# Photoinduced High-Frequency Charge Oscillations in Dimerized Systems

Kenji Yonemitsu E-mail<EMAIL_ADDRESS>of PhysicsDepartment
of Physics Chuo University Chuo University Bunkyo Bunkyo Tokyo 112-8551 Tokyo
112-8551 Japan Japan

###### Abstract

Photoinduced charge dynamics in dimerized systems is studied on the basis of
the exact diagonalization method and the time-dependent Schrödinger equation
for a one-dimensional spinless-fermion model at half filling and a two-
dimensional model for $\kappa$-(bis[ethylenedithio]tetrathiafulvalene)2X
[$\kappa$-(BEDT-TTF)2X] at three-quarter filling. After the application of a
one-cycle pulse of a specifically polarized electric field, the charge
densities at half of the sites of the system oscillate in the same phase and
those at the other half oscillate in the opposite phase. For weak fields, the
Fourier transform of the time profile of the charge density at any site after
photoexcitation has peaks for finite-sized systems that correspond to those of
the steady-state optical conductivity spectrum. For strong fields, these peaks
are suppressed and a new peak appears on the high-energy side, that is, the
charge densities mainly oscillate with a single frequency, although the
oscillation is eventually damped. In the two-dimensional case without
intersite repulsion and in the one-dimensional case, this frequency
corresponds to charge-transfer processes by which all the bonds connecting the
two classes of sites are exploited. Thus, this oscillation behaves as an
electronic breathing mode. The relevance of the new peak to a recently found
reflectivity peak in $\kappa$-(BEDT-TTF)2X after photoexcitation is discussed.

##  1 Introduction

Photoinduced dynamics and phase transitions in itinerant electron systems [1,
2, 3, 4] have received renewed interest with the observation of strengthened
or weakened orders and even transient suppression of charge motion,[5, 6, 7]
enhancing the prospect for controlling the electronic phase.[8, 9] Here,
pictures of photoinduced states are not conventional ones obtained after the
absorption of photons but rather electrons directly and coherently driven by
an oscillating electric field.[10, 11] In this context, the concept of
dynamical localization may play an important role, although it is basically
applicable to systems that are driven by continuous waves.[12, 13, 14, 15]
Even after a pulse excitation, states induced by a strong electric field have
been shown to be similar to states expected for dynamical localization.[5, 7,
16, 17]

Dynamical localization describes the long-time behavior obtained by time-
averaging. Negative-temperature states and inverted interactions have also
been discussed by time-averaging after photoexcitation.[18, 19, 20, 21] For
the long-time behavior, continuous-wave- and pulse-induced phenomena have been
compared in a quantitative manner from a broad perspective[22, 23, 24].
However, the picture for short-time behavior has not been discussed in a
systematic manner. Thus, it is desirable to present a concrete example.

Quite recently, a new reflectivity peak has been discovered in photoexcited
$\kappa$-(bis[ethylenedithio]tetrathiafulvalene)2Cu[N(CN)2]Br [$\kappa$-(BEDT-
TTF)2Cu[N(CN)2]Br] on the high-energy side of the main reflectivity
spectrum.[25] This unprecedented peak is narrow. Its energy is independent of
the excitation strength and it survives for a while after photoexcitation;
thus, it is not due to the optical Stark effect. The associated charge
oscillation has been shown to be enhanced near criticality in the
“pressure”-temperature phase diagram.[26] A mechanism of the emergence of such
a high-energy peak is theoretically studied using the exact diagonalization
method for small clusters in this paper. Thus, the influence of criticality is
beyond the scope of this paper.

The object material is one of the $\kappa$-(BEDT-TTF)2X, which are quasi-two-
dimensional three-quarter-filled dimerized organic conductors. Photoinduced
insulator-metal transitions are known to take place in these materials.[27,
28, 29] The intradimer charge degrees of freedom are studied in reference to
anomalous dielectric permittivity,[30] which is associated with polar charge
distributions inside dimers.[31, 32, 33, 34, 35] However, the high-frequency
charge oscillation mode had not been discussed before Ref. iwai_unpub. Thus,
the mechanism and condition for the appearance of this mode are yet to be
clarified.

Here, we show that such a charge oscillation mode emerges in different
dimerized systems after the application of a strong pulse of an oscillating
electric field. Numerical results are presented in a one-dimensional spinless-
fermion “$t_{1}$-$t_{2}$-$V$” model at half filling and a two-dimensional
extended Hubbard model for $\kappa$-(BEDT-TTF)2X at three-quarter filling,
which is photoexcited along the $a$\- or $c$-axis. The high-frequency charge-
oscillation mode is shown to appear in a wide parameter space of ground states
with a uniform charge distribution. A close association with time-averaged
properties is also revealed.

##  2 Dimerized Models in One and Two Dimensions

In one dimension, we use one of the simplest models, i.e., a spinless fermion
model at half filling,

$\displaystyle H_{\mbox{1D}}$ $\displaystyle=$ $\displaystyle
t_{1}\sum_{n}(c^{\dagger}_{2n}c_{2n+1}+c^{\dagger}_{2n+1}c_{2n})+t_{2}\sum_{n}(c^{\dagger}_{2n-1}c_{2n}+c^{\dagger}_{2n}c_{2n-1})$
(1)
$\displaystyle+V\sum_{j}\left(n_{j}-\frac{1}{2}\right)\left(n_{j+1}-\frac{1}{2}\right)\;,$

where $c^{\dagger}_{j}$ creates a spinless fermion at site $j$ and
$n_{j}$=$c^{\dagger}_{j}c_{j}$. The parameter $V$ represents the nearest-
neighbor repulsion. Large and small transfer integrals, $t_{1}$ and $t_{2}$,
are alternated, as shown in Fig. 1(a).

Figure 1: (Color online) (a) One-dimensional lattice for spinless-fermion
model and (b) two-dimensional lattice for $\kappa$-(BEDT-TTF)2X.

A 24-site system with a periodic boundary condition is used. The distance
between neighboring sites is set to be equal and unity. We use $t_{2}=-0.1$
and vary $t_{1}$ and $V$.

In two dimensions, we use an extended Hubbard model at three-quarter filling,

$\displaystyle H_{\mbox{2D}}$ $\displaystyle=$ $\displaystyle\sum_{\langle
ij\rangle\sigma}t_{ij}(c^{\dagger}_{i\sigma}c_{j\sigma}+c^{\dagger}_{j\sigma}c_{i\sigma})+U\sum_{i}\left(n_{i\uparrow}-\frac{3}{4}\right)\left(n_{i\downarrow}-\frac{3}{4}\right)$
(2) $\displaystyle+\sum_{\langle
ij\rangle}V_{ij}\left(n_{i}-\frac{3}{2}\right)\left(n_{j}-\frac{3}{2}\right)\;,$

where $c^{\dagger}_{i\sigma}$ creates an electron with spin $\sigma$ at site
$i$, $n_{i\sigma}$=$c^{\dagger}_{i\sigma}c_{i\sigma}$, and
$n_{i}$=$\sum_{\sigma}n_{i\sigma}$. The parameter $U$ represents the on-site
Coulomb repulsion. The transfer integral $t_{ij}$ and the intersite Coulomb
repulsion $V_{ij}$ depend on the bond $ij$, as shown in Fig. 1(b). A 16-site
system with periodic boundary conditions is used. The intermolecular distances
and angles are taken from the structural data for $\kappa$-(BEDT-
TTF)2Cu[N(CN)2]Cl.[36, 37] Unless stated otherwise, we use, in units of eV,
$t_{b1}=-0.3006$, $t_{b2}=-0.1148$, $t_{p}=-0.1107$, and $t_{q}=0.0424$, which
are estimated from the extended Hückel calculation,[37] $U=0.8$,
$V_{b1}=0.40$, $V_{b2}=0.24$, $V_{p}=0.28$, and $V_{q}=0.24$. In Eq. (2), the
constant term is subtracted in such a way that the total energy becomes zero
in equilibrium at infinite temperature.

The initial state is the ground state obtained by the exact diagonalization
method. Photoexcitation is introduced through the Peierls phase

$c_{i\sigma}^{\dagger}c_{j\sigma}\rightarrow\exp\left[\frac{ie}{\hbar
c}\mbox{\boldmath$r$}_{ij}\cdot\mbox{\boldmath$A$}(t)\right]c_{i\sigma}^{\dagger}c_{j\sigma}\;,$
(3)

which is substituted into Eq. (2) [and its spinless analog is substituted into
Eq. (1)] for each combination of sites $i$ and $j$ with relative position
$\mbox{\boldmath$r$}_{ij}=\mbox{\boldmath$r$}_{j}-\mbox{\boldmath$r$}_{i}$. We
employ symmetric one-cycle electric-field pulses[17, 20, 21] and use the time-
dependent vector potential

$\mbox{\boldmath$A$}(t)=\frac{c\mbox{\boldmath$F$}}{\omega_{c}}\left[\cos(\omega_{c}t)-1\right]\theta(t)\theta\left(\frac{2\pi}{\omega_{c}}-t\right)\;,$
(4)

where $F$ describes the amplitude ($F=\mid\mbox{\boldmath$F$}\mid$) and
polarization of the electric field. Unless stated otherwise, the central
frequency $\omega_{c}$ is chosen to be $\omega_{c}=0.7$, which is above the
main charge-transfer excitations, as shown below. Hereafter, frequencies
$\omega$ including $\omega_{c}$ are also shown in units of eV (with
$\hbar$=1). The optical conductivity spectra are calculated for the ground
states as before.[28] The time-dependent Schrödinger equation is numerically
solved by expanding the exponential evolution operator with a time slice
$dt$=0.02 to the 15th order and by checking the conservation of the norm.[38]

##  3 Intradimer and Interdimer Bond Densities in Two-Dimensional Case

In this section, we use the two-dimensional model for $\kappa$-(BEDT-TTF)2X.
Some quantities time-averaged after photoexcitation along the $c$-axis are
shown in Fig. 2.

Figure 2: (Color online) (a) Time-averaged bond densities
$\sum_{\sigma}\langle\langle\left(c^{\dagger}_{i\sigma}c_{j\sigma}+c^{\dagger}_{j\sigma}c_{i\sigma}\right)\rangle\rangle$
for different combinations of $ij$, (b) time-averaged kinetic energy, and
total energy after photoexcitation along $c$-axis as functions of field
amplitude $F$.

In this paper, averages are taken over the interval of $5T<t<10T$ with
$T=2\pi/\omega_{c}$ as before.[17] Figure 2(a) shows the time-averaged bond
densities
$\sum_{\sigma}\langle\langle\left(c^{\dagger}_{i\sigma}c_{j\sigma}+c^{\dagger}_{j\sigma}c_{i\sigma}\right)\rangle\rangle$
for $ij$ along the $b_{1}$, $b_{2}$, $p$, and $q$ bonds as functions of the
field amplitude $F$. Here, $F$ is shown in units of V/Å. If we take the length
scale $a$=2.6 Åfrom the component parallel to the $c$-axis of the intradimer
intermolecular relative position in $\kappa$-(BEDT-TTF)2X, $F$=0.20
corresponds to $eaF/\hbar\omega_{c}$=0.74. As $F$ increases, the time-averaged
bond densities decrease in magnitude, simultaneously vanish at $F$=0.20, and
invert their signs.

This behavior is inconsistent with effective transfer integrals, which are
renormalized by the zeroth-order Bessel functions with bond-dependent
arguments[22] and vanish at different values of $F$. This fact suggests that
electrons on different bonds are transferred in a concerted manner for large
$F$, which will be discussed in later sections. This behavior is universally
observed for large $F$. For $\omega_{c}$=0.9, the time-averaged bond densities
simultaneously invert their signs again at $eaF/\hbar\omega_{c}$=0.74. For
$\omega_{c}$=0.5, which is inside the main charge-transfer excitations in the
conductivity spectrum, they are substantially suppressed at
$eaF/\hbar\omega_{c}$=0.74, but they do not invert their signs for larger
values of $F$ (not shown).

Figure 2(b) shows the time-averaged kinetic energy, i.e., the expectation
value of the first term in Eq. (2), and the total energy after
photoexcitation. A negative-temperature state is realized when the total
energy is positive (i.e., larger than that in equilibrium at infinite
temperature). Note that a negative-temperature state is formed after the
application of a one-cycle pulse of the electric field in different
models.[20, 21] For $\omega_{c}$=0.9, the time-averaged kinetic energy and the
total energy behave similarly to those for $\omega_{c}$=0.7 here. For
$\omega_{c}$=0.5, however, they do not invert their signs; thus, a negative-
temperature state is not realized.

To see why the $F$ dependences of the time-averaged bond densities are not
described by the corresponding effective transfer integrals, we need to
observe the short-time behavior of transient states. Thus, the time evolution
of the charge density $2-\langle n_{0}\rangle$ is shown in Fig. 3 for
photoexcitation along the $c$-axis with small values of $F$.

Figure 3: (Color online) Time evolution of charge density $2-\langle
n_{0}\rangle$ at zeroth site in Fig. 1(b) during and after photoexcitation
along $c$-axis with small $F$.

For photoexcitation along the $a$-axis or along the $c$-axis, all sites are
classified into two groups according to their charge densities for a reason of
symmetry. Because the total charge is conserved, the charge distribution among
sites is determined once the charge density at one site is known.
Consequently, under the condition of Fig. 3, the charge densities at the even-
numbered sites in Fig. 1(b) vary with time as shown here, and those at the
odd-numbered sites vary in the opposite phase.

A charge distribution deviating from the initial uniform distribution is
caused by photoexcitation. Thus, the time profile of the charge density above
is expected to have similar information to the steady-state optical
conductivity spectrum. We calculate the absolute values of the Fourier
transforms of the time profiles ($T<t<50T$ after photoexcitation) of the
charge density in Fig. 3 and show them in Fig. 4.

Figure 4: (Color online) Absolute values of Fourier transforms of time
profiles ($T<t<50T$) of charge density in Fig. 3 compared with optical
conductivity spectrum with polarization parallel to $c$-axis in ground state.

For comparison, we also show the optical conductivity spectrum with
polarization parallel to the $c$-axis in the ground state. To facilitate the
comparison, we take a rather long time span for the Fourier transforms with a
small frequency slice. It is clearly shown that, for small $F$, the absolute
values of the Fourier transforms have peaks with large weights at energies
where the conductivity spectrum has peaks.

The above result shows how charge densities are modulated by photoexcitation
and implies that their characteristic time profiles can be observed by optical
measurements. In the following sections, we investigate Fourier spectra for
large $F$ more systematically in the one- and two-dimensional models (for
specifically polarized fields in the two-dimensional case) where there are two
inequivalent sites with respect to charge densities.

Before discussing specific models, we mention a general fact in noninteracting
systems. In any noninteracting system,
$H_{\mbox{NI}}=\sum_{\lambda}\epsilon_{\lambda}c^{\dagger}_{\lambda}c_{\lambda}$,
we have
$\left(id/dt\right)c^{\dagger}_{\lambda_{1}}c_{\lambda_{2}}=\left(\epsilon_{\lambda_{2}}-\epsilon_{\lambda_{1}}\right)c^{\dagger}_{\lambda_{1}}c_{\lambda_{2}}$
for any $\lambda_{1}$ and $\lambda_{2}$; thus, the quantities
$Q_{\lambda_{1}\lambda_{2}}\equiv
c^{\dagger}_{\lambda_{1}}c_{\lambda_{2}}+c^{\dagger}_{\lambda_{2}}c_{\lambda_{1}}$
and $P_{\lambda_{1}\lambda_{2}}\equiv-
ic^{\dagger}_{\lambda_{1}}c_{\lambda_{2}}+ic^{\dagger}_{\lambda_{2}}c_{\lambda_{1}}$
for $\lambda_{1}\neq\lambda_{2}$ oscillate with
$\omega=\mid\epsilon_{\lambda_{2}}-\epsilon_{\lambda_{1}}\mid$ similarly to
the position and momentum operators of a harmonic oscillator.

##  4 Photoinduced Charge Oscillations in One Dimension

To study photoinduced charge oscillations in dimerized systems from a broad
perspective, we here use the one-dimensional spinless fermion model in Eq. (1)
at half filling. Here and in the following section, we take a shorter time
span of $T<t<10T$ after photoexcitation for Fourier transforms and refer to
the absolute value of the Fourier transform of the time profile of the charge
density simply as a Fourier spectrum. For large $F$ ($F\geq 0.10$), Fourier
spectra are found to be insensitive to the frequency slice. In other words,
the broad peak for large $F$ ($F\geq 0.10$) is an intrinsic one, except for in
the noninteracting case. A larger frequency slice (i.e., a shorter time span)
than that used for Fig. 4 makes them easy to read. The Fourier spectra thus
obtained are shown for different values of $t_{1}$ and $V$ in Fig. 5.

Figure 5: (Color online) Absolute values of Fourier transforms of time
profiles ($T<t<10T$) of charge density at any site in Fig. 1(a) with different
$F$ and optical conductivity spectrum in ground state for (a) $t_{1}$=$-$0.2,
$V$=0.0, (b) $t_{1}$=$-$0.2, $V$=0.2, (c) $t_{1}$=$-$0.3, $V$=0.2, and (d)
$t_{1}$=$-$0.3, $V$=0.4.

If the model parameters are in units of eV, $F$ is shown in units of
V/[intersite distance].

In the noninteracting case ($V$=0.0) [Fig. 5(a)] and for $\mid t_{1}\mid>\mid
t_{2}\mid$, the one-electron states of energies from $-\mid t_{1}\mid-\mid
t_{2}\mid$ to $-\mid t_{1}\mid+\mid t_{2}\mid$ are occupied, and those from
$\mid t_{1}\mid-\mid t_{2}\mid$ to $\mid t_{1}\mid+\mid t_{2}\mid$ are
unoccupied. Then, the optical conductivity spectrum has weights in the energy
range from $2(\mid t_{1}\mid-\mid t_{2}\mid)$ to $2(\mid t_{1}\mid+\mid
t_{2}\mid)$ [from 0.2 to 0.6 in Fig. 5(a)]. They form discrete peaks owing to
the finite-size effect, and their oscillator strengths become larger as the
energy is lowered. For small $F$ ($F$=0.05), the weights of the Fourier
spectrum are in this energy range. As $F$ increases, the weights first
increase as a whole ($F$=0.30), but the low-energy part is then decreased,
while the high-energy part is further increased ($0.30<F<0.50$). For large $F$
($F$=0.50), the charge density oscillates with $\omega_{\mbox{osc}}=2(\mid
t_{1}\mid+\mid t_{2}\mid)$ (i.e., with a period of $\pi/(\mid t_{1}\mid+\mid
t_{2}\mid)$).

In general, on a bipartite lattice (with sublattices “e” and “o”) without
interactions,
$H_{\mbox{BL}}=\sum_{i,j}t_{i,j}\left(c^{\dagger}_{e,i}c_{o,j}+c^{\dagger}_{o,j}c_{e,i}\right)$,
where all sites are equivalent in terms of the network of transfer integrals,
the quantity
$Q\equiv\sum_{i,j}\left(c^{\dagger}_{e,i}c_{e,j}-c^{\dagger}_{o,i}c_{o,j}\right)$
behaves as a harmonic oscillator, $\left(id/dt\right)^{2}Q=\omega^{2}Q$, which
is derived through the double commutator. In the ground state, its expectation
value is zero. If such a system has $N$ sites, we define
$c_{e,k}=\sqrt{2/N}\sum_{j}e^{ikj}c_{e,j}$ and
$c_{o,k}=\sqrt{2/N}\sum_{j}e^{ikj}c_{o,j}$ to have
$Q=(N/2)\left(c^{\dagger}_{e,k=0}c_{e,k=0}-c^{\dagger}_{o,k=0}c_{o,k=0}\right)$,
whose motion is governed by the term
$H_{\mbox{BL},k=0}=(t_{1}+t_{2})\left(c^{\dagger}_{e,k=0}c_{o,k=0}+c^{\dagger}_{o,k=0}c_{e,k=0}\right)$
in $H_{\mbox{BL}}=\sum_{k}H_{\mbox{BL},k}$ in the present case. If
$t_{1}t_{2}<0$, we multiply the creation and annihilation operators for sites
2, 3, 6, 7, etc., by $(-1)$ to have $t_{1}t_{2}>0$ at first, or equivalently
we use
$Q=(N/2)\left(c^{\dagger}_{e,k=\pi}c_{e,k=\pi}-c^{\dagger}_{o,k=\pi}c_{o,k=\pi}\right)$
and
$H_{\mbox{BL},k=\pi}=(t_{1}-t_{2})\left(c^{\dagger}_{e,k=\pi}c_{o,k=\pi}+c^{\dagger}_{o,k=\pi}c_{e,k=\pi}\right)$
instead of the above. Then, it is straightforward to show that
$\omega=\omega_{\mbox{osc}}$ and the charge oscillation is undamped in the
noninteracting case. However, with interactions,
$\left(id/dt\right)^{2}Q\neq\omega^{2}Q$ and the charge oscillation is damped.

The above behavior is basically maintained in an interacting case [Fig. 5(b)].
In the optical conductivity spectrum, the oscillator strength is concentrated
on the low-energy peak owing to the exciton effect. The weight of the Fourier
spectrum for small $F$ ($F$=0.05) is also concentrated on a peak at the
corresponding energy. However, for large $F$, the low-energy part is decreased
and the high-energy part is increased as $F$ increases ($0.40<F<0.80$). The
frequency of the strong-field-induced charge oscillation is given by
$\omega_{\mbox{osc}}=2(\mid t_{1}\mid+\mid t_{2}\mid)$. Compared with the
noninteracting case [Fig. 5(a)], where the charge oscillation is undamped
owing to integrability, it is damped; thus, the peak height is smaller.

Numerical results for a different value of $t_{1}$ are shown in Figs. 5(c) and
5(d). The $F$ dependence of the Fourier spectrum is similar to above,
especially for large $F$ ($0.30<F<0.50$). The frequency of the strong-field-
induced charge oscillation is given by $\omega_{\mbox{osc}}=2(\mid
t_{1}\mid+\mid t_{2}\mid)$ again, which is 0.8 for Figs. 5(c) and 5(d). For
$t_{1}$=$-$0.4, we find a peak at $\omega_{\mbox{osc}}$=1.0 for $V$=0.4 and
$V$=0.6 (not shown), which also satisfies the above equation for
$\omega_{\mbox{osc}}$. When we further increase $\mid t_{1}\mid$, we find that
it is occasionally necessary to increase the central frequency $\omega_{c}$ as
well (e.g., $\omega_{c}$=1.4 for $t_{1}$=$-$0.6 and $t_{1}$=$-$0.8 for
$\omega_{c}$ to resonate with $\omega_{\mbox{osc}}$) to obtain the strong-
field-induced charge oscillation and that it always appears at a frequency
near $\omega_{\mbox{osc}}=2(\mid t_{1}\mid+\mid t_{2}\mid)$. This relation is
maintained even when the nearest-neighbor repulsion strengths are alternated.
This relation is also maintained even away from half filling. This charge
oscillation is not observed in the charge-ordered phase with $V>2\mid
t_{1}\mid$.

The above relation between $\omega_{\mbox{osc}}$ and the transfer integrals
implies that this charge oscillation is realized by simultaneous charge
transfers through the $t_{1}$ and $t_{2}$ bonds, as shown in Fig. 6.

Figure 6: (Color online) Schematic view of photoinduced charge oscillation.

At a site, fermions are simultaneously transferred from the neighboring sites
on both sides to this site and back to these two sites. Because these charge-
transfer processes are realized coherently everywhere in the system, the whole
process is effectively described by a two-site model,
$H_{0}=t_{0}\left(c^{\dagger}_{e}c_{o}+c^{\dagger}_{o}c_{e}\right)=t_{0}\left(c^{\dagger}_{b}c_{b}-c^{\dagger}_{a}c_{a}\right)$,
where $c_{b(a)}=(c_{e}\pm c_{o})/\sqrt{2}$ with one fermion
$c^{\dagger}_{e}c_{e}+c^{\dagger}_{o}c_{o}=c^{\dagger}_{b}c_{b}+c^{\dagger}_{a}c_{a}=1$,
or equivalently by a one-spin model, $H_{0}=t_{0}\sigma^{x}$, where
$\sigma^{x}$ is the Pauli matrix with eigenvalues $\pm 1$. Its time evolution
operator is described (with $\hbar$=1) by
$e^{-itH_{0}}=e^{-itt_{0}\sigma^{x}}$. At $t=\pi/(2\mid t_{0}\mid)$, the
operator $e^{\mp i(\pi/2)\sigma^{x}}=\mp i\sigma^{x}$ gives a complete charge
transfer (spin flip). At $t=\pi/(\mid t_{0}\mid)$, the operator $e^{\mp
i\pi\sigma^{x}}=-1$ completes a cycle of the charge oscillation. These
operators are reminiscent of local unitary operators discussed for a discrete
time crystal[39, 40] in many-body-localized driven systems[41, 42]. Here, we
do not use a continuous wave but a pulse; thus, we do not need a many-body-
localized system to avoid thermalization, which is similar to the situation in
Refs. yonemitsu_jpsj17b and poletti_pra11. Of course, the pulse-induced charge
oscillation decays with time and the Fourier spectra show a broad peak; thus,
only the short-time behavior is approximately described by this two-site one-
fermion model.

##  5 Photoinduced Charge Oscillations in Two Dimensions

Now, we return to the two-dimensional model for $\kappa$-(BEDT-TTF)2X in Eq.
(2) at three-quarter filling and take the time span of $T<t<10T$ again for
Fourier transforms. The Fourier spectra thus obtained are shown for different
strengths of intersite repulsive interactions and polarizations of
photoexcitation in Fig. 7.

Figure 7: (Color online) Absolute values of Fourier transforms of time
profiles ($T<t<10T$) of charge density at any site in Fig. 1(b) with different
$F$ and optical conductivity spectrum in ground state for (a), (b)
$V_{b1}$=$V_{b2}$=$V_{p}$=$V_{q}$=0.0 with polarizations along (a) $a$\- and
(b) $c$-axes, and (c), (d) $V_{b1}$=0.40, $V_{b2}$=0.24, $V_{p}$=0.28, and
$V_{q}$=0.24 with polarizations along (c) $a$\- and (d) $c$-axes. In all
cases, $U$=0.8.

In the case without intersite repulsion ($V_{ij}$=0.0), the Fourier spectra
are shown in Fig. 7(a) [Fig. 7(b)] for polarization along the $a$\-
($c$-)axis. For small $F$ ($F$=0.01), their weights are mainly distributed in
the energy range $\omega<0.8$, similarly to the corresponding conductivity
spectra. As $F$ increases, the weights first increase as a whole ($F$=0.10),
but the low-energy part ($\omega<0.8$) is then decreased, while the high-
energy part is further increased ($0.10<F<0.16$). The peak energy for large
$F$ ($F$=0.16) appears around $\omega$=1.2 for polarization along the $a$-axis
[Fig. 7(a)], which is slightly lower than $\omega_{\mbox{osc}}=2(\mid
t_{b1}\mid+\mid t_{b2}\mid+2\mid t_{p}\mid)$, amounting to 1.27 here, and
around $\omega$=1.0 for polarization along the $c$-axis [Fig. 7(b)], which is
close to $\omega_{\mbox{osc}}=2(\mid t_{b1}\mid+\mid t_{b2}\mid+2\mid
t_{q}\mid)$, amounting to 1.00 here. The peak energy for large $F$ generally
well matches the equation for $\omega_{\mbox{osc}}$ above for each
polarization in cases with different values of $t_{ij}$ and $U$ as long as the
intersite interactions are absent, $V_{ij}$=0.0.

The above relation between $\omega_{\mbox{osc}}$ and the transfer integrals is
similar to that in the one-dimensional spinless fermion model. The strong-
field-induced charge oscillation is realized by simultaneous charge transfers
through the $t_{b1}$, $t_{b2}$, and two $t_{p}$ bonds for polarization along
the $a$-axis, as shown in Figs. 8(a) and 8(b), and those through the $t_{b1}$,
$t_{b2}$, and two $t_{q}$ bonds for polarization along the $c$-axis, as shown
in Figs. 8(c) and 8(d).

Figure 8: (Color online) Schematic view of charge oscillation between (a) and
(b) after photoexcitation along $a$-axis and that between (c) and (d) after
photoexcitation along $c$-axis.

Through these bonds, the charge-rich sites are connected to the charge-poor
sites. At a charge-poor site, holes are simultaneously transferred from all of
the four neighboring charge-rich sites to this site and then back to these
four sites. Thus, this oscillation looks like an electronic breathing mode or
a checkerboard pattern where black and white are time-periodically exchanged.

The fact that $\omega_{\mbox{osc}}$ is independent of the sign of $t_{p}$ or
that of $t_{q}$ is due to the particular structure shown in Fig. 1(b). If we
multiply the creation and annihilation operators for red dimers (sites 2, 3,
6, 7, 10, 11, 14, and 15) by $(-1)$, it corresponds to inverting the signs of
$t_{p}$ and $t_{q}$. In general, the equation for $\omega_{\mbox{osc}}$ is
sensitive to the sign of the transfer integral. For instance, if we add a
third-neighbor hopping term,
$t_{4}\sum_{j}\left(c^{\dagger}_{j}c_{j+3}+c^{\dagger}_{j+3}c_{j}\right)$, to
the spinless model of Eq. (1), $\omega_{\mbox{osc}}$ depends on the relative
signs of the transfer integrals and it does not always correspond to the
highest-energy peak of the optical conductivity spectrum in the noninteracting
case: $\omega_{\mbox{osc}}=2\mid t_{1}+t_{2}+2t_{4}\mid$ if $t_{1}t_{2}>0$ and
$\omega_{\mbox{osc}}=2\mid t_{1}-t_{2}\mid$ if $t_{1}t_{2}<0$ irrespective of
a second-neighbor hopping term, at least for $\mid t_{1}\mid,\mid
t_{2}\mid\geq 2\mid t_{4}\mid$.

The above behaviors for large $F$ are partly changed by intersite repulsive
interactions. Examples are shown in Fig. 7(c) [Fig. 7(d)] for polarization
along the $a$\- ($c$-)axis. The parameters used for Fig. 7(d) are the same as
those used for Fig. 2. For small $F$ ($F$=0.01), their weights are again
mainly distributed in the energy range $\omega<0.8$, similarly to the
corresponding conductivity spectra. As $F$ increases, the weights first
increase as a whole (with a maximum around $F$=0.10). These behaviors so far
are similar to those in the case without intersite repulsion. However, as $F$
further increases, the low-energy part is significantly decreased, and the
high-energy part is also decreased to some degree ($0.10<F<0.20$). As a
consequence, a very broad peak is formed in the spectra. For both
polarizations, the peak energies are significantly lower than
$\omega_{\mbox{osc}}$ in the case without intersite repulsion. The intersite
repulsive interactions (attractive interactions between charge-rich and
charge-poor sites) slow down this charge oscillation. Even in one dimension,
if we add a second-neighbor hopping term to the spinless model of Eq. (1) to
allow an exchange of fermions, the frequency of the strong-field-induced
charge oscillation is altered by interactions (not shown).

Thus, immediately after the application of a strong pulse of an oscillating
electric field, charge densities coherently oscillate with a finite lifetime.
This fact is consistent with the finding in Fig. 2(a), that is, the time-
averaged bond densities are synchronized with each other as functions of $F$.
During the charge oscillation, holes are transferred through different bonds
simultaneously with a common frequency; thus, the time-averaged bond densities
on different bonds behave similarly as functions of $F$. These simultaneous
charge transfers are necessary to suppress the rise in the entropy and to
realize a negative-temperature state in Fig. 2(b).

Now, we consider what is necessary for this synchronized behavior. Each of the
effective transfer integrals is renormalized by the zeroth-order Bessel
function with the bond-dependent argument proportional to the inner product of
$F$ and $\mbox{\boldmath$r$}_{ij}$. If the effective transfer integrals
governed the long-time behavior, the synchronized behavior would not be
obtained. The picture based on the effective transfer integrals should
basically be applied to noninteracting and weakly interacting systems. Then,
in the case without intersite repulsion and with different values of $U$, we
show Fourier spectra for large $F$ ($F$=0.16) and polarization along the
$c$-axis [as used in Fig. 7(b)] in Fig. 9(a).

Figure 9: (Color online) (a) Fourier spectra similar to Fig. 7 but for
$F$=0.16 and different $U$, and (b),(c) time-averaged bond densities similar
to Fig. 2(a) but for (b) $U$=0.2 and (c) $U$=0.4. The parameters used here are
$V_{b1}$=$V_{b2}$=$V_{p}$=$V_{q}$=0.0, and the polarization is along the
$c$-axis.

In general, charge oscillations are undamped in the noninteracting case; thus,
their lifetimes are long for weakly interacting systems. That is why, as $U$
decreases, the overall weights in the Fourier spectra increase. Roughly
speaking, these spectra consist of a low-energy part ($\omega<0.8$)
corresponding to the conductivity spectrum and a high-energy peak around
$\omega$=1.0 due to the strong-field-induced charge oscillation. As $U$
decreases, the suppression of the low-energy part becomes weak; thus, the
strong-field-induced charge oscillation becomes relatively weaker. This
implies that the tendency for simultaneous charge transfers becomes weak as
$U$ decreases.

Then, we show the time-averaged bond densities
$\sum_{\sigma}\langle\langle\left(c^{\dagger}_{i\sigma}c_{j\sigma}+c^{\dagger}_{j\sigma}c_{i\sigma}\right)\rangle\rangle$
in Figs. 9(b) and 9(c) for the small-$U$ cases of Fig. 9(a). Indeed, for small
$U$ ($U$=0.2), the time-averaged bond densities are not synchronized as
functions of $F$ [Fig. 9(b)]. As $F$ increases, the effective transfer
integral for the $b_{2}$ bond first vanishes, then that for the $p$ bond
vanishes, and so on. Thus, the behaviors of the time-averaged bond densities
are not simply described by the corresponding effective transfer integrals.
For larger $U$, the time-averaged bond densities are almost [Fig. 9(c) for
$U$=0.4] or fully [as in Fig. 2(a) for $U$=0.6 (not shown)] synchronized as
functions of $F$. Thus, interactions are found to be essential for the
simultaneous charge transfers: the interactions cause damping of charge
oscillations and synchronization of the charge transfers at the same time,
which result in the strong damping of low-frequency charge oscillations and
the relatively weak damping of the high-frequency charge oscillation. This
fact is reminiscent of the situation in a discrete time crystal, where
interactions are essential for collective synchronization in strongly
disordered systems.[40]

The emergence of the strong-field-induced charge oscillation is basically
limited to the polarizations along the $a$\- and $c$-axes, which guarantee
that there are only two groups of sites according to their charge densities
and that the Fourier spectra are common to all sites. When the polarization
deviates from these axes, the charge densities at the four sites in the unit
cell become nonequivalent after photoexcitation; thus, the time profiles
depend on the site and their Fourier spectra become different. Even for large
$F$, substantially large weights remain at low energies.

Thus, the presence of only two nonequivalent sites with respect to charge
density is important. Dimerized structures are favorable in this sense.
Indeed, similar results are obtained in a one-dimensional spin-1/2 model at
three-quarter filling with a similar degree of dimerization, where the
frequency of the strong-field-induced charge oscillation is given by
$\omega_{\mbox{osc}}=2(\mid t_{1}\mid+\mid t_{2}\mid)$ for different values of
$U$ and $V$ in the uniform-charge-density phase.

##  6 Conclusions and Discussion

Instead of the long-time behavior often explained by concepts such as
dynamical localization and modified effective interactions, we pay attention
to a short-time behavior that is nonlinear with respect to the field amplitude
$F$ and varies on a time scale of the period of the external field. This study
is motivated by a recent experiment suggesting the importance of short-time
behavior.[25] Because of the dimerized structures and specifically polarized
fields considered in this study, all sites are classified into two groups
according to their charge densities. Charge densities take a common value
within a group after photoexcitation. Numerical results are presented for a
one-dimensional spinless-fermion model at half filling and a two-dimensional
model for $\kappa$-(BEDT-TTF)2X at three-quarter filling, but the main
conclusions are not limited to these models as long as all sites are
classified into two groups and the ground state is in the uniform-charge-
density phase.

For small $F$, Fourier spectra for charge-density time profiles after
photoexcitation have peaks at energies where the corresponding conductivity
spectra have peaks. For large $F$, the spectral distribution is changed and
has a peak at a single energy on the high-energy side. For the models we use
in this paper and without intersite repulsion in two dimensions, the peak
energy is given by twice the sum of the absolute values of the transfer
integrals between a site and all neighboring sites with different charge
density. In two dimensions, this peak energy is lowered by intersite
repulsion. However, we can construct a model where the long-range hopping
increases, decreases, or maintains the peak energy depending on the relative
signs of transfer integrals. This field-induced charge oscillation appears
only when $F$ is large; thus, it is a nonlinear phenomenon that emerges when a
strong pulse of an oscillating electric field is applied to dimerized systems
and charge densities are shaken coherently. This strong-field-induced charge
oscillation is considered to be closely related to the newly observed
reflectivity peak in photoexcited $\kappa$-(BEDT-TTF)2Cu[N(CN)2]Br on the
high-energy side of the main reflectivity spectrum.[25]

The coherence associated with the strong-field-induced charge oscillation is
responsible for the behaviors of the time-averaged bond densities, which
decrease in magnitude, simultaneously vanish at a particular value of $F$, and
invert their signs as $F$ increases. After the sign inversion, a negative-
temperature state is realized, which implies that the rise in the entropy is
suppressed. This suppression is enabled by coherently shaking charge
densities. Note that a negative-temperature state is more generally realized
even without dimerization.[21]

For continuous waves, thermalization is suppressed in a many-body-localized
system, which is necessary to realize a discrete time crystal.[39, 40] The
time evolution operators for the two-site one-fermion model that is referred
to for the discussion of the frequency of the strong-field-induced charge
oscillation in Sect. 4 are similar to local unitary operators discussed for a
discrete time crystal. This similarity may be helpful when considering the
possibility of emergent charge oscillations in different situations.

The author is grateful to S. Iwai and Y. Tanaka for various discussions. This
work was supported by Grants-in-Aid for Scientific Research (C) (Grant No.
16K05459) and Scientific Research (A) (Grant No. 15H02100) from the Ministry
of Education, Culture, Sports, Science and Technology of Japan.

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#  APPROXIMATE ORTHOGONALITY, BOURGAIN’S PINNED DISTANCE THEOREM AND
EXPONENTIAL FRAMES

Alex Iosevich and Azita Mayeli  Department of Mathematics, University of
Rochester, Rochester, NY<EMAIL_ADDRESS>City University of New
York, NY<EMAIL_ADDRESS>
###### Abstract.

Let $A$ be a countable and discrete subset of ${\mathbb{R}}^{d}$, $d\geq 2$,
of positive upper Beurling density. Let $K$ denote a bounded symmetric convex
set with a smooth boundary and everywhere non-vanishing Gaussian curvature. It
is known that ${\mathcal{E}}(A)=\\{e^{2\pi ix\cdot a}\\}_{a\in A}$ cannot
serve as an orthogonal basis for $L^{2}(K)$ [12]. In this paper, we prove that
even approximate average orthogonality is an obstacle to the existence of an
exponential frame in the following sense. Let $A$ be as above and $\phi\geq 0$
be a continuous monotonically non-increasing function on $[0,\infty)$ such
that the approximate orthogonality condition holds

${\left(\frac{1}{2^{j}}\int_{2^{j}}^{2^{j+1}}\phi^{p}(t)dt\right)}^{\frac{1}{p}}\leq
c_{j}2^{-j\frac{d+1}{2}}\quad\text{and}\quad\
|\widehat{\chi}_{K}(a-a^{\prime})|\leq\phi(\rho^{*}(a-a^{\prime}))\ \ \forall
a\not=a,a,a^{\prime}\in A,$

where $\rho^{*}$ is the Minkowski functional on $K^{*}$, the dual body of $K$.
Then, if

$\limsup_{j\to\infty}c_{j}=0,$

then the upper density of $A$ is equal to $0$, hence
${\mathcal{E}}(A)={\\{e^{2\pi ix\cdot a}\\}}_{a\in A}$ is not a frame for
$L^{2}(K)$. The case $p=\infty$ was previously established by the authors of
this paper in [9]. The point is that if ${\mathcal{E}}(A)$ is a frame for
$L^{2}(K)$, then very few pairs of distinct exponentials $e^{2\pi ix\cdot a}$,
$e^{2\pi ix\cdot a^{\prime}}$ from ${\mathcal{E}}(A)$ come anywhere near being
orthogonal. Our proof uses a generalization of Bourgain’s result on pinned
distances determined by sets of positive Lebesgue upper density in
${\mathbb{R}}^{d}$, $d\geq 2$. We also improve the $L^{\infty}$ version of
this result originally established in [9]. By using an extension of the
combinatorial idea from [13], we prove that under the $L^{\infty}$ hypothesis,
$A$ is finite if $d\not=1\mod 4$. If $d=1$ mod $4$, $A$ may be infinite, but
if it is, then it must be a subset of a line.

###### Key words and phrases:

Approximate orthogonality, exponential systems, distances in sets of positive
density, maximal theorems

The research of the first listed author was supported in part by the National
Science Foundation under grant no. HDR TRIPODS - 1934962 and by the NSF DMS -
2154232

##  1\. Introduction

The primary goal of this paper is to continue the investigation of the
structure of discrete sets $A$ such that ${\\{e^{2\pi ix\cdot a}\\}}_{a\in A}$
is a frame for $L^{2}(K)$, where $K$ is a bounded symmetric convex body with a
smooth boundary and non-vanishing curvature. Recall that such a collection of
exponentials is a frame if there exist $0<C_{L}\leq C_{U}<\infty$ so that for
every $f\in L^{2}(K)$,

(1.1) $C_{L}|K|\cdot{||f||}^{2}_{L^{2}(K)}\leq\sum_{a\in
A}{\left|\widehat{f}(a)\right|}^{2}\leq C_{U}|K|\cdot{||f||}^{2}_{L^{2}(K)}.$

Let us recall some definitions.

######  Definition 1.1.

Let $S$ be a separated subset of ${\mathbb{R}}^{d}$, in the sense that there
exists $\delta>0$ such that $|s-s^{\prime}|\geq\delta>0$ for all
$s,s^{\prime}\in S$, $s\not=s^{\prime}$. The upper density of $S$, is given by

$\overline{dens}(S)=\limsup_{R\to\infty}\frac{\\#\\{S\cap{[-R,R]}^{d}\\}}{2^{d}R^{d}}.$

Similarly, the lower density of $S$ is given by

$\underline{dens}(S)=\liminf_{R\to\infty}\frac{\\#\\{S\cap{[-R,R]}^{d}\\}}{2^{d}R^{d}}.$

If $\overline{dens}(S)=\underline{dens}(S)$, then we say that the Beurling
density $dens(S)$ exists and is equal to this quantity.

It is not difficult to construct a frame for $L^{2}(K)$ by enclosing $K$ in a
rectangular box and restricting the orthonormal exponential basis for this box
to $K$. In this case, the frame is tight, i.e., in the definition of the frame
in (1.1) we have $C_{L}=C_{U}$. One unfortunate feature of such a frame is
that its density is much higher than the volume of $K$.

Also recall that ${\mathcal{E}}(A)={\\{e^{2\pi ix\cdot a}\\}}_{a\in A}$ is a
Riesz basis for $L^{2}(K)$ if it is equivalent (via an invertible and bounded
map) to an orthonormal basis (not necessarily exponential) for $L^{2}(K)$. It
is an old open question of whether $L^{2}(B_{d})$, $d\geq 2$, possesses a
Riesz basis of exponentials, where $B_{d}$ denotes the unit ball.

It is well-known that if ${\mathcal{E}}(A)$ is a Riesz basis for $L^{2}(K)$,
then $dens(A)=|K|$. In hopes of shedding some light on the question of a
possible Riesz basis for $L^{2}(K)$, where $K$ is bounded, symmetric, convex
and has a smooth boundary with non-vanishing curvature, we prove a result that
puts a significant restriction on the structure of a set $A$ that generates a
frame for $L^{2}(K)$.

A well-known stationary phase formula due to Herz [7] (see also [5] and Lemma
3.1 below for a precise formulation) says that

(1.2)
$\widehat{\chi}_{K}(\xi)=\kappa^{-\frac{1}{2}}\left(\frac{\xi}{|\xi|}\right)\sin\left(2\pi\left(\rho^{*}(\xi)-\frac{d-1}{8}\right)\right){|\xi|}^{-\frac{d+1}{2}}$

plus an error term, where $\rho^{*}$ is the Minkowski functional on $K^{*}$,
the dual body of $K$. We are going to show in Theorem 2.1 below that if
${\mathcal{E}}(A)$ is a frame, $|\widehat{\chi}_{K}(a-a^{\prime})|$, with
$a\not=a^{\prime}$, $a,a^{\prime}\in A$, cannot be much smaller than
${|a-a^{\prime}|}^{-\frac{d+1}{2}}$ very often. In other words, the difference
set $A-A$ must stay away from the zeroes of $\widehat{\chi}_{K}$.

It was shown in [12] (see also the case of the ball in [11]) that with $K$ as
above, $L^{2}(K)$ does not possess an orthogonal basis of exponentials.
Orthogonality implies that $(A-A)\backslash\\{\vec{0}\\}$ is strictly
contained in the zero set of $\widehat{\chi}_{K}$. What we prove in this paper
is that $A-A$ cannot be close to the zeroes of $\widehat{\chi}_{K}$ very
often, even on average, in order for ${\mathcal{E}}(A)$ to be a frame for
$L^{2}(K)$. More precisely, the main results of the current paper are the
following.

####  1.0.1. Our Contributions

######  Theorem 1.2.

Let $A$ be a countable and discrete subset of ${\mathbb{R}}^{d}$, $d\geq 2$,
and let $p\in[1,\infty]$. Let $K$ denote a bounded symmetric convex set with a
smooth boundary and everywhere non-vanishing Gaussian curvature. Let $\phi\geq
0$ be a continuous monotonically non-increasing function on $[0,\infty)$ such
that

(1.3)
${\left(\frac{1}{2^{j}}\int_{2^{j}}^{2^{j+1}}\phi^{p}(t)dt\right)}^{\frac{1}{p}}\leq
c_{j}\left(2^{-j\frac{d+1}{2}}\right)\ \ \ \forall j\in\mathbb{Z},$

for some constants $c_{j}\geq 0$.

Further suppose that

(1.4) $|\widehat{\chi}_{K}(a-a^{\prime})|\leq\phi(\rho^{*}(a-a^{\prime}))$

for all $a\not=a^{\prime}$, $a,a^{\prime}\in A$.

Then if

$\limsup_{j\to\infty}c_{j}=0,$

the upper density of $A$ is equal to $0$, and, consequently,
${\mathcal{E}}(A)$ is not a frame for $L^{2}(K)$.

######  Remark 1.3.

Note that if ${\mathcal{E}}(A)$ is an orthogonal system, then
$\widehat{\chi}_{K}(a-a^{\prime})=0$ if $a\not=a^{\prime}$. Therefore, Theorem
1.2 is a generalization of the fact that $L^{2}(K)$ does not possess any
orthogonal basis of exponentials if $K$ is a symmetric convex body with a
smooth boundary and non-vanishing Gaussian curvature [12].

######  Remark 1.4.

An immediate consequence of Theorem 1.2 is that if ${\mathcal{E}}(A)$ is a
frame for $L^{2}(K)$, with $K$ as above, then for any subset
$A^{\prime}\subset A$ on which (1.3) and (1.4) hold, with
$\limsup_{j\to\infty}c_{j}=0$, $A^{\prime}$ has zero positive density. In
other words, these properties can hold only on a subset of zero density. It
would be interesting to determine if a stronger statement is true, namely that
if ${\mathcal{E}}(A)$ is a frame for $L^{2}(K)$, and the above hypotheses hold
on $A^{\prime}\subset A$, then for some $0<\alpha<d$,

$\limsup_{R\to\infty}\frac{\\#\\{A^{\prime}\cap B_{R}\\}}{R^{d-\alpha}}=0$

where $B_{R}$ denote the ball of radius $R$. For example, if $A$ were
contained in a line, we could take any $\alpha<d-1$. We partially address this
question in Theorem 1.7.

######  Remark 1.5.

It is proved in ([15]; see also [8]) that if $\Omega$ is a bounded domain of
positive Lebesgue measure in ${\mathbb{R}}^{d}$, $d\geq 2$, and ${\\{e^{2\pi
ix\cdot a}\\}}_{a\in A}$ is a frame for $L^{2}(\Omega)$ with upper and lower
frame constants $C_{U},C_{L}$, respectively, then every $d$-dimensional cube
of side-length $\ell$ with

$\ell>{\left(\frac{C_{U}{|\partial\Omega|}_{\alpha}}{C_{L}|\Omega|}\right)}^{\frac{1}{d-\alpha}}$

contains at least one point of $A$, where $0<\alpha<d$ and
${|\partial\Omega|}_{\alpha}$ denotes the $\alpha$-dimensional Minkowski
content of $\partial\Omega$, the boundary of $\Omega$. In particular, if
$\Omega$ is convex, as it is assumed in Theorem 1.2, every cube of side-length
$\ell$ with

$\ell>\frac{C_{U}|\partial\Omega|}{C_{L}|\Omega|}$

contains at least one point of $A$, where $|\partial\Omega|$ denotes the
surface measure of $\partial\Omega$. This can be viewed as a quantitative
version of the fact (due to Beurling ([1])), which states that if $\Omega$ has
positive Lebesgue measure and ${\mathcal{E}}(A)$ is frame for $L^{2}(\Omega)$,
then $A$ has positive upper density.

######  Remark 1.6.

The case $p=\infty$, $K=B_{d}$, the unit ball in ${\mathbb{R}}^{d}$, is
essentially contained in the prequel to this paper ([9]). More precisely, the
authors showed that if $\phi:[0,\infty)\to[0,\infty)$ is any bounded and
positive measurable function such that

(1.5) $\lim_{t\to\infty}{(1+t)}^{\frac{d+1}{2}}\phi(t)=0,\ \
\text{and}\quad|\widehat{\chi}_{B_{d}}(a-a^{\prime})|\leq\phi(|a-a^{\prime}|)\quad\forall\
a,a^{\prime}\in A,\ a\neq a^{\prime},$

then $A$ has zero upper density. Making a change of variables
$s=\frac{t}{2^{j}}$ in (1.3), it turns the inequality (1.3) into

${\left(\int_{1}^{2}\phi^{p}(2^{j}s)ds\right)}^{\frac{1}{p}}\leq
c_{j}2^{-j\frac{d+1}{2}}.$

The $L^{\infty}$ version of this statement is

$\sup_{s\in[1,2]}2^{j\frac{d+1}{2}}\phi(2^{j}s)\leq c_{j},$

so in the case $p=\infty$, (1.3) is a slightly weaker assumption than the
(1.5), except that the monotonicity assumption is not needed in the
$L^{\infty}$ setting. However, as the reader is about to see, we can
strengthen the $L^{\infty}$ result considerably by using different methods.

Our second result is a significant improvement of a previous theorem by the
authors of this paper described in Remark 1.6 above.

######  Theorem 1.7.

Suppose that if $\phi:[0,\infty)\to[0,\infty)$ is any bounded positive
measurable function such that (1.5) holds for all $a,a^{\prime}\in A$,
$a\not=a^{\prime}$. Then $A$ is contained in a line. Moreover, if $d\not=1\mod
4$, then $A$ is finite.

######  Remark 1.8.

As the reader will see, the proof of Theorem 1.7 is based on the techniques
developed in [13]. The re-examantion of these techniques in the context of
Theorem 1.7 allowed the authors of this paper to realize that the approximate
version of the Erdős Integer Distance Principle proved in [13] holds under
considerably weaker hypotheses.

##  2\. Outline of the proofs and the structure of the paper

In this section we outline the proof of Theorems 1.2 and 1.7.

###  2.1. Outline of the proof of Theorem 1.2

By the method of stationary phase (see Lemma 3.1 below), with $K$ as above,

$|\widehat{\chi}_{K}(a-a^{\prime})|\leq
C{(1+|a-a^{\prime}|)}^{-\frac{d+1}{2}}$

for some constant $C>0$. It follows that (2.1) always holds with $c=C$. The
point is that if it holds with a small enough constant, we can invoke the
asymptotic formula (3.2) (see below) and argue that

$\sin\left(2\pi\left(\rho^{*}(a-a^{\prime})+\frac{d-1}{8}\right)\right)$

is small quite often for $a\not=a^{\prime}$, $a,a^{\prime}\in A$. This will be
accomplished by reducing the proof of Theorem 1.2 to the following statement.

######  Theorem 2.1.

Let $A$ be a discrete and countable subset of ${\mathbb{R}}^{d}$, $d\geq 2$.
Let $K$ denote a bounded symmetric convex set with a smooth boundary and
everywhere non-vanishing Gaussian curvature. Let ${\mathcal{A}}_{j}$ denote
annuli of inner radius $2^{j}$, and outer radius $2^{j+1}$, centered at
$a^{\prime}\in A$. Suppose that

(2.1) ${\left(\frac{1}{2^{dj}}\sum_{a\in
A\cap{\mathcal{A}}_{j}}{|\widehat{\chi}_{K}(a-a^{\prime})|}^{p}\right)}^{\frac{1}{p}}\leq
c_{j}2^{-j\frac{d+1}{2}}$

with $c_{j}$ independent of $a^{\prime}$. Then if

(2.2) $\limsup_{j\to\infty}c_{j}=0,$

then the upper density of $A$ is equal to $0$, hence ${\mathcal{E}}(A)$ is not
a frame for $L^{2}(K)$.

By pigeonholing, we will be able to conclude that there exists a point $x_{0}$
living in a small neighborhood of an $a_{0}\in A$, such that pinned
$\rho^{*}$-distances from $x_{0}$ to the small neighborhood of a positive
proportion of $A$ is clustered around $\frac{k}{2}+\frac{d-1}{8}$, where $k$
ranges over the integers. This violates a variant of the result due to
Bourgain which says that in sets of positive upper Lebesgue density, a pinned
distance set with respect to almost every point contains every sufficiently
large positive real number. In order to invoke Bourgain’s result we will
further have to refine the previously pigeon-holed subset of $A$.

###  2.2. Outline of the proof of Theorem 1.7

The authors of [13] proved that if ${\mathcal{E}}(A)$ is an infinite
orthogonal set in $L^{2}(K)$, where $K\subset\mathbb{R}^{d}$ is a symmetric
convex body with a smooth boundary and everywhere non-vanishing curvature,
then $A$ is contained in a line, thus, ${\mathcal{E}}(A)$ is incomplete in
$L^{2}(K)$. Moreover, they showed that if $d\not=1\mod 4$, then $A$ must be
finite. We are going to see that the proof of this result still goes through
under the weaker assumption given by (1.5) above.

###  2.3. Structure of this paper

This paper is organized as follows. In the Section 3 below, we prove Theorem
2.1. At the end of that section, we comment on the modification of the pinned
distance set result due to Bourgain that is needed for our proof. In Section 4
we deduce Theorem 1.2 from Theorem 2.1.

##  3\. Proof of Theorem 2.1

Suppose that (2.1) and (2.2) hold and $A$ has finite upper density
$\alpha_{0}>0$. By assumption, for a fixed $a^{\prime}\in A$

(3.1) $\sum_{a\in
A\cap{\mathcal{A}}_{j}}{|\widehat{\chi}_{K}(a-a^{\prime})|}^{p}\leq
c^{p}_{j}2^{j(d-p\frac{d+1}{2})}.$

We shall need the following well-known stationary phase result.

######  Lemma 3.1.

([7]) Let $K$ be a bounded symmetric convex body with a smooth boundary and
everywhere non-vanishing Gaussian curvature. Let

$\rho^{*}(\xi)=\sup_{x\in K}x\cdot\xi.$

Given $\omega\in S^{d-1}$, let $\kappa(\omega)$ denote the Gaussian curvature
of $\partial K$ at the (unique) point where the unit normal is $\omega$. Then

(3.2)
$\widehat{\chi}_{K}(\xi)=\kappa^{-\frac{1}{2}}\left(\frac{\xi}{|\xi|}\right)\sin\left(2\pi\left(\rho^{*}(\xi)-\frac{d-1}{8}\right)\right){|\xi|}^{-\frac{d+1}{2}}+{\mathcal{D}}_{K}(\xi),$

where

$|{\mathcal{D}}_{K}(\xi)|\leq C_{K}{|\xi|}^{-\frac{d+3}{2}}.$

Plugging this into (3.1) yields

(3.3) $\sum_{a\in
A\cap{\mathcal{A}}_{j}}\kappa^{-\frac{p}{2}}\left(\frac{a-a^{\prime}}{|a-a^{\prime}|}\right){\left|\sin\left(2\pi\left(\rho^{*}(a-a^{\prime})-\frac{d-1}{8}\right)\right)\right|}^{p}{|a-a^{\prime}|}^{-p\frac{d+1}{2}}\leq
c^{p}_{j}C_{1}2^{j(d-p\frac{d+1}{2})}$

for some fixed $C_{1}>0$, and it follows that

$\sum_{a\in
A\cap{\mathcal{A}}_{j}}{\left|\sin\left(2\pi\left(\rho^{*}(a-a^{\prime})-\frac{d-1}{8}\right)\right)\right|}^{p}\leq
C_{2}c^{p}_{j}2^{dj},$

for some fixed $C_{2}>0$ that depends on the maximum value of the curvature
function on $\partial K$. Note that the error term ${\mathcal{D}}_{K}$ from
(3.2), which is $O(2^{-j\frac{d+3}{2}})$ when $a\in{\mathcal{A}}_{j}$, can be
absorbed into the constant $C_{1}$ on the right hand side of (3.3) if $j$ is
sufficiently large.

We write the left hand side in the form $I+II$, where $I$ is the sum over

$\left\\{a\in
A\cap{\mathcal{A}}_{j}:{\left|\sin\left(2\pi\left(\rho^{*}(a-a^{\prime})-\frac{d-1}{8}\right)\right)\right|}^{p}\leq\delta\right\\}$

and $II$ is the sum over the complement of this set, with $0<\delta<<1$ to be
determined later. We have

$II>\delta\cdot\\#\left\\{a\in
A\cap{\mathcal{A}}_{j}:{\left|\sin\left(2\pi\left(\rho^{*}(a-a^{\prime})-\frac{d-1}{8}\right)\right)\right|}^{p}>\delta\right\\},$

hence

(3.4) $\displaystyle\\#\left\\{a\in
A\cap{\mathcal{A}}_{j}:{\left|\sin\left(2\pi\left(\rho^{*}(a-a^{\prime})-\frac{d-1}{8}\right)\right)\right|}^{p}>\delta\right\\}<C_{3}c^{p}_{j}\delta^{-1}2^{dj},$

for some fixed $C_{3}>0$.

Suppose that $j\geq j_{0}$, to be determined later. If $\sup_{j\geq
j_{0}}c^{p}_{j}$ is sufficiently small, choosing $\delta=\frac{1}{1000}$ we
have that the right hand side of (3.4) is
$<(1-r)\\#\\{A\cap{\mathcal{A}}_{j}\\}$, where $r\in(0,1)$, close to $1$, to
be chosen later. This can be done since $A$ has positive upper density and

$\limsup_{j\to\infty}\frac{\\#\\{A\cap{\mathcal{A}}_{j}\\}}{|{\mathcal{A}}_{1}|2^{dj}}=\alpha_{0}>0.$

Note that the upper density is not affected by taking only the annuli
${\mathcal{A}}_{j}$ with $j\geq j_{0}$ for any finite $j_{0}$. More precisely,
the upper density of $A$ with $\cup_{j\leq j_{0}}\\{{\mathcal{A}}_{j}\cap
A\\}$ removed is the same as the upper density of $A$,

It follows that if we throw these points of $A$ which belong to the set in
(3.4) out, we will have an $r$ fraction of the points of
$A\cap{\mathcal{A}}_{j}$ left. Doing this for every $j\geq j_{0}$, with
$j_{0}$ to be determined, and taking the union, we obtain a set
$A_{good,j_{0}}(a^{\prime})$ of positive upper density, such that

(3.5)
${\left|\sin\left(2\pi\left(\rho^{*}(a-a^{\prime})-\frac{d-1}{8}\right)\right)\right|}^{p}\leq\delta\
\text{for all}\ a\in A_{good,j_{0}}(a^{\prime}).$

It follows from (3.5) that if $E$ denote the $\delta$-neighborhood of
$A_{good,j_{0}}(a^{\prime})$, with $\delta$ as above, then there does not
exist $L_{0}$ such that for every $L>L_{0}$, $\rho^{*}(x-a^{\prime})=L$. If
$a^{\prime}$ is replaced by any point in the ball of radius $\delta$ centered
at $a^{\prime}$, the conclusion remains the same. We shall use a
generalization of the following result due to Bourgain.

######  Theorem 3.2.

Let $E$ be a set of positive upper Lebesgue density in ${\mathbb{R}}^{2}$.
Then for almost every $x_{0}\in E$, there exists $L_{0}=L_{0}(x_{0})>0$ such
that for all $L>L_{0}$, there exists $x\in E$ with $|x-x_{0}|=L$.

As we shall sketch below in Section 5, Bourgain result can be generalized as
follows. Let $E$ be a set of positive upper Lebesgue density in
${\mathbb{R}}^{d}$, $d\geq 2$. Let $K$ and $\rho^{*}$ be as above. Then for
almost every $x_{0}\in E$, there exists $L_{0}>0$ such that for all $L>L_{0}$,
there exists $x\in E$ with $\rho^{*}(x-x_{0})=L$.

This result was established by Bourgain in [2] (see a related result in [4]).
However, as we shall indicate below, the generalization follows from
Bourgain’s method with only minor modifications. The proof only relies on the
$L^{p}$ estimate for the circular maximal function, which Bourgain proved in
1986 ([3]). However, the same estimate holds if the circle is replaced with
any smooth curve with non-vanishing curvature. Since

$\\{x\in{\mathbb{R}}^{d}:\rho^{*}(x)=1\\}$

is a smooth surface, with $\rho^{*}$ as above, the two-dimensional argument
follows. In higher dimensions, the argument is much easier as the needed
maximal estimate was proved by Stein ([19]) in the case of the sphere, and by
Greenleaf ([6]) for general compact surfaces with non-vanishing Gaussian
curvature. We shall sketch the proof of this in Section 5 below.

In order to use Bourgain’s result, we need to make sure that if $E$ is as in
the previous paragraph, then one of the points in the $\frac{1}{1000}$
neighborhood of $a^{\prime}$ can serve as a pin in Theorem 3.2. The proof of
Bourgain’s theorem yields a slightly stronger results than the conclusion of
Theorem 3.2, namely that there exists $x_{0}\in E$ such that for every subset
of $E$ of relative density $r$ sufficiently close to $1$, there exists
$L_{0}>0$ such that for all $L>L_{0}$, there exists $x\in E$ with
$\rho^{*}(x-x_{0})=L$. In view of this, we complete the proof by finding such
an $x_{0}$ in the $\delta$-neighborhood of $A$ and choosing $a^{\prime}$ to be
the point in $A$ in the $\delta$-neighborhood of which $x_{0}$ resides.

##  4\. Proof of Theorem 1.2

Let ${\mathcal{A}}_{j}$ be as above. By the assumption of Theorem 1.2, we have

$\sum_{a\in
A\cap{\mathcal{A}}_{j}}{|\widehat{\chi}_{K}(a-a^{\prime})|}^{p}\leq\sum_{a\in
A\cap{\mathcal{A}}_{j}}\phi^{p}(\rho^{*}(a-a^{\prime})).$

Since $\phi$ is monotonic, the right expression above is bounded by

$C\int_{c_{1}2^{j}\leq\rho^{*}(u-a^{\prime})\leq
c_{2}2^{j+1}}\phi^{p}(\rho^{*}(u-a^{\prime}))du=C\int_{c_{1}2^{j}\leq\rho^{*}(u)\leq
c_{2}2^{j+1}}\phi^{p}(\rho^{*}(u))du,$

or

$C\int_{c_{1}2^{j}\leq\rho^{*}(r\omega)\leq
c_{2}2^{j+1}}\phi^{p}(\rho^{*}(r\omega))r^{d-1}d\omega
dr=C\int_{c_{1}2^{j}\leq r\rho^{*}(\omega)\leq
c_{2}2^{j+1}}\phi^{p}(r\rho^{*}(\omega))r^{d-1}d\omega dr,$

where $C$ is an independent constant.

Let $t=r\rho^{*}(\omega)$. We obtain

$C\int_{c_{1}2^{j}\leq t\leq
c_{2}2^{j+1}}\int_{S^{d-1}}\phi^{p}(t)t^{d-1}\frac{d\omega}{{(\rho^{*}(\omega))}^{d}}dt$
(4.1) $=C\int_{c_{1}2^{j}\leq t\leq
c_{2}2^{j+1}}\int_{S}\phi^{p}(t)t^{d-1}dt\frac{d\sigma_{\rho^{*}}(\eta)}{|\nabla\rho^{*}(\eta)|},$

where $d\sigma_{\rho^{*}}$ is the surface measure on

$S=\\{x:\rho^{*}(x)=1\\},$

the $\rho^{*}$-unit sphere.

The expression (4.1) follows from the classical co-area formula. See, for
example, [14], for a similar calculation.

Since $\rho^{*}(\eta)=\eta\cdot\nabla\rho^{*}(\eta),$ it follows that
$|\nabla\rho^{*}(\eta)|\geq C>0$, when $\eta$ is on the $\rho^{*}$-unit
sphere, for some different $C>0$. It follows that (4.1) is bounded by

$\displaystyle(\ref{almost})\leq C^{\prime}2^{j(d+1)}\cdot
2^{-j}\int_{c_{1}2^{j}}^{c_{2}2^{j+1}}\phi^{p}(t)t^{d-1}dt\leq
C^{\prime\prime}c_{j}2^{dj}\cdot 2^{-j\frac{d+1}{2}p},$

where the first inequality above follows from the fact that all norms on
finite dimensional normed spaces are comparable.

The conclusion follows after dividing both sides by $2^{dj}$ and taking $p$’th
root. This recovers Theorem 1.2 from Theorem 2.1.

##  5\. Sketch of proof of a slight strengthening of Theorem 3.2

In this section we sketch the proof of the following result, which essentially
follows from Bourgain’s argument in [2].

######  Definition 5.1.

We say that $E\subset{\mathbb{R}}^{d}$ has positive upper Lebesgue density if

$\overline{dens}_{{\mathcal{L}}}(E)=\limsup_{R\to\infty}\frac{|E\cap
B_{R}|}{|B_{R}|}>0,$

where $B_{R}$ denote the ball of radius $R$.

######  Definition 5.2.

Suppose that $E\subset{\mathbb{R}}^{d}$ with
$\overline{dens}_{{\mathcal{L}}}(E)>0$. We say that $E^{\prime}$ is a
refinement of $E$ of relative density $r$ if $E^{\prime}\subset E$ and

$\overline{dens}_{{\mathcal{L}}}(E^{\prime})\geq
r\cdot\overline{dens}_{{\mathcal{L}}}(E).$

######  Theorem 5.3.

Let $E\subset{\mathbb{R}}^{d}$ of positive upper Lebesgue density
$\overline{dens}_{{\mathcal{L}}}(E)$. Then there exists a threshold $L_{0}>0$,
a threshold $r\in(0,1)$, and $x_{0}\in E$, such that for any refinement
$E^{\prime}$ of $E$ of relative density $r$, and any $L>L_{0}$, there exists
$x^{\prime}\in E^{\prime}$ such that $\rho^{*}(x_{0}-x^{\prime})=L$.

Bourgain’s proof of the case $d=2$, $\rho^{*}(x)=|x|$, has two key
ingredients. The first is the following maximal theorem which follows from
Bourgain’s proof of the circular maximal theorem.

######  Theorem 5.4.

Let $\sigma$ be the arc-length measure on the circle $S^{1}$, and let
$\sigma_{s}$ be defined by the relation
$\widehat{\sigma}_{s}(\xi)=\widehat{\sigma}(s\xi)$. Let $P_{t}$ denote the
Poisson semi-group kernel on ${\mathbb{R}}^{2}$, so
$\widehat{P}_{t}(\xi)=e^{-t|\xi|}$. For $p>2$, there exist constants
$C(p)<\infty$ and $\alpha(p)>0$, such that

${\left|\left|\sup_{s\geq
t_{0}}|(f-(f*P_{t})*\sigma_{s})|\right|\right|}_{p}\leq
C(p){\left(\frac{t}{t_{0}}\right)}^{\alpha(p)}{||f||}_{p}.$

The fact that the same conclusion is achieved if $S^{1}$ is replaced by any
symmetric smooth convex curve with non-vanishing curvature follows from [18].
In higher dimensions, the following result follows from [6].

######  Theorem 5.5.

Let $\rho$ be the Minkowski functional on the convex symmetric body $K$ with a
smooth boundary and everywhere non-vanishing curvature. Let $\rho^{*}$ be as
above. Let $S_{\rho}=\\{x\in{\mathbb{R}}^{d}:\rho^{*}(x)=1\\}$, and let
$\sigma_{\rho}$ denotes the surface measure. Let $\sigma_{\rho,s}$ denote the
dilated surface measure, as above. For $d>2$, there exist $C<\infty$, such
that

${\left|\left|\sup_{s\geq
t_{0}}|(f-(f*P_{t})*\sigma_{\rho,s})|\right|\right|}_{2}\leq
C{\left(\frac{t}{t_{0}}\right)}^{\frac{d-2}{2}}{||f||}_{2}.$

The second ingredient in Bourgain’s proof is the observation that if the
conclusion of Theorem 3.2 is false in the case $d=2,\rho^{*}(\xi)=|\xi|$, then
there exists $U\subset{[0,1]}^{2}$, $|U|>\epsilon$, and a sequence of positive
numbers

$s_{1}>t_{1}>s_{2}>t_{2}>\dots>s_{J}>t_{J},$

$J$ arbitrarily large, satisfying

$s_{j+1}<\frac{1}{2}t_{j},$

and $x\in U\cap[s_{j},1-s_{j}]$, $s_{j}\in(0,1/2)$, implies that
$\sup_{s_{j}<t<s_{j+1}}|(1_{{[0,1]}^{2}}-f)*\sigma_{t}|=1$. The same reduction
can be performed in dimensions $d\geq 3$ with the Euclidean metric replaced by
$\rho^{*}$. The contradiction is then obtained using Theorem 5.4. In
dimensions three and higher, the contradiction is obtained in exactly the same
way using Theorem 5.5.

Moreover, the same argument yields the slightly stronger conclusion, claimed
in Theorem 5.3, that there exists a threshold $L_{0}>0$, a threshold
$r\in(0,1)$, and $x_{0}\in E$, such that for any refinement $E^{\prime}$ of
$E$ of relative density $r$, and any $L>L_{0}$, there exists $x^{\prime}\in
E^{\prime}$ such that $\rho^{*}(x_{0}-x^{\prime})=L$. This is because assuming
that this statement is false leads to exactly the same statement as above and
the contradiction is obtained in the same way.

##  6\. Proof of Theorem 1.7

Suppose that ${\mathcal{E}}(A)$ is an orthogonal collection in $L^{2}(K)$ in
the sense that

$\int_{K}e^{2\pi ix\cdot(a-a^{\prime})}dx=0\ \text{for all}\
a\not=a^{\prime}\in A,$

where $K$ is as in Theorem 1.7.

The first listed author and Rudnev proved in [13] that if $A$ is infinite, $A$
is contained in a line. Moreover, they showed that if $d\not=1\mod 4$, then
$A$ must be finite.

The proof relies on two estimates that follow from the orthogonality
assumption for ${\mathcal{E}}(A)$. The first says that if ${\mathcal{E}}(A)$
is an orthogonal set in $L^{2}(K)$, then for all $a,a^{\prime}\in A$,

(6.1)
$\rho^{*}(a-a^{\prime})=\frac{k}{2}+\frac{d-1}{8}+O({|a-a^{\prime}|}^{-1})\
\text{for some integer}\ k\ \text{as}\ |a-a^{\prime}|\to\infty.$

The second estimate says that if $a_{0},a_{1}\in A$, $a_{0}\not=a_{1}$, then
for all $a\in A$,

(6.2) $\rho^{*}(a_{0}-a)-\rho^{*}(a_{1}-a)=\frac{k}{2}+O({|a|}^{-2})\
\text{for some integer}\ k\ \text{as}\ |a|\to\infty.$

The first estimate follows directly from Lemma 3.1. The second estimate uses
more refined Bessel expansions (see Lemma 1.4 in [13]). Using these estimates,
the authors established their claim.

Under the hypotheses of Theorem 1.7, the same arguments yield the following
form of (6.1) and (6.2):

(6.3)
$\rho^{*}(a-a^{\prime})=\frac{k}{2}+\frac{d-1}{8}+O\left(\frac{\phi(|a-a^{\prime}|)}{{|a-a^{\prime}|}^{\frac{d+1}{2}}}\right)\
\text{for some integer}\ k\ \text{as}\ |a-a^{\prime}|\to\infty,$

where (1.5) holds, and (6.2) becomes

(6.4)
$\rho^{*}(a_{0}-a)-\rho^{*}(a_{1}-a)=\frac{k}{2}+O\left(\frac{\phi(|a|)}{{|a|}^{\frac{d+1}{2}}}\cdot{|a-a^{\prime}|}^{-1}\right)\
\text{for some integer}\ k\ \text{as}\ |a|\to\infty.$

Observe that

$O\left(\frac{\phi(|a-a^{\prime}|)}{{|a-a^{\prime}|}^{\frac{d+1}{2}}}\right)=o(1),\
\text{as}\ |a-a^{\prime}|\to\infty,$

by assumption. Similarly,

$O\left(\frac{\phi(|a|)}{{|a|}^{\frac{d+1}{2}}}\cdot{|a-a^{\prime}|}^{-1}\right)=o({|a-a|}^{-1})\
\text{as}\ |a-a^{\prime}|\to\infty.$

The geometric observation used to establish the conclusion of the main result
in [13], namely Lemma 1.5, does not require anything stronger than (6.3) and
(6.4) above. These are all that are needed to establish Lemma 1.5 and
Proposition 1.6 in that paper. This results in the following version of the
Erdős Integer Distance Principle.

######  Theorem 6.1.

(Approximate version of the Erdős Integer Distance Principle) Let $\rho^{*}$
be as above, corresponding to a symmetric convex body $K$ with a smooth
boundary and non-vanishing Gaussian curvature. Let $A\subset{\mathbb{R}}^{d}$,
$d\geq 2$, and suppose that for $a,a_{0},a_{1}\in A$,

(6.5) $\rho^{*}(a-a_{0})-\rho^{*}(a-a_{1})=o({|a|}^{-1}),$

and there exist constants $c_{1},c_{2}$ such that for $a,a^{\prime}\in A$,

(6.6) $\rho^{*}(a-a^{\prime})=c_{1}k+c_{2}+o(1).$

Then $A$ is a subset of a line. If $d\not=1\mod 4$, then $A$ is finite.

##  7\. Remarks and open problems

The purpose of this section is to describe some open problems that may be
accessible using the methods of this paper and related ideas.

  * i)

The authors of this paper previously proved ([10]) that if
$g(x)=\chi_{B_{d}}(x)$, where $B_{d}$ is the unit ball in dimensions two and
higher, and $d\not=1\mod 4$, then there does not exist
$S\subset{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$, such that ${\\{g(x-a)e^{2\pi
ix\cdot b}\\}}_{(a,b)\in S}$ is an orthogonal basis for
$L^{2}({\mathbb{R}}^{d})$. The case when $d=1\mod 4$ is work in progress. A
reasonable question to ask is whether the same conclusion can be established
under the assumption of approximate orthogonality.

  * ii)

In view of [12], it should be possible to show that the conclusion of Theorem
1.2 still holds if we assume that $\partial K$ has one point, in the
neighborhood of which it is smooth and has non-vanishing Gaussian curvature.

  * iii)

It is reasonable to conjecture that if $K$ is a convex body in
${\mathbb{R}}^{d}$, $d\geq 2$, then there exists $S\subset{\mathbb{R}}^{2d}$
such that ${\\{\chi_{K}(x-a)e^{2\pi ix\cdot b}\\}}_{(a,b)\in S}$ is an
orthogonal basis for $L^{2}(\mathbb{R}^{d})$ if and only if $K$ tiles
${\mathbb{R}}^{d}$ by translation, i.e iff $K$ is a symmetric polytope of the
type classified by Venkov [20] and McMullen [17]. It may be possible to
approach this problem by combining the techniques of this paper and those of
[16] who recently proved that if $K\subset{\mathbb{R}}^{d}$, $d\geq 2$, is
convex, then $L^{2}(K)$ has an orthogonal basis of exponentials if and only if
$K$ tiles ${\mathbb{R}}^{d}$ by translation. The two dimensional case was
previously established in [12].

  * iv)

The contrast between the proof of Theorem 1.2 and Theorem 1.7 raises the
question of whether there is a strengthening of the approximate version of the
Erdős Integer Distance Principle, proved in [13], where the point-wise (near)
integer distance condition is replaced by an average version. For example, is
it true that if $A$ is an infinite separated subset of ${\mathbb{R}}^{d}$,
$d\geq 2$, such that there exist constants $c_{1},c_{2}$ and $p\in[1,\infty)$
so that

$|a-a^{\prime}|=c_{1}k+c_{2}+O({\mathcal{D}}(|a-a^{\prime}|)),$

with

$\limsup_{j\to\infty}\frac{1}{2^{j}}\int_{2^{j}}^{2^{j+1}}{|{\mathcal{D}}(t)|}^{p}dt=0,$

then $A$ is a subset of a line?

If anything like this is true, it should lead to a considerable strengthening
of Theorem 1.2. Heuristically, the idea (described in two dimensions for the
sake of simplicity) is the following. Fix $a_{0},a_{1}$, $a_{0}\not=a_{1}$.
Then $90$ percent of the points in $A$ are nearly $c_{1}k+c_{2}$ distance away
from $a_{0}$, and the same is true with respect to $a_{1}$. It follows that
$81$ percent of the points lie on finitely many hyperbolas with focal points
at $a_{0}$ and $a_{1}$. Now choose $a_{2}$ that does not lie on a line that
passes through $a_{0}$ and $a_{1}$ and play the same game with respect to
$a_{1},a_{2}$ and the points that are nearly shifted scaled integers with
respect to $a_{2}$. The hyperbolas with focal points at $a_{0},a_{1}$
intersect the hyperbolas with focal points at $a_{1},a_{2}$ finitely many
times (depending on the distance between those pairs of points) and shows that
the resulting subset $A^{\prime}\subset A$ (having at least $72.9$ percent of
the points) is a subset of a line. We now replay the same game, using the same
points $a_{0},a_{1},a_{2}$ with respect to the complement of $A^{\prime}$.
This approach can probably be used to show that $A=A_{good}+A_{bad}$, where
$A_{good}$ is a subset of a line, and $A_{bad}$ is arbitrarily sparse. This
may lead to an alternate proof of Theorem 1.2. Whether one can conclude that
$A_{bad}$ is empty is an interesting question that should be investigated.

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