\dots,f(x_{k})$ are
linearly independent.

We offer simple examples for small $d$ or $k$:

  1. (1)

$d=1$: the moment curve $\mathbb{R}\to\mathbb{R}^{k}\colon
t\mapsto(1,t,t^{2},\dots,t^{k-1})$ is $k$-regular, due to the nonvanishing of
the Vandermonde determinant on the configuration space $F_{k}(\mathbb{R})$;

  2. (2)

$k=2$: the map
$\mathbb{R}^{d}\to\mathbb{R}^{d+1}=\mathbb{R}^{d}\times\mathbb{R}\colon
x\mapsto(x,1)$ is $2$-regular;

  3. (3)

$k=3$: if $h\colon\mathbb{R}^{d}\to S^{d}$ is an embedding, then
$\mathbb{R}^{d}\to\mathbb{R}^{d+2}\colon x\mapsto(h(x),1)$ is $3$-regular.

Aware of these basic examples, Borsuk posed the question: given $d\geq 1$ and
$k\geq 2$, what is the smallest dimension $n=n(d,k)$ such that
$\mathbb{R}^{d}$ admits a $k$-regular map to $\mathbb{R}^{n}$? It is not
difficult to check that the target dimensions are optimal for the given values
of $d$ or $k$ in the above examples, so that $n(1,k)=k$, $n(d,2)=d+1$, and
$n(d,3)=d+2$.

In addition to the obvious geometric appeal, the question historically
attracted interest due to connections with approximation theory and Chebyshev
polynomials. For $k\geq 4$ the question has been studied by a number of
mathematicians and has proven to be notoriously difficult.

The first nontrivial result, that $n(d,2k)\geq(d+1)k$, appeared in a 1960
paper of Boltyansky, Ryzhkov, and Shashkin [8]. This can be shown by
considering, for $f\colon\mathbb{R}^{d}\to\mathbb{R}^{n}$, the test function

$\displaystyle\varphi\colon D_{1}\times\cdots\times
D_{k}\times(\mathbb{R}-\left\\{0\right\\})^{k}\to\mathbb{R}^{n}\colon(x_{1},\dots,x_{k},\lambda_{1},\dots,\lambda_{k})\mapsto\sum\lambda_{i}f(x_{i}),$
(2)

where the $D_{i}$ are disjoint disks in $\mathbb{R}^{d}$. The bound follows
from the observation that if $f$ is $2k$-regular, $\varphi$ embeds its
$((d+1)k)$-dimensional domain into $\mathbb{R}^{n}$.

In 1978, Cohen and Handel observed in [11] that a $k$-regular map
$f\colon\mathbb{R}^{d}\to\mathbb{R}^{n}$ induces an $S_{k}$-equivariant map

$F_{k}(\mathbb{R}^{d})\to
V_{k}(\mathbb{R}^{n})\colon(x_{1},\dots,x_{k})\mapsto(f(x_{1}),\dots,f(x_{k}));$

here $V_{k}(\mathbb{R}^{n})$ is the Stiefel manifold of $k$-tuples of linearly
independent vectors in $\mathbb{R}^{n}$, and the symmetric group $S_{k}$ acts
on each space by permuting elements of the $k$-tuple. This observation
highlighted the equivariant nature of the problem, and in some sense, the
subsequent results for $k$-regular maps can be viewed as a yardstick by which
to measure the advances in equivariant cohomology theory over the following
decades.

For example, in the same paper, Cohen and Handel showed that $n(2,k)\geq
2k-\alpha(k)$, where $\alpha(k)$ denotes the number of ones in the dyadic
presentation of $k$. Chisholm [10] generalized this by showing that
$n(2^{\ell},k)\geq 2^{\ell}(k-\alpha(k))+\alpha(k)$. Other results on
$k$-regular maps, including some for other manifolds or simplicial complexes,
were contributed by Bogatyi [7], Handel [15, 16, 17], and Handel and Segal
[18]. The first strong existence results were obtained in 2019 using methods
of algebraic geometry [9].

Finally in 2021, strong obstructions were computed by Blagojević, Cohen,
Crabb, Lück, and Ziegler, as a counterpart to their massive breakthrough in
understanding the mod-$2$ equivariant cohomology of configuration spaces:

######  Theorem 9 ([4], Theorem 6.16).

Let $d\geq 2$, $k\geq 1$, and write $d=2^{t}+e$ for some $t\geq 1$ and $0\leq
e\leq 2^{t}-1$. Let $\epsilon(k)$ denote the remainder of $k$ mod $2$. Then
$n(d,k)\geq(d-e-1)(k-\alpha(k))+e(\alpha(k)-\epsilon(k))+k$.

The proof of this theorem relies on a lengthy technical argument which
corrects a proof of a theorem of Hung; the details surrounding the error, as
well as a history of relevant configuration space computations, are well-
chronicled in the introduction of [4].

We are now equipped to discuss the relationship between regular maps and
Theorem 1. It is convenient to introduce some intermediate language: a
continuous map $f\colon\mathbb{R}^{d}\to\mathbb{R}^{n}$ is called _affinely_
$(k-1)$-_regular_ if for every $k$ distinct points of $\mathbb{R}^{d}$, the
images do not lie in any affine $(k-2)$-plane. The relationship with
$k$-regularity is simple:

######  Lemma 10.

There is a $k$-regular map $\mathbb{R}^{d}\to\mathbb{R}^{n}$ if and only if
there is an affinely $(k-1)$-regular map $\mathbb{R}^{d}\to\mathbb{R}^{n-1}$.

###### Proof.

If $f\colon\mathbb{R}^{d}\to\mathbb{R}^{n-1}$ is affinely $(k-1)$-regular,
then $\mathbb{R}^{d}\to\mathbb{R}^{n},x\mapsto(f(x),1)$ is $k$-regular.
Conversely, given a $k$-regular map $f\colon\mathbb{R}^{d}\to\mathbb{R}^{n}$,
we may arrange (by restricting the domain if necessary) that $f$ misses some
affine hyperplane $H$. We may assume that $H$ is given by ${x_{n}=0}$.
Identify $\mathbb{R}^{n-1}$ with the affine hyperplane ${x_{n}=1}$, and define
$g\colon\mathbb{R}^{d}\to\mathbb{R}^{n-1}$ by letting $g(x)$ be the
intersection of the line spanned by $f(x)$ with the hyperplane ${x_{n}=1}$.
Then $g$ is affinely $(k-1)$-regular. ∎

Therefore Theorem 3, which prohibits affinely $3$-regular maps
$\mathbb{R}^{d}\to\mathbb{R}^{n}$ when $n\leq 2d+2^{\gamma(d)}-1$, can be
viewed as an obstruction to the existence of $4$-regular maps.

######  Corollary 11.

$n(d,4)\geq 2d+2^{\gamma(d)}+1$.

We compare this bound to those described above. First, note that the bound is
never worse than that obtained by using the test function $\varphi$.
Chisholm’s bound, which applies only when $d$ is a power of $2$, gives
$n(d,4)\geq 3d+1$, which is better than our bound for $d\geq 32$. Similarly,
the bound of Theorem 9 is frequently better than our bound, although there are
infinitely many values of $d$ for which the bounds agree. For example, when
$d=2^{\ell}-2=2(2^{\ell-1}-1)$, $\rho(d)=2=2^{\gamma(d)}$, our bound yields
$n(2^{\ell}-2,4)\geq 2^{\ell+1}-1$, which matches the bound obtained by
Theorem 9. In this sense Theorem 3, which not only prohibits $4$-regular
embeddings, but imposes further geometric constraints on $4$ points,
strengthens the results of Theorem 9 for infinitely many dimensions $d$.

##  4\. The development and geometry of the test function

Here we explain the steps leading to the construction of the test function
$\Phi$, and we describe the sense in which $\Phi$ captures the geometry more
adequately than the test function $\varphi$ defined in Equation (2).

Consider two disjoint disks $D_{1}$, $D_{2}$ in $\mathbb{R}^{d}$. An affinely
$3$-regular map $f$ induces an embedding of the space $A$ of non-zero affine
combinations of $D_{1}$ and $D_{2}$ into $\mathbb{R}^{n}$, which gives a lower
bound for $n$ for dimension reasons; this is the geometric idea captured by
the test function $\varphi$.

Now observe that we can improve this by moving the disks around in a special
way to generate whole families of embeddings. In particular, using a
nonsingular bilinear map, we can find an $S^{\rho-1}$-family of maps of $X$
into $\mathbb{R}^{n}$:

$\Psi_{z}\colon X\to\mathbb{R}^{n}\colon(t,x,y)\mapsto
t[f(x+y+B(z,x-y))-f(x+y-B(z,x-y))],\ \ z\in S^{\rho-1}$

Note that $\Psi_{-z}=-\Psi_{z}$, and $\Psi_{z}$ fails to be an embedding if
and only if $\Phi(\cdot,z)$ hits zero. Thus the proof of Theorem 1 relies not
on obstructing a single embedding, but obstructing the existence of
$\mathbb{Z}/2$-equivariant families of embeddings; this is the geometric idea
captured by the test function $\Phi$. Similarly, the technical modification of
$\Phi$ used in the proof of Theorem 3 relies on obstructing
$(\mathbb{Z}/2)^{2}$-equivariant families of embeddings.

The idea to try to obstruct equivariant families of embeddings was inspired by
the authors’ recent development and study of the $\mathbb{Z}/2$-_coindex of
spaces of embeddings_ ; see [13]. It is our belief that we have only scratched
the surface of results of this form, and it would be interesting to see
whether similar techniques could yield strong bounds for other values of $d$
and $k$. We emphasize that the tools used in our proofs of Theorems 1 and 3
predated much of the study of $k$-regular embeddings, especially the recent
results which rely on technical advances in equivariant topology. We hope that
our simple argument demonstrates that strong obstructions can be computed for
$k$-regular embeddings and other nondegenerate functions without the use of
sophisticated algebraic techniques; it is only important that the test
functions adequately capture the geometry of the situation.

We conclude by noting that Theorems 1 and 3 also connect to the studies of
inscribed shapes in various objects, perhaps most notably the square/rectangle
peg problem in $\mathbb{R}^{2}$ (see [14] or [2] for description and history).
According to Greene and Lobb in [14], there is “a long line of attack on these
problems which involves identifying the inscribed feature with the
(self-)intersection of an associated geometric-topological object. The
arguments tend to be quite short, once the appropriate outlook and auxiliary
result is identified.” Our short proof of Theorem 3, which capitalizes on the
fact that a self-intersection occurs in one of a
$(\mathbb{Z}/2)^{2}$-equivariant ($S^{2^{\gamma(d)}-\rho(d)}\times
S^{\rho(d)-1}$)-parameter family of maps $X\to\mathbb{R}^{n}$ when $n\leq
2d+2^{\gamma(d)}-1$, provides one more example of such an attack.

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# Gold, currencies and market efficiency

Ladislav Kristoufek<EMAIL_ADDRESS>Miloslav Vosvrda
<EMAIL_ADDRESS>Institute of Information Theory and Automation, Academy of
Sciences of the Czech Republic, Pod Vodarenskou Vezi 4, 182 08, Prague, Czech
Republic, EU

###### Abstract

Gold and currency markets form a unique pair with specific interactions and
dynamics. We focus on the efficiency ranking of gold markets with respect to
the currency of purchase. By utilizing the Efficiency Index (EI) based on
fractal dimension, approximate entropy and long-term memory on a wide
portfolio of 142 gold price series for different currencies, we construct the
efficiency ranking based on the extended EI methodology we provide. Rather
unexpected results are uncovered as the gold prices in major currencies lay
among the least efficient ones whereas very minor currencies are among the
most efficient ones. We argue that such counterintuitive results can be partly
attributed to a unique period of examination (2011-2014) characteristic by
quantitative easing and rather unorthodox monetary policies together with the
investigated illegal collusion of major foreign exchange market participants,
as well as some other factors discussed in some detail.

###### keywords:

efficient market hypothesis , gold , currencies , fractal dimension , entropy
, long-term memory
JEL codes: C58, G14, Q02
PACS codes: 05.45.Df, 05.45.Tp, 89.65.Gh, 89.70.Cf

††journal:  Physica A

##  1 Introduction

For decades, the efficient market hypothesis (EMH) has been a building block
of financial economics. In his fundamental paper, Fama (1970) summarizes the
then-current empirical findings following the theoretical papers of Fama
(1965) and Samuelson (1965). Fama (1991) then recalls various issues of the
hypothesis and reviews the newer literature on the topic. The capital market
efficiency is standardly parallelized with the informational efficiency so
that the markets are efficient as long as all the available information is
fully reflected into market prices (Fama, 1970). Depending on the level of
information availability, the EMH is usually separated into three forms – weak
(historical prices), semi-strong (public information), and strong (all
information, even private) (Fama, 1991). The theory has been challenged on
both theoretical (Malkiel, 2003) and empirical (Cont, 2001) grounds regularly,
yet still it remains a popular and fruitful topic of financial research.

The empirical testing of capital markets efficiency has a long history across
various assets. The already-mentioned review study of Fama (1970) focuses
mainly on stock markets. In commodity markets, Roll (1972) and Danthine (1977)
are among the first ones to study their efficiency arriving at contradicting
results. In the same timeline, foreign exchange rates are investigated as well
(Frenkel, 1976; Cornell and Dietrich, 1978). The termination of the Bretton
Woods system in 1971 made the detachment of gold and currency prices
interesting for research of the separate phenomena (Booth and Kaen, 1979).
Nonetheless, the two still remain tightly connected. Koutsoyiannis (1983)
focuses on the efficiency of gold prices and argues that the market efficiency
cannot be refuted. Nevertheless, the author finds a tight connection between
gold prices and the strength of the US dollar as well as the inflation,
interest rates and a general state of the US economy. The gold prices and
foreign exchange rates are thus found to be firmly interconnected, which is
supported by another early study of Ho (1985). Frank and Stengos (1989)
further suggest that simple linear testing of the gold (and silver) market
efficiency need not be sufficient.

The efficiency studies of foreign exchange rates are quite unique compared to
the mentioned stocks and commodities as the foreign exchange rates pricing has
solid macroeconomic foundations such as the balance of payment theory, the
purchasing power parity, the interest rate parity, the Fisher effect and
others (Dunn Jr. and Mutti, 2004; Levi, 2005; Feenstra, 2008). These theories
lead to different ways of efficiency treatment and testing.

Charles et al. (2012) examine the return predictability of major foreign
exchange rates between 1975 and 2009. Using various tests, the authors show
that the exchange rates are unpredictable most of the time. Short-term
inefficiencies are attributed to major events such as coordinated central bank
interventions and financial crises. The crises perspective is further studied
by Ahmad et al. (2012) who focus on the Asia-Pacific region. They argue that
the 1997-1998 Asian crisis was more disturbing compared to the 2008-2009
global financial crisis. In addition, the floating currency markets are found
to be more resilient than the countries with managed currencies. Al-Khazali et
al. (2012) further examine the Asia-Pacific region using the random walk and
martingale definitions of the market efficiency. Out of 8 studied currencies,
only three (Australian dollar, Korean won and Malaysian ringgit) are found to
be efficient while the other exchange rates offer profitable trading
opportunities.

Olmo and Pilbeam (2011) review the literature on the foreign exchange rate
efficiency testing based on the uncovered interest rate parity. They suggest
that the rejection of efficiency in this area of research may be due to
significant differences in volatilities of the logarithmic changes of exchange
rates and the forward premium, in addition to conditional heteroskedasticity
of the data. The authors introduce a set of profitability-based tests of
market efficiency based on the uncovered interest rate parity and they show
that the foreign exchange rates are much closer to market efficiency than
usually claimed. Chen and Tsang (2013) inspect whether interest rates
structure (yield curve) can be used for foreign exchange rate forecasting.
They show that it is the case on time horizons between one month and two
years. They also argue that these results can help explaining the uncovered
interest rate parity puzzle by relating currency risk premium to inflation and
business cycle risks. Bianco et al. (2012) further discuss the potential of
using economic fundaments for foreign exchange rates forecasting. Their
fundamentals-based econometric model for weekly euro-dollar rates is shown to
beat the random walk model for time horizons between one week and one month.
Engel et al. (2015) construct factors from exchange rates and they use their
idiosyncratic deviations for forecasting. Combining these with the Taylor
rule, and monetary and purchasing power parity models, they improve the
forecasting power of the model compared to the random walk benchmark for the
periods between 1999 and 2007 but not for earlier periods down to 1987.

Chaboud et al. (2014) inspect the effect of algorithmic trading on efficiency
of the foreign exchange markets in the high-frequency domain. They show that
algorithmic trading improves market efficiency in two aspects – triangular
arbitrage opportunities and autocorrelation of high-frequency returns. On the
contrary, they argue that this may impose higher adverse selection costs on
slower traders.

Studies of the foreign exchange rates efficiency, in the same way as of the
other assets, primarily focus on testing whether a given currency or a set of
currencies may or may not be considered efficient. To reflect this point,
Kristoufek and Vosvrda (2013) introduce the Efficiency Index (EI) which can be
used to rank assets according to their efficiency. In addition, the index is
very flexible and it can incorporate various measures of the market
efficiency. In the original study, Kristoufek and Vosvrda (2013) study 41
stock indices and find the Japanese NIKKEI to be the most efficient one. From
a geographic perspective, the most efficient indices are localized in Europe
and the least efficient ones in Asia and Latin America. Kristoufek and Vosvrda
(2014b) further focus on the index specification and show that approximate
entropy adds a significant informative value to the index. Kristoufek and
Vosvrda (2014a) then study efficiency across various commodity futures and
uncover that energy commodities are the most efficient ones whereas the
livestock commodities such as cattle and hogs are the least efficient ones.
Here we focus on efficiency ranking of the gold market with respect to a
currency used for the purchase, and we also contribute to the discussion on
statistical properties of the Efficiency Index.

##  2 Methods

Coming back to the roots of the efficient market hypothesis in 1965, the
treatment has been split into two main branches – based on the random random
walk hypothesis (Fama, 1965) and following the martingale specification
(Samuelson, 1965). We follow the latter approach as it is less restrictive and
it assumes the returns of the efficient market to be only serially
uncorrelated and with finite variance. This straightforward treatment enables
us to use various measures of market efficiency and use them to construct the
Efficiency Index, which allows to rank financial assets according to their
efficiency. In this section, we briefly describe the Efficiency Index, its
components and its statistical treatment. Introducing a procedure to assess
statistical features of the Efficiency Index is an important and novel
contribution to this line of research.

###  2.1 Capital market efficiency measure

Kristoufek and Vosvrda (2013, 2014a, 2014b) define the Efficiency Index (EI)
as

$EI=\sqrt{{\sum_{i=1}^{n}{\left(\frac{\widehat{M_{i}}-M_{i}^{\ast}}{R_{i}}\right)^{2}}}},$
(1)

where $M_{i}$ is the $i$th measure of efficiency, $\widehat{M_{i}}$ is an
estimate of the $i$th measure, $M_{i}^{\ast}$ is an expected value of the
$i$th measure for the efficient market and $R_{i}$ is a range of the $i$th
measure. EI is thus a distance from the efficient market situation. The index
can include various efficiency measures but these need to be bounded, which
turns out to be rather restrictive. We utilize three efficiency measures,
which meet such criterion and which are frequently used in market efficiency
studies (Cajueiro and Tabak, 2004, 2005; Di Matteo et al., 2005; Di Matteo,
2007; Zunino et al., 2010, 2011; Ortiz-Cruz et al., 2012) – Hurst exponent $H$
with an expected value of 0.5 for the efficient market ($M_{H}^{\ast}=0.5$),
fractal dimension $D$ with an expected value of 1.5 ($M_{D}^{\ast}=1.5$), and
the approximate entropy with an expected value of 1 ($M_{AE}^{\ast}=1$). As
discussed later in this section, Hurst exponent and fractal dimension share
their range for stationary processes whereas approximate entropy does not. For
this point, we need to rescale the approximate entropy part of the Efficiency
Index so that we have $R_{AE}=2$ and $R_{D}=R_{H}=1$.

###  2.2 Long-range dependence and its estimators

Long-range dependent series can be formally described as the ones with a
power-law decaying autocorrelation function (in time domain) and/or a
divergent at origin spectrum (in frequency domain). Specifically, the
autocorrelation function $\rho(k)$ with time lag $k$ of a long-range dependent
process decays as $\rho(k)\propto k^{2H-2}$ for $k\rightarrow+\infty$, and
spectrum $f(\lambda)$ with frequency $\lambda$ scales as
$f(\lambda)\propto\lambda^{1-2H}$ for $\lambda\rightarrow 0+$ (Geweke and
Porter-Hudak, 1983; Beran, 1994; Robinson, 1995). The characteristic parameter
$H$ is Hurst exponent which has several interesting values and intervals of
existence. For $H<0.5$, the processes are anti-persistent and switch their
sign frequently compared to an uncorrelated process. For $H=0.5$, the
processes are not long-range dependent, and for $H>0.5$, the processes are
persistent. The last group of processes can be further categorized according
to stationarity and (non)existence of variance. For stationary processes, it
holds that $H<1$. For the purposes of the Efficiency Index construction, it is
important that for an efficient market, we have $H=0.5$, as well as is the
fact that the index is bounded for stationary processes. Out of plethora of
Hurst exponent estimators (Beran, 1994; Taqqu et al., 1995; Taqqu and
Teverovsky, 1996; Robinson, 1995; Geweke and Porter-Hudak, 1983; Di Matteo et
al., 2003; Di Matteo, 2007; Barunik and Kristoufek, 2010; Teverovsky et al.,
1999), we choose the local Whittle estimator and the GPH estimator as they are
suitable for short time series with possible weak short-term memory, and they
are consistent and asymptotically normal (Geweke and Porter-Hudak, 1983;
Beran, 1994; Robinson, 1995; Taqqu et al., 1995; Taqqu and Teverovsky, 1996;
Phillips and Shimotsu, 2004).

###  2.3 Fractal dimension

Long-range dependence can be seen as a global characteristic of a time series.
Contrary to this view, fractal dimension $D$ can be interpreted as a measure
of local memory of the series since it captures roughness of the series
(Kristoufek and Vosvrda, 2013). Fractal dimension ranges between $1<D\leq 2$
for univariate series and this range is separated by the value of $D=1.5$ for
uncorrelated processes, which represents the efficient markets value. Low
fractal dimension signifies lower roughness and thus local persistence.
Reversely, high fractal dimension characterizes rougher series and thus
locally negatively correlated. Fractal dimension is thus well defined for an
efficient market and it is bounded for univariate series, which makes it a
perfect candidate to be included into the Efficiency Index. Specifically, we
utilize two estimators of fractal dimensions which share desirable statistical
properties for short time series – Hall-Wood and Genton estimators (Gneiting
and Schlather, 2004; Gneiting et al., 2010).

###  2.4 Approximate entropy

Entropy is considered as a measure of complexity. High entropy suggests little
or no information in the system and thus high uncertainty whereas low entropy
is characteristic for deterministic systems (Pincus and Kalman, 2004). From
the efficiency perspective, systems with maximum entropy can be seen as
efficient as these are serially uncorrelated. The lower the entropy level, the
less efficient the market is. For the construction of the Efficiency Index, we
utilize the approximate entropy which is bounded and thus well suited for the
index (Pincus, 1991).

###  2.5 Statistical inference

The original Efficiency Index (Kristoufek and Vosvrda, 2013) is a point
estimate of the true index value. This poses problems when discussing the
results and their statistical validity. We tackle this issue by introducing a
new approach to estimating EI which stems in the following steps:

  1. 1.

Obtain the estimated components $\widehat{M}_{i}$ of the Efficiency Index
according to Eq. 1.

  2. 2.

Shuffle the underlying return series.

  3. 3.

Estimate the components of the Efficiency Index for the shuffled series, and
label these as $\widehat{M}_{i,shuffle}$.

  4. 4.

Use $\widehat{M}_{i,shuffle}$ in place of $M^{\ast}_{i}$ in Eq. 1.

  5. 5.

Obtain $\widehat{EI}$ based on the previous steps.

  6. 6.

Repeat $N$ times.

  7. 7.

Obtain necessary statistics based on these $N$ estimates.

This way, we obtain an estimate of the Efficiency Index which controls for the
potential finite sample bias and the influence of distributional properties of
the analyzed series. For purposes of our study, we set $N=100$.

##  3 Results and discussion

We study the efficiency ranking of the gold111Gold is selected as a numéraire
due to its historical reputation as a safe haven as well as its reserve status
and a relative long-term price stability. prices quoted in different
currencies. The portfolio of study comprises 142 worldwide currencies, which
are described in Table 1. The dataset has been obtained from oanda.com, which
provides a large set of FX pairs as well as gold (and other precious metals)
prices in various currencies. The covered period ranges between 1.1.2011 and
30.11.2014, which totals 1430 observations for each of the 142 analyzed
currencies222We prefer a width of the portfolio to its depth to be able to
compare as many currencies as possible.. These currencies cover almost all
available and traded fiat currencies in addition to Bitcoin, the most popular
and used cryptocurrency.

For the efficiency ranking, we use the Efficiency Index (Eq. 1) with
adjustments described in Sec. 2.5. Specifically, we utilize two measures of
long-range dependence – the local Whittle estimator and the GPH estimator –,
two measures of fractal dimension – the Hall-Wood estimator and the Genton
estimator – and the approximate entropy as proposed by Pincus and Kalman
(2004). In the procedures, we follow the standard procedure of using the
logarithmic returns for the Hurst exponent and approximate entropy estimators,
and logarithmic prices for the fractal dimension estimation. Using 100
repetitions (shuffling), we obtain the estimated Efficiency Index as a median
value with a corresponding standard error for more information about precision
of the estimate.

The resulting ranking of gold prices with respect to the used currency is
presented in Table 2. The ranking is rather unexpected or even surprising.
Practically all of the most liquid currencies – the US dollar, the British
pound, the Australian dollar, the New Zealand dollar, the Japanese yen, the
Euro, the South Korean won, the Norwegian krone – are among the least
efficient gold markets (the least efficient third of the sample). Among these,
also the Bitcoin currency lays at the very bottom of the ranking. On the other
side of the ranking, the Top 5 is formed by the the Liberian dollar, the
Seychellois rupee, the Maldivian rufiyaa, the Comorian franc, and the Somali
shilling. The differences between levels of EI are stunning as the most
efficient markets share the index between 0.1 and 0.2 whereas the least
efficient ones jump close to 0.4. Such divergence is further accentuated by
very low standard errors of the estimates usually below 0.01 (medians and
standard errors are reported in Table 2).

To further investigate the contribution of the three different parts of the
Efficiency Index, i.e. Hurst exponent, fractal dimension, and approximate
entropy, we present Fig. 1. We observe that overall Hurst exponent is the
biggest contributor to the index. However, the strength of contribution varies
with the efficiency ranking. For the most efficient currencies, Hurst exponent
and approximate entropy play a similar role in the index. The influence of the
latter declines with a decreasing efficiency, and vice versa for the former.
The role of fractal dimension is also efficiency dependent. For the most
efficient markets, it forms only a small fraction of the index but its role
slightly increases for the less efficient currencies. For approximately the
lower two thirds of the ranking, the contributions are rather stable with
Hurst exponent at around 50%, and fractal dimension and approximate entropy
each at around 25%. All three components of the Efficiency Index thus form its
important parts. But apart from the most efficient markets, the long-term
memory plays a prominent role. The gold prices in various currencies exhibit a
persistent behavior with long-term trends even from the global perspective.
Such stable results accentuate the advantages of using the adjusted
methodology proposed here.

Returning back to the overall results which can be labelled as unexpected ones
(contrary to the quite expected results found for the stock markets
(Kristoufek and Vosvrda, 2013, 2014b) and other commodities (Kristoufek and
Vosvrda, 2014a)), we highlight the specific connection between the gold market
and the currency markets and further discuss potential causes.

The analyzed period of 2011 and 2014 covers very unorthodox times with regards
to monetary policies of the developed world as reactions to the Global
financial crisis, the Eurozone crisis, the Greek crisis and connected
phenomena. Various waves of the quantitative easing (QE) in the USA and the
UK, together with parallel actions of the European Central Bank eventually
leading to the quantitative easing as well, have formed an enormous pressure
on the relevant currencies and their depreciation. The first two waves of QE
in the USA pushed the gold prices upwards as these rallied till the end of
2011. The last wave of the USA QE, which was much weaker than the previous two
had no significant effect on the USD gold prices. The connection between the
currency depreciation and the consequent gold price (in the given currency)
boosting, together with a long-term effect of QE known in advance forms a
perfect environment for inefficiency of the gold market. This is well in hand
with most of the currencies the central banks of which participated in QE or
other forms of practical money-printing being among the least efficient
markets. It also further puts forward the gold’s speculative asset status
during the QE periods.

Such reasoning is further supported by gold being used as a hedge against
inflation (Narayan et al., 2010). During the initial stages of QE, there was a
serious concern about uncontrolled inflation as a reaction to the virtual
money-printing. As investors were hedging against expected inflation by
purchasing gold, its price was pushed further up. In time, the concern slowly
vanished as there were no signs of dangerous inflation pressures. Nonetheless,
the predictability and inefficiency of the gold markets under QE currencies
have come out as the final effect.

The following complementary explanation of the ranking structure can be quite
counterintuitive in the efficient market logic. The fact that a central bank
or a central authority of a country is transparent and holds up to its word
can actually lead to market inefficiency. Consider a central bank announcing a
new wave of QE. If the central bank is trustworthy, the investors will start
behaving accordingly and maximize their profit by acting upon it. However, the
QE process is a gradual one and it thus does not affect the market instantly
but in steps. Putting these factors together leads us to a quite well
predictable market behavior with relatively low risk assuming the authority
holds up to its promises. From the other side, the authorities which are not
too trustworthy are prone to change the announced policy repeatedly so that
the shocks to the currency market are unpredictable. Such unpredictability
leads to higher efficiency.

Additionally, the current situation at the foreign exchange markets has
uncovered another potential source of inefficiency in the most traded
currencies. In 2013, Bloomberg News reported that global regulators had
started investigation of major banks in the foreign exchange markets for
front-running orders and colluding to rig the foreign exchange rate benchmarks
(Bank of England, 2014). This affair is now referred to as the “forex probe”
and it has been claimed that the exchange rates manipulation had been realized
for about a decade. Such collusion goes majorly against the notion of market
efficiency and it provides a firm ground to the reported results of currency
efficiency ranking.

To summarize, the combination of gold prices and currencies forms a very
interesting and unique structure, the dynamics of which is much different
compared to other assets such as stock or commodity markets. We have shown
that the least efficient gold prices are mostly the ones quoted in major
currencies such the US dollar, the Euro, and the British pound. On the other
side of the spectrum, the most efficient gold prices are the ones quoted in
smaller and less traded currencies. From the practitioners’ perspective, we
have two possibilities of utilizing the results. We can either speculate on
gold prices in the major currencies, or we can hedge gold prices using the
minor currencies to obtain stable and efficient market position. Only the time
will tell whether the “forex probe” scandal and its resolution as well as the
end of the quantitative easing(s) will bring major currencies closer to
efficiency.

## Acknowledgements

The research leading to these results has received funding from the European
Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement No.
FP7-SSH-612955 (FinMaP) and the Czech Science Foundation project No.
P402/12/G097 “DYME – Dynamic Models in Economics”.

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Figure 1: Contributions to the Efficiency Index. Contributions of the three factors – Hurst exponent, fractal dimension, and approximate entropy – are illustrated here in percentage ($y$-axis). The currencies are ranked from the most efficient ones (from the left) to the least efficient ones (to the right). Note that not all the labels are visible on the $x$-axis due to the high number of analyzed currencies. Nonetheless, the values and contributions are present for all the currencies. The contributions are stacked. The bottom part represents the Hurst exponent contribution, the middle part illustrates the fractal dimension contribution, and the upper part shows the approximate entropy contribution. Hurst exponent plays an important role for all currencies and its contribution around 50% is quite stable across all currencies. For the most efficient ones, the contributions is slightly lower. The fractal dimension contribution is quite small for the most efficient markets and its value increases for the less efficient markets. The reverse is true for the approximate entropy contribution.  Table 1: Analyzed currencies Currency name | Code | Currency name | Code | Currency name | Code
---|---|---|---|---|---
Afghan afghani | AFN | Ghana cedi | GHS | Pakistani rupee | PKR
Albanian lek | ALL | Gibraltar pound | GIP | Panamanian balboa | PAB
Algerian dinar | DZD | Guatemalan quetzal | GTQ | Papua New Guinean kina | PGK
Angolan kwanza | AOA | Guyanese dollar | GYD | Paraguayan guarani | PYG
Argentine peso | ARS | Haitian gourde | HTG | Peruvian nuevo sol | PEN
Armenian dram | AMD | Honduran lempira | HNL | Philippine peso | PHP
Aruban florin | AWG | Hong Kong dollar | HKD | Polish zloty | PLN
Australian dollar | AUD | Hungarian forint | HUF | Qatari riyal | QAR
Azerbaijani manat | AZN | Icelandic krona | ISK | Romanian leu | RON
Bahamian dollar | BSD | Indian rupee | INR | Russian rubble | RUB
Bahraini dinar | BHD | Indonesian rupee | IDR | Rwandan franc | RWF
Bangladeshi taka | BDT | Iraqi dinar | IQD | Saint Helena pound | SHP
Barbadian dollar | BBD | Israeli new shekel | ILS | Samoan tala | WST
Belarusian ruble | BYR | Jamaican dollar | JMD | Sao Tome and Principe dobra | STD
Belize dollar | BZD | Japanese yen | JPY | Saudi riyal | SAR
Bitcoin | BTC | Jordanian dinar | JOD | Serbian dinar | RSD
Botswana pula | BWP | Kazakhstani tenge | KZT | Seychellois rupee | SCR
Brazilian real | BRL | Kenyan shilling | KES | Sierra Leonean leone | SLL
British pound | GBP | Kuwaiti dinar | KWD | Singapore dollar | SGD
Brunei dollar | BND | Kyrgyzstani som | KGS | Solomon Islands dollar | SBD
Bulgarian lev | BGN | Lao kip | LAK | Somali shilling | SOS
Burundian franc | BIF | Lebanese pound | LBP | South African rand | ZAR
Cambodian riel | KHR | Lesotho loti | LSL | South Korean won | KRW
Canadian dollar | CAD | Liberian dollar | LRD | Sri Lanka rupee | LKR
Cape Verdean escudo | CVE | Libyan dinar | LYD | Surinamese dollar | SRD
Cayman Islands dollar | KYD | Lithuanian litas | LTL | Swazi lilangeni | SZL
CFP franc | XPF | Macanese pataca | MOP | Swedish krona | SEK
Chilean peso | CLP | Macedonia denar | MKD | Swiss franc | CHF
Chinese yuan | CNY | Malagasy ariary | MGA | Syrian pound | SYP
Colombian peso | COP | Malaysian riggit | MYR | Tajikistani somoni | TJS
Comorian franc | KMF | Maldivian rufiyaa | MVR | Tanzanian shilling | TZS
Congolese franc | CDF | Mauritanian ouguiya | MRO | Thai baht | THB
Costa Rican colon | CRC | Mauritian rupee | MUR | Tongan pa’anga | TOP
Croatian kuna | HRK | Mexican peso | MXN | Trinidad and Tobago dollar | TTD
Cuban convertible peso | CUC | Moldovan leu | MDL | Tunisian dinar | TND
Czech koruna | CZK | Mongolian togrog | MNT | Turkish lira | TRY
Danish krone | DKK | Moroccan dirham | MAD | Turkmenistan manat | TMT
Djiboutian franc | DJF | Mozambican metical | MZN | Ugandan shilling | UGX
Dominican peso | DOP | Namibian dollar | NAD | Ukrainian hryvnia | UAH
East Caribbean dollar | XCD | Nepalese rupee | NPR | United Arab Emirates dirham | AED
Egyptian pound | EGP | Netherlands Antillean guilder | ANG | United States dollar | USD
Eritrean nakfa | ERN | New Taiwan dollar | TWD | Uruguayan peso | UYU
Ethiopian birr | ETB | New Zealand dollar | NZD | Uzbekistani som | UZS
Euro | EUR | Nicaraguan cordoba | NIO | Vanuatu vatu | VUV
Falkland Islands pound | FKP | Nigerian naira | NGN | Vietnamese dong | VND
Fijian dollar | FJD | North Korean won | KPW | Yemeni rial | YER
Gambian dalasi | GMD | Norwegian krone | NOK |  |
Georgian lari | GEL | Omani rial | OMR |  |
Table 2: Estimated Efficiency Index for gold prices in worldwide currencies Rank | Country | EI | Rank | Country | EI | Rank | Country | EI
---|---|---|---|---|---|---|---|---
1 | Liberia |  0.1064$\pm$0.0083 | 49 | Albania |  0.2233$\pm$0.0069 | 97 | Dominica |  0.2598$\pm$0.0068
2 | Seychelles |  0.1167$\pm$0.0067 | 50 | Pakistan |  0.2233$\pm$0.0064 | 98 | Sweden |  0.2601$\pm$0.0054
3 | Maldives |  0.1447$\pm$0.0060 | 51 | Paraguay |  0.2251$\pm$0.0055 | 99 | Uganda |  0.2606$\pm$0.0056
4 | Comoros |  0.1473$\pm$0.0076 | 52 | Haiti |  0.2264$\pm$0.0096 | 100 | Barbados |  0.2623$\pm$0.0066
5 | Somalia |  0.1525$\pm$0.0064 | 53 | Laos |  0.2283$\pm$0.0068 | 101 | Angola |  0.2624$\pm$0.0078
6 | Tonga |  0.1571$\pm$0.0094 | 54 | Solomon Isl. |  0.2307$\pm$0.0111 | 102 | Denmark |  0.2627$\pm$0.0059
7 | Mauritania |  0.1572$\pm$0.0063 | 55 | Lebanon |  0.2318$\pm$0.0060 | 103 | Neth. Antilles |  0.2638$\pm$0.0075
8 | Rwanda |  0.1656$\pm$0.0098 | 56 | Phillipines |  0.2323$\pm$0.0065 | 104 | Saudi Arabia |  0.2639$\pm$0.0081
9 | Chile |  0.1663$\pm$0.0101 | 57 | Tajikistan |  0.2339$\pm$0.0054 | 105 | Cuba |  0.2640$\pm$0.0064
10 | S. T. & Princ |  0.1680$\pm$0.0083 | 58 | Papua N. Guin. |  0.2341$\pm$0.0122 | 106 | Turkey |  0.2659$\pm$0.0055
11 | Mozambique |  0.1709$\pm$0.0100 | 59 | Mexico |  0.2344$\pm$0.0053 | 107 | India |  0.2665$\pm$0.0053
12 | Domin. Rep. |  0.1727$\pm$0.0109 | 60 | Bangladesh |  0.2370$\pm$0.0053 | 108 | Croatia |  0.2674$\pm$0.0058
13 | Libya |  0.1729$\pm$0.0088 | 61 | Peru |  0.2401$\pm$0.0081 | 109 | Tanzania |  0.2684$\pm$0.0075
14 | Belize |  0.1743$\pm$0.0082 | 62 | Malaysia |  0.2419$\pm$0.0069 | 110 | Lithuania |  0.2687$\pm$0.0056
15 | Madagascar |  0.1752$\pm$0.0067 | 63 | Bahrain |  0.2436$\pm$0.0064 | 111 | Bulgaria |  0.2696$\pm$0.0049
16 | Costa Rica |  0.1758$\pm$0.0076 | 64 | Swaziland |  0.2449$\pm$0.0066 | 112 | Azerbaijan |  0.2700$\pm$0.0066
17 | Fr. Polynesia |  0.1768$\pm$0.0059 | 65 | Taiwan |  0.2450$\pm$0.0099 | 113 | Czech Rep. |  0.2704$\pm$0.0049
18 | Indonesia |  0.1773$\pm$0.0068 | 66 | Jamaica |  0.2459$\pm$0.0071 | 114 | Canada |  0.2716$\pm$0.0054
19 | Burundi |  0.1794$\pm$0.0103 | 67 | Eritrea |  0.2461$\pm$0.0069 | 115 | Morocco |  0.2727$\pm$0.0047
20 | Mauritius |  0.1798$\pm$0.0073 | 68 | Qatar |  0.2461$\pm$0.0068 | 116 | Macau |  0.2732$\pm$0.0062
21 | Djibouti |  0.1828$\pm$0.0084 | 69 | Trin. & Tob. |  0.2468$\pm$0.0071 | 117 | Armenia |  0.2733$\pm$0.0095
22 | Macedonia |  0.1834$\pm$0.0078 | 70 | Argentina |  0.2472$\pm$0.0100 | 118 | Guyana |  0.2742$\pm$0.0074
23 | Uzbekistan |  0.1836$\pm$0.0075 | 71 | Aruba |  0.2473$\pm$0.0077 | 119 | Serbia |  0.2757$\pm$0.0057
24 | Iceland |  0.1840$\pm$0.0055 | 72 | Yemen |  0.2476$\pm$0.0070 | 120 | Congo (DRC) |  0.2760$\pm$0.0078
25 | Cape Verde |  0.1841$\pm$0.0074 | 73 | Ethiopia |  0.2478$\pm$0.0060 | 121 | Singapore |  0.2770$\pm$0.0073
26 | Sierra Leone |  0.1855$\pm$0.0089 | 74 | Botswana |  0.2479$\pm$0.0092 | 122 | EU |  0.2773$\pm$0.0051
27 | Ghana |  0.1870$\pm$0.0061 | 75 | Kuwait |  0.2479$\pm$0.0076 | 123 | Norway |  0.2775$\pm$0.0058
28 | Mongolia |  0.1876$\pm$0.0090 | 76 | Kenya |  0.2481$\pm$0.0058 | 124 | Poland |  0.2781$\pm$0.0069
29 | Nicaragua |  0.1894$\pm$0.0086 | 77 | Jordan |  0.2490$\pm$0.0065 | 125 | Saint Helena |  0.2818$\pm$0.0111
30 | Cambodia |  0.1896$\pm$0.0065 | 78 | Thailand |  0.2497$\pm$0.0073 | 126 | Hungary |  0.2819$\pm$0.0055
31 | Nepal |  0.1905$\pm$0.0055 | 79 | China |  0.2500$\pm$0.0069 | 127 | Nigeria |  0.2824$\pm$0.0092
32 | Bahamas |  0.1925$\pm$0.0086 | 80 | Namibia |  0.2509$\pm$0.0070 | 128 | Australia |  0.2839$\pm$0.0067
33 | Brazil |  0.1936$\pm$0.0063 | 81 | Georgia |  0.2513$\pm$0.0078 | 129 | South Korea |  0.2846$\pm$0.0079
34 | Israel |  0.1950$\pm$0.0068 | 82 | Tunisia |  0.2515$\pm$0.0076 | 130 | Honduras |  0.2904$\pm$0.0082
35 | Iraq |  0.1987$\pm$0.0078 | 83 | Panama |  0.2528$\pm$0.0069 | 131 | Japan |  0.2905$\pm$0.0077
36 | Colombia |  0.2010$\pm$0.0076 | 84 | Sri Lanka |  0.2539$\pm$0.0046 | 132 | Switzerland |  0.2921$\pm$0.0056
37 | Samoa |  0.2042$\pm$0.0091 | 85 | UAE |  0.2551$\pm$0.0072 | 133 | South Africa |  0.2997$\pm$0.0059
38 | Moldova |  0.2061$\pm$0.0062 | 86 | Turkmenistan |  0.2555$\pm$0.0075 | 134 | New Zealand |  0.3101$\pm$0.0102
39 | Algeria |  0.2063$\pm$0.0063 | 87 | Lesotho |  0.2555$\pm$0.0065 | 135 | Falkland Isl. |  0.3118$\pm$0.0101
40 | Ukraine |  0.2103$\pm$0.0105 | 88 | Oman |  0.2559$\pm$0.0061 | 136 | UK |  0.3172$\pm$0.0108
41 | Cayman Isl. |  0.2105$\pm$0.0059 | 89 | Romania |  0.2560$\pm$0.0070 | 137 | Gibraltar |  0.3177$\pm$0.0098
42 | Suriname |  0.2120$\pm$0.0090 | 90 | Vietnam |  0.2565$\pm$0.0117 | 138 | Vanuatu |  0.3196$\pm$0.0078
43 | Fiji |  0.2137$\pm$0.0099 | 91 | Guatemala |  0.2572$\pm$0.0056 | 139 | Belarus |  0.3373$\pm$0.0069
44 | Kazakhstan |  0.2146$\pm$0.0065 | 92 | Hong Kong |  0.2576$\pm$0.0075 | 140 | Gambia |  0.3530$\pm$0.0095
45 | Syria |  0.2149$\pm$0.0067 | 93 | Kyrgyzstan |  0.2578$\pm$0.0059 | 141 | Egypt |  0.3574$\pm$0.0116
46 | Afghanistan |  0.2179$\pm$0.0060 | 94 | Russia |  0.2589$\pm$0.0066 | 142 | Bitcoin |  0.3846$\pm$0.0076
47 | Brunei |  0.2221$\pm$0.0083 | 95 | North Korean |  0.2596$\pm$0.0064 |  |  |
48 | Uruguay |  0.2226$\pm$0.0056 | 96 | USA |  0.2597$\pm$0.0069 |  |  |