# Magnetic phase diagram of a 2-dimensional triangular lattice antiferromagnet
Na2BaMn(PO4)2

Jaewook Kim1, Kyoo Kim1, Eunsang Choi2, Young Joon Ko3, Dong Woo Lee1, Sang Ho
Lim1, Jong Hoon Jung3, and Seungsu Lee1 1 Korea Atomic Energy Research
Institute, Daejeon 34057, Republic of Korea  2 National High Magnetic Field
Laboratory, Florida State University, Tallahassee, FL 32310-3706, U.S.A.  3
Department of Physics, Inha University, Incheon 22212, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract

We report the magnetic phase transitions of a spin-5/2, 2-dimensional
triangular lattice antiferromagnet (AFM) Na2BaMn(PO4)2. From specific heat
measurements, we observe two magnetic transitions at temperatures 1.15 K and
1.30 K at zero magnetic field. Detailed AC magnetic susceptibility
measurements reveal multiple phases including the
$\uparrow$$\uparrow$$\downarrow$ (up-up-down)-phase between 1.9 T and 2.9 T at
47 mK when magnetic field is applied along the $c$ axis, implying that
Na2BaMn(PO4)2 is a classical 2$d$ TL Heisenberg AFM with easy-axis anisotropy.
However, it deviates from an ideal model as evidenced by a hump region with
hysteresis between the $\uparrow$$\uparrow$$\downarrow$ and $V$-phases and
weak phase transitions. Our work provides another experimental example to
study frustrated magnetism in 2$d$ TL AFM which also serves as a reference to
understand the possible quantum spin liquid behavior and anomalous phase
diagrams observed in sibling systems Na2Ba$M$(PO4)2 ($M$ = Co, Ni).

  * August 2022

Keywords: Na2BaMn(PO4)2, Magnetic phase diagram, Triangular lattice
antiferromagnet

##  1 Introduction

The 2-dimensional triangular lattice (2$d$ TL) antiferromagnetic (AFM) system
has served as a cornerstone in the field of geometrical frustrated magnetism
[1, 2, 3]. Pioneering works on Ising AFM system showed that it maintains a
disordered state with sizable residual entropy down to zero temperature ($T$)
[4, 5]. The idea of “resonating valence bond (RVB)” has been proposed on
Heisenberg system with spin $S$ = 1/2 by Anderson [6], where magnetic long-
range-order (LRO) is suppressed by quantum fluctuations. Although later works
have shown that its ground state is similar to its classical-spin counterpart,
an ordered state with a coplanar 120 ∘ phase [7, 8, 9], the RVB model has
evolved to the concept of “quantum spin liquid” (QSL) [10, 11, 12, 13], which
led to researches on geometrically frustrated systems such as pyrochlore and
kagome lattice systems [14, 15, 16].

The evolution of the ordered phase of 2$d$ TL Heisenberg AFM under temperature
($T$) and magnetic field ($H$) has been extensively studied theoretically.
Under application of magnetic field, the coplanar 120 ∘ phase, which is the
$H$ = 0 ground state of both quantum ($S$ = 1/2) and classical ($S$
$\rightarrow$ $\infty$) spin systems, evolves to the $Y$,
$\uparrow$$\uparrow$$\downarrow$ (up-up-down), and $V$-phases (see Fig. 1)
upon increasing the magnetic field, stabilized by quantum and thermal
fluctuation, respectively [17, 18, 19, 20]. Considering the classical
Heisenberg spin case, at $T$ = 0, the $\uparrow$$\uparrow$$\downarrow$-phase
is stable only at the 1/3 of saturation magnetic field but expands to a finite
range of magnetic field at non-zero $T$ and exhibits the highest transition
temperature among other phases. At zero magnetic field, earlier studies
suggested that the system would undergo a single phase transition from
paramagnetic (PM) to the coplanar 120 ∘ phase [18]. However, recent numerical
works have shown that there are actually two transitions when magnetic field
is sufficiently close to zero; from high $T$ PM phase to
$\uparrow$$\uparrow$$\downarrow$ and then to $Y$-phase upon cooling [20, 19],
each transitions corresponding to different symmetry breakings.

Figure 1:  Schematic representations of the (a) 120 ∘, (b) $Y$, (c) up-up-down
($\uparrow$$\uparrow$$\downarrow$), and (d) $V$-phases.

Surprisingly, there are only a few systems that realize the classical 2$d$ TL
Heisenberg AFM model discussed above, despite the fact that its simple
structural motif is realized in many materials [21, 22, 23, 24, 25]. Moreover,
systems with magnetically easy-axis anisotropy are scarcer compared to easy-
plane ones: as far as we know, only GdPd2Al3 [21] and Rb4Mn(MoO4)3 [23] fall
into this category. In practice, various perturbations, such as anisotropic
[26], next-nearest-neighbor (NNN) [27, 28], and nearest-neighbor (NN)
interlayer [29] exchange interactions intervene and modify the ideal/simple
ground state.

Recently, Na2Ba$M$(PO4)2 system (NB$M$PO, $M$ = Co, Ni) was reported to show
frustrated magnetism within the 2$d$ TL structure. NBCoPO was proposed as a
candidate for QSL since it consists of Co2+ ions with effective spin-1/2.
First experimental work demonstrated that NBCoPO is magnetically disordered
down to 0.05 K with a broad continuum in inelastic neutron scattering,
probably due to spinon excitations [30]. However, another study showed that an
AFM phase transition occurs at the Néel temperature ($T_{N}$) of 0.148 K in
$H$ = 0 in the same compound [31]. Although this observation excludes the QSL
as its ground state, thermal conductivity ($\kappa$) above $T_{N}$ follows a
$T$-linear behavior and exhibits a non-zero value of $\kappa$/$T$ at the $T$ =
0 intercept, implying that there are itinerant thermal carriers, also possibly
due to spinons [31]. Another system, NBNiPO, which has a slightly distorted
structure but maintains the 2$d$ TL structure, displays a phase transition at
0.43 K in $H$ = 0 [32, 33]. Both systems mentioned above show successive phase
transitions under applied magnetic field, including the
$\uparrow$$\uparrow$$\downarrow$-phase [31, 33].

Here, we report the physical properties and magnetic phase diagram of
Na2BaMn(PO4)2 with spin-5/2 of Mn2+ ions, realizing the classical 2$d$ TL
Heisenberg AFM in the NB$M$PO series. Low temperature specific heat ($C_{p}$)
measurement shows two phase transitions at 1.15 and 1.30 K under $H$ = 0.
Entropy analysis suggests that almost half of the measured entropy is released
above the ordering temperature, suggesting strong magnetic frustration
persists in the PM phase. Combining the $C_{p}$ and AC magnetic susceptibility
data, we map the complete $H$ – $T$ phase diagram of this material. When
magnetic field is applied along the $c$ axis, we find three magnetic field-
induced phases ($Y$, $\uparrow$$\uparrow$$\downarrow$, and $V$) that are in
good agreement with those expected for a classical 2$d$ TL Heisenberg AFM
model with easy-axis anisotropy. Surprisingly, we find an unusually wide
hysteresis hump of $\Delta$$H$ $\sim$ 0.7 T between the
$\uparrow$$\uparrow$$\downarrow$ and $V$-phase. While both phase boundaries of
the $\uparrow$$\uparrow$$\downarrow$-phase in the model 2$d$ TL Heisenberg AFM
is known to be a continuous transition of the Berezinskii-Kosterlitz-Thouless
(BKT) type, observation of hysteresis in NBMnPO suggests that the system
involves additional exchange interactions. Our work adds a rare example in
classical 2$d$ TL Heisenberg AFM systems with easy-axis anisotropy which can
be fully mapped in a typical laboratory magnetic field scale and serves as a
reference to understand quantum phenomena and complex phase diagram that are
observed in other NB$M$PO systems.

##  2 Methods

Single crystals of NB$M$PO ($M$ = Mg, Mn, Co, and Ni) were grown by high
temperature flux method similar to a previous report [30]. Typical dimension
of the samples is $\sim$ 1.0 – 3.0 mm in edge length and $\sim$ 0.3 mm in
thickness. NBMnPO single crystals are transparent in color, as shown in Fig.
2(c), and are planar in shape with the planes perpendicular to the $c$ axis.
Single crystal X-ray diffraction (XRD) data were collected by using graphite-
monochromated Mo K$\alpha$ radiation (PHOTON II X-ray diffractometer, Bruker)
and were refined with SHELXL-13 [34]. DC magnetic susceptibility ($\chi$) and
magnetization ($M$) were measured using a superconducting quantum
interferometer (Magnetic Properties Measurement System, Quantum design).
Specific heat ($C_{p}$) was measured by using a relaxation method in a
cryostat equipped with a superconducting magnet (Dynacool-14, Quantum Design).
A dilution refrigerator (DR) was used to access low temperature down to 0.08
K. AC magnetic susceptibility measurements were performed by using a homemade
susceptometer with a voltage controlled current source (CS580, Stanford
Research) and lock-in amplifier (SR830, Stanford Research). The phase of the
lock-in amplifier is set to measure the first harmonic signal. We used the
SCM1 magnet equipped with a DR in National High Magnetic Field Laboratory
(Tallahassee, FL, U.S.A.).

Figure 2:  Crystal structure of Na2BaMn(PO4)2 viewed along (a) [001] and (b)
[110] direction. Na, Ba, Mn, P, and O ions are shown in yellow, green, blue,
violet, and red spheres, respectively. Mn and P ions are enclosed in oxygen
octahedra and tetrahedra, respectively. (c) Photograph of a single crystal on
1 $\times$ 1 mm2 scale grid. (d) Calculated structure factor squared ($F^{2}$)
versus measured $F^{2}$ according to the single crystal XRD fitting. Dashed
line is a guide to the eye.

##  3 Results

NBMnPO was first reported as a naturally occurring mineral with space group
$P\bar{3}$ [35]. A subsequent study, using a chemically pure powder form,
showed that its actual space group is $P\bar{3}m1$ [36]. Our single crystal
XRD refinement result (Fig. 2(d)) is consistent with space group $P\bar{3}m1$
with lattice parameters $a$ = 5.3761(9) Å and $c$ = 7.0999(12) Å(see
Supplemental Material [37] for best fitting results and structural parameters)
and previous reports including other isostructural NB$M$PO systems ($M$ = Co
and Mg) [39, 30]. As shown in Fig. 2(a,b), Mn2+ ions are enclosed in perfect
$M$O6 octahedra connected by distorted PO4 tetrahedra, forming equilateral
triangular layers. Mn2+ ion is a half-filled 3$d^{5}$ electron system with
high spin-5/2 and zero angular momentum, representing a Heisenberg spin. These
layers are stacked along the $c$ axis, separated by Ba ions. The dominant
interaction between these spins are presumably the NN intralayer superexchange
interaction through Mn–O–P–O–Mn.

Next, we discuss the magnetic properties of NBMnPO single crystal.
Temperature-dependence of DC magnetic susceptibility data were taken while
warming under $\mu_{0}$$H$ = 0.2 T after zero field cooling. $\chi$($T$)
curves show monotonic increase upon cooling from 320 to 1.8 K without any
signature of LRO (Fig. 3(a)). The system is more-or-less magnetically
isotropic down to 1.8 K with anisotropy ratio ($\equiv$
$\chi_{\parallel}$c/$\chi_{\parallel}$ab) of 0.95 at 2 K. We applied the
Curie-Weiss (CW) model at 20 K $\leq$ $T$ $\leq$ 100 K range and found that
the CW temperature ($\Theta_{CW}$) and effective magnetic moment
($\mu_{eff.}$) values of -7.20 K (-7.61 K) and 5.61 $\mu_{B}$/Mn (5.60
$\mu_{B}$/Mn) for $H$ $\parallel$ $c$ ($H$ $\parallel$ $ab$), respectively.
The $\mu_{eff.}$ values are slightly smaller than that expected for spin-only
Mn2+ (high-spin, 5.92 $\mu_{B}$). Magnetization measurements at 2 K also show
isotropic behavior for both directions (Fig. 3(b)). The slope changes above
$\sim$ 6 T, but does not saturate up to the maximum magnetic field measured.

Figure 3:  Magnetic properties of Na2BaMn(PO4)2 single crystal. (a)
Temperature-dependences of magnetic susceptibility measured along two
crystallographic directions. (b) Magnetic field-dependences of magnetization
at 2 K. Red circle and blue square symbols represent the data measured along
$ab$ plane and $c$ axis, respectively. Inset in (a) shows the inverse of
magnetic susceptibility as a function of temperature. Dashed line is a guide
to the eye.

The frustration ratio ($\equiv$ $|\Theta_{CW}|$/$T_{N}$) of NBMnPO is 5.85,
similar to other classical 2$d$ TL Heisenberg AFM systems, $cf.$ Rb4Mn(MoO4)3
(7.14) [23], RbFe(MoO4)2 (5.79) [22], and NBNiPO (4.3) [33], signaling a
moderate magnetic frustration. Assuming the mean-field theory, the NN
intralayer exchange interaction $J$ of the Heisenberg Hamiltonian
$\mathcal{H}$ = $\sum_{i,j}J\vec{S_{i}}\cdot\vec{S_{j}}$ can be estimated from
the equation $\Theta_{CW}$ = $-zJS(S+1)/3k_{B}$, where $z$ is the number of
nearest-neighbors. For NBMnPO, we find $J$ $\sim$ 0.43 K which is comparable
to that of NBNiPO (0.48 K) [33].

To further study the phase transition at lower $T$, we measured the specific
heat down to 0.08 K. Fig. 4(a) shows the $C_{p}$($T$) curves measured for
NB$M$PO single crystals ($M$ = Mg, Mn, Co, and Ni). NBMgPO is measured as a
non-magnetic reference for lattice contribution of specific heat. All three
systems with magnetic ions show similar behaviors: upturns (i.e., local
minimum) in $C_{p}$($T$) upon cooling below 6–8 K and peak features at low $T$
indicating magnetic LRO, consistent with previous reports [32, 31, 33]. Among
them, NBMnPO shows the highest upturn temperature at 7.7 K.

Figure 4:  (a) Specific heat of different Na2Ba$M$(PO4)2 ($M$ = Mg, Mn, Co,
and Ni) single crystals as a function of temperature in log-log scale measured
at zero magnetic field. Specific heat of Na2BaMn(PO4)2 as a function of
temperature below 2 K under different magnetic fields applied along the (b)
$c$ axis and (c) $ab$ plane. All curves in (b) ((c)), except for $H$ = 0, are
sequentially shifted by 6 J/mol K (5.5 J/mol K) for clarity. (d) Change of
magnetic entropy ($\Delta$$S_{mag.}$) as a function of temperature in semi-log
scale. Dashed line in (d) indicates the $R$ln6 value as explained in text.
Open squares for $\mu_{0}$$H$ = 3, 4 T below 0.5 K, are extrapolated data that
follow a power law fit below $T_{N}$, for comparison with the 0 T curve.

A closer look at the $C_{p}$($T$) data below 2 K reveals two phase transitions
in NBMnPO (Fig. 4(b)) with a strong magnetic field-dependence. At $H$ = 0, two
sharp peaks appear at 1.30 K ($T_{N1}$) and 1.15 K ($T_{N2}$) indicative of
magnetic phase transitions . Under $H$ $\parallel$ $c$, the peak at $T_{N2}$
reduces concomitantly in size and shifts to lower $T$ and disappears at
$\mu_{0}$$H$ $>$ 2 T, while that of $T_{N1}$ initially shifts to low $T$ at
0.5 T then gradually moves to higher $T$, becomes sharpest at 2 T, and finally
shifts to lower $T$ and reduces in size. Another small upturn is observed
below 0.23 K in $H$ = 0 curve, which originates from nuclear Schottky anomaly
due to large nuclear spin of Mn ($I$ = 5/2). When magnetic field is applied
along the $ab$ plane (Fig. 4(c)), $T_{N1}$ decreases slightly at 1 T, stays
nearly constant in $T$ up to 3 T and gradually shifts to lower $T$ above 4 T,
whereas that of $T_{N2}$ shifts to 1.0 K at 1 T and shows a weakly smeared
behavior at the same temperature at 2 T.

The magnetic specific heat $C_{mag.}$ of NBMnPO is obtained by subtracting the
$C_{p}$ data of NBMgPO as phonon contribution and nuclear Schottky
contribution ($\propto$ $T^{-2}$) at very low temperature from that of NBMnPO.
Then, $C_{mag.}$/$T$ was integrated as a function of $T$ from 0.15 K to obtain
the magnetic entropy ($\Delta$$S_{mag.}$). Fig. 4(d) displays the
$\Delta$$S_{mag.}$($T$) in several magnetic fields under $H$ $\parallel$ $c$.
For comparison, data for 3 and 4 T between 0.5 K and 0.15 K were extrapolated
(shown as open symbols) assuming a power-law dependence of $C_{p}$($T$) curve
below $T_{N}$. At $H$ = 0, $\Delta$$S_{mag.}$ saturates to 12.0 J/mol K at 30
K, which is 80.5 % of $R$ln(2$S$ \+ 1) = 14.90 J/mol K value that is expected
for Mn2+ spins with $S$ = 5/2. The $\Delta$$S_{mag.}$($T$) curves increase
abruptly at two points: at $T_{N1}$ and at $\sim$ 20 K. A large fraction of
observed entropy (51.2 % of $\Delta$$S_{mag.}$($T$ = 30 K)) is released in
$T_{N1}$ $<$ $T$ $<$ 30 K range suggesting that spin fluctuation persists up
to temperature that is roughly 20 times larger than $T_{N1}$.

Figure 5:  AC susceptibility and integrated magnetization of Na2BaMn(PO4)2.
Temperature dependence of $\chi^{\prime}$ measured with fixed magnetic field
along the (a) $c$ axis and (b) $ab$ plane. (c) Magnetic-field dependence of
$\chi^{\prime}$ measured with increasing magnetic field applied along the $c$
axis at constant temperatures. Inset in (b) shows the AC susceptibility curve
at 0.047 K measured up to 18 T. (d) Enlarged graph of (b), showing hysteresis
regions at different temperatures. (e) Magnetic-field dependence curves
measured with increasing magnetic field applied along the [100] direction at
constant temperatures. (f) Magnetization along two directions at 0.076 K
obtained by integrating the AC susceptibility curves as described in text.
$\blacklozenge$, $\bullet$, *, and $\lozenge$ symbols indicate relatively weak
transition features described in text. Curves shown in (a–e) are shifted along
the $y$ axis for clarity. Temperatures for each curves in (c) and (e) are,
from bottom to top, 0.047, 0.076, 0.109, 0.137, 0.185, 0.251, 0.342, 0.370,
0.480, 0.651, 0.751, 0.952, and 1.44 K, respectively. Frequencies of AC
magnetic field for the $c$ axis and $ab$ plane directions are 471 Hz and 717
Hz, respectively.

To further track the phase transitions, we measured the low temperature AC
magnetic susceptibility ($\chi^{\prime}$) in detail. Fig. 5(a) and (b) show
the $\chi^{\prime}$ vs. $T$ curves with both DC and AC magnetic field applied
along the $c$ axis and [100] direction, respectively. Both zero field curves
show increase in $\chi^{\prime}$ upon cooling, reaching a saturated behavior
at 1.28 K and exhibits a sharp drop at 1.23 K (marked with black arrows in
Fig. 5(b)), indicating the two phase transitions observed in $C_{p}$($T$).
Below 0.28 K, $\chi^{\prime}$ increases slightly which is more pronounced in
$H$ $\parallel$ $ab$ plane data. $\chi^{\prime}$($T$) data with applied
magnetic fields were measured below $\sim$ 0.9 K to preserve the 3He/4He
mixture at low temperature during the experiment. In the $H$ $\parallel$ $c$
arrangement (Fig. 5(a)), a peak appears at 0.82 K at 1 T which shifts to lower
$T$ at 1.5 T and in the $H$ $\parallel$ $ab$ arrangement (Fig. 5(b)), a small
peak appears at 0.60 K at 1.5 T.

Fig. 5(c) shows the $\chi^{\prime}$ versus $H$ curves with increasing the DC
magnetic field along the $c$ axis. At the lowest temperature, 0.047 K, we
observe a sharp peak at $\mu_{0}$$H_{c1}$ = 1.9 T followed by a large dip,
with its minimum at 2.3 T, which terminates as a broad maximum at
$\mu_{0}$$H_{c2}$ = 2.9 T. With further increase in magnetic field,
$\chi^{\prime}$ shows another peak at $\mu_{0}$$H_{c3}$ = 6.45 T and quickly
decreases and shows a linear behavior above $\mu_{0}$$H_{sat.}$ $\sim$ 7.4 T
(inset in Fig. 5(c)). By increasing the temperature, the dip feature becomes
broader and $H_{c1}$ shifts to lower magnetic field with broadening in the
peak shape, while the one at $H_{c2}$ becomes sharper without significant
change in peak position.

In Fig. 5(d), we show the enlarged view of $\chi^{\prime}$($H$) curves shown
in Fig. 5(c) with both up- and down-sweeps of magnetic field at selected
temperatures. There are three noticeable characteristics. First, the broad
maximum in $H_{c2}$ shows a strong hysteresis in magnetic field: in the up-
sweep it manifests as a broad hump between 2.7 and 3.4 T ($T$ = 0.047 K), but
it disappears in down-sweep and $\chi^{\prime}$($H$) curve changes smoothly
then enters the dip with a shoulder-like shape. At higher temperatures, the
hump observed in the up-sweep becomes narrow in $H$, and a peak develops above
0.251 K. Second, a small shoulder (marked with $\blacklozenge$ in Fig. 5(d))
is observed below the $\uparrow$$\uparrow$$\downarrow$-phase which occur at
0.6 T (0.5 T) at 0.076 K in the up-sweep (down-sweep). The position of this
anomaly slightly shifts to higher magnetic field upon increase in temperature.
Third, another small shoulder ($\bullet$ in Fig. 5(d)) is observed above the
$\uparrow$$\uparrow$$\downarrow$-phase, e.g., at 5.2 T (4.8 T) in the up-sweep
(down-sweep) data at 0.076 K. With increase in temperature, this feature
quickly shifts to lower magnetic-field.

On the other hand, when magnetic field is applied along the $ab$ plane,
$\chi^{\prime}$($H$) curves show a simpler behavior compared to that of the
$H$ $\parallel$ $c$ case (Fig. 5(e)). For example, at $T$ = 0.076 K, three
anomalies are observed; small peaks at $\mu_{0}$$H_{c4}$ = 1.7 T (* symbol),
$\mu_{0}$$H_{c5}$ = 2.4 T ($\lozenge$ symbol), and a sharp peak at
$\mu_{0}$$H_{c6}$ = 6.3 T. Measurements under different $T$ show that the
former one is almost independent in $T$ while the latter one shifts to lower
$H$ as $T$ approaches $T_{N}$.

Since the shape of the $\chi^{\prime}$($H$) curves are almost frequency-
independent between 147 Hz to 991 Hz (See Supplementary Information [37]), it
can be referred to as the differential magnetic susceptibility (d$M$/d$H$) in
the DC limit. Then, by integrating the $\chi^{\prime}$($H$) curves with
respect to $H$ and assuming that the magnetization saturates above $H_{sat.}$,
we obtain a normalized $M$/$M_{sat.}$($H$) curve at 0.076 K as shown in Fig.
5(f). Changes of slope in $M$/$M_{sat.}$($H$) is clearly visible when $H$ is
parallel to the $c$ axis, marking the 1/3 magnetization plateau with
$M$/$M_{sat.}$ $\sim$ 0.34 at 2.3 T, which behavior is expected for a
$\uparrow$$\uparrow$$\downarrow$-phase. This behavior also confirms that the
system belongs to the easy-axis magnetic anisotropy. On the contrary, when $H$
is applied along the $ab$ plane direction, magnetization increase linearly up
to saturation magnetic field.

We note that, although the broad hump feature at $H_{c2}$ is clearly visible
in the $\chi^{\prime}$($H$) curves (Fig. 5(d)), its contribution to
magnetization, i.e., the area of the hump between up- and down-sweep
$\chi^{\prime}$($H$) curves, is relatively small. For example, at 0.076 K, the
hump feature is $\sim$ 3.1 % of the $M_{sat.}$ and not apparent in the
integrated $M$($H$) curve.

Combining $C_{p}$($T$), $\chi^{\prime}$($T$), and $\chi^{\prime}$($H$) data,
we obtain full $H$ – $T$ phase diagrams of NBMnPO along two applied magnetic
field directions as shown in Fig. 6. When magnetic field is applied along the
$c$ axis (Fig. 6(a)), we find a series of ordered phases denoted as I, II,
$\uparrow$$\uparrow$$\downarrow$, III, and IV. The peaks at $T_{N2}$ in
$C_{p}$($T$) data smoothly connects to the peaks in $\chi^{\prime}$($T$) at 1
and 1.5 T and $\chi^{\prime}$($H$) at $H_{c1}$, constituting the lower field
phase boundary of the $\uparrow$$\uparrow$$\downarrow$-phase while the peaks
at $T_{N1}$ connects to the sharp peak features at $H_{c3}$, delineating the
LRO to PM phase boundary (solid lines). The expansion of
$\uparrow$$\uparrow$$\downarrow$-phase is due to the thermal fluctuations [18,
19, 20], which is also observed in classical spin systems with other structure
[38]. The higher field phase boundary of the
$\uparrow$$\uparrow$$\downarrow$-phase is marked with $H_{c2}$ and is nearly
linear in $H$ –$T$ phase diagram (solid line). The region of the hump in the
up-sweep of $\chi^{\prime}$($H$), colored in grey, is located adjacent to this
phase boundary on the high field side and terminates at the point where the
hysteresis disappears (magenta circles). The hysteresis feature persists up to
0.75 K.

When magnetic field is applied along the $ab$ plane (Fig. 6(b)), the phase
diagram consists of three phases: I′, II′, and II′′. The
$\uparrow$$\uparrow$$\downarrow$-phase is absent in this configuration,
demonstrating the easy-axis anisotropy of this system. The peaks at $T_{N1}$
in $C_{p}$($T$) curve shift to lower $T$ with increasing field, merging with
the peaks at $H_{c6}$ observed in $\chi^{\prime}$($H$). The peak at $T_{N2}$
decreases quickly in magnetic field and seem to merge with the peak in
$\chi^{\prime}$($T$) at 1.5 T and weak $\chi^{\prime}$ anomalies at $H_{c4}$.
Phase II′ lies in between the weak transition features observed at $H_{c4}$
and $H_{c5}$.

Figure 6:  $H$ – $T$ phase diagrams of Na2BaMn(PO4)2 with magnetic field
applied along the (a) $c$ axis and (b) $ab$ plane. Red squares indicate the
phase boundaries from peak in $C_{p}$($T$) data. Blue triangles and magenta
circles indicate anomalies observed in $\chi^{\prime}$($H$). Green squares
indicate anomalies observed in $\chi^{\prime}$($T$) data. Grey area in (a)
describes the hump region with hysteresis observed in AC susceptibility
measurement as described in text. Phase boundaries marked by solid lines are
determined from peak positions in $C_{p}$($T$) and $\chi^{\prime}$($H$) data.
Dashed lines are determined from weak anomalies in $\chi^{\prime}$($H$).

##  4 Discussion

Having assigned the $\uparrow$$\uparrow$$\downarrow$-phase from the 1/3
magnetization plateau, we can assume that the low- (phase I, II) and high-
field phases (phase III, IV) that appear in $H$ $\parallel$ $c$ configuration
are similar to the $Y$\- and $V$-phases, respectively, as expected for the
classical 2$d$ TL Heisenberg AFM system. The splitting of these phases occur
due to additional interactions, such as NN interlayer interactions [29]. While
the exact magnetic structure of these phases are not studied to date, we infer
them by taking analogy from other 2$d$ TL Heisenberg AFM systems with similar
features. We note that Rb4Mn(MoO4)3, a well-studied classical 2$d$ TL
Heisenberg AFM system with easy-axis anisotropy, does not show any sign of
phase transitions inside both the $Y$\- and $V$-phases. Rather, splitting of
the phases are observed in easy-plane anisotropy systems RbFe(MoO4)2 and
Ba3CoSb2O6 in which additional interactions, such as NNN intralayer
($J^{\prime}$) and/or NN interlayer ($J_{\perp}$) interactions are present. In
RbFe(MoO4)2, the low-field ground state is a 120 $\degree$ structure that
modulates with an incommensurate (IC) wave vector along the $c$ axis. It
evolves to a $Y$-phase with magnetic field applied along the easy-plane
direction through a first-order phase transition [40, 41]. RbFe(MoO4)2 shows
another first-order phase transition within the $V$-phase: a “fan”-phase
appears close to saturation magnetic field, which is a variation of the
$V$-phase with IC modulation between the neighboring layers [42, 41, 43]. The
easy-plane anisotropic system Ba3CoSb2O6 shows a first-order phase transition
within the $V$-phase at $H$ $\sim$ 0.7 $H_{sat.}$ that display a sharp peak in
d$M$/d$H$ curve that signals the onset of another phase ($V^{\prime}$) [44].

Thus, taking analogy with the RbFe(MoO4)2 case, we assume that phases I and II
in NBMnPO both form 120 $\degree$ structures within the layers but differ in
magnetic ordering pattern along the $c$ axis. This idea is partially supported
by the weak phase transitions between I′, II′, and II′′ in the $H$ $\parallel$
$ab$ configuration (Fig. 6(b)). If adjacent layers share the same 120
$\degree$ pattern at zero field, with in-plane magnetic fields, the system
would form an inverted $Y$ structure having one of the spins pointing towards
the magnetic field direction and evolve continuously to a fully polarized
state by canting of spins without any phase transitions. Phases III and IV can
be considered as $V$\- and its variant ($V^{\prime}$), respectively. Indeed
further studies, such as neutron diffraction, are necessary to exactly
understand the phases diagram of NBMnPO.

But the model including the NNN intralayer ($J^{\prime}$) and/or NN interlayer
($J_{\perp}$) interactions fails to capture the emergence of a wide hysteresis
region (colored region in Fig. 6(a)) in NBMnPO, which manifests as a hump in
the up-sweep in $\chi^{\prime}$($H$) curves, adjoining the
$\uparrow$$\uparrow$$\downarrow$-phase on the high field side, in $H$
$\parallel$ $c$ configuration.

A plausible scenario for the hump region can be deduced from numerical
simulation works by Seabra $et~{}al.$ [45, 46], where they studied the
classical 2$d$ TL AFM system by including additional terms $J^{\prime}$,
$J_{\perp}$, and single-ion anisotropy to explain the complex low-field phase
diagram, including a “spin-supersolid” phase [47], observed in a metallic 2$d$
TL AFM system AgNiO2 [48]. For a certain set of exchange interaction and
single-ion anisotropy parameters, they obtained a phase diagram that consists
of first-order phase transitions and phase coexistence regions adjacent to the
$\uparrow$$\uparrow$$\downarrow$-phase (for example, see Fig. 24 in [46]). But
this model relies on relatively strong next-nearest-neighbor and interlayer
interactions ($J^{\prime}$/$J$ = 0.15, $J_{\perp}$/$J$ = -0.15) and single-ion
anisotropy ($D$/$J$ = 0.5) that stabilizes a stripe phase at $H$ = 0 and a
half-magnetization plateau under applied magnetic field which is absent in
NBMnPO.

We note that the magnetic dipolar interaction in NBMnPO is non-negligible due
to large magnetic moment of Mn2+. By using the magnetic moment of Mn2+
obtained from CW analysis (5.60 $\mu_{B}$), the dipolar interaction energy in
NBMnPO between NN, interlayer NNN, and intralayer NNN spins are calculated as
0.13 K, 0.024 K, and 0.055 K, respectively. These values are comparable to $J$
$\sim$ 0.43 K, obtained from mean-field theory, and may play an important role
in understanding its unusual phase diagram.

Finally, we compare the phase diagrams of NB$M$PO series ($M$ = Mn, Co, and
Ni). All three systems exhibit easy-axis magnetic anisotropy with 120
$\degree$ phase as the zero field state and, by application of $H$ along $c$
axis, evolve to $\uparrow$$\uparrow$$\downarrow$-, $V$-, and
$V^{\prime}$-phases. The shapes of the phase diagrams are different in the
classical (NBMnPO) and quantum spin (NBCoPO) limit. In NBMnPO, phase
boundaries between II–$\uparrow$$\uparrow$$\downarrow$, III–IV, and IV–PM
display strong temperature dependences. Especially, the
$\uparrow$$\uparrow$$\downarrow$-phase region shrinks upon cooling, indicating
that it is a thermally stabilized state. On the other hand, phase boundaries
of NBCoPO are nearly constant in $T$ [31]. This difference clearly manifests
the role of thermal and quantum fluctuation in stabilizing the ordered phases
in 2$d$ TL AFM systems.

##  5 Conclusion

Our experimental study on single crystals of NBMnPO shows that it display
salient features of the classical 2$d$ TL Heisenberg AFM with easy-axis
anisotropy, including the $\uparrow$$\uparrow$$\downarrow$-phase with 1/3
magnetization plateau when magnetic field is applied along the $c$ axis. A
detailed study of AC magnetic susceptibility reveals a unique hysteresis hump
feature between the $\uparrow$$\uparrow$$\downarrow$ and $V$-phases as well as
weak phase transitions. While the weak phase transitions can be understood in
terms of small additional NNN intralayer and/or NN interlayer interactions,
the hysteresis hump require at least an order of magnitude large interaction
scale and warrants further theoretical and experimental works in the future.
Our findings may also trigger to establish a model to understand the
complicated QSL-like behavior and complex phase diagrams observed in sibling
compounds NBCoPO and NBNiPO.

We thank T. H. Jang at MPK-POSTECH for carrying out the $C_{p}$ measurement in
DR. JK acknowledge Sungdae Ji, Bongjae Kim, Ara Go, Heung-Sik Kim, and Jiyeon
Kim for helpful discussions and M. Kim for critical reading of the manuscript.
This work is supported by the KAERI Internal R&D program (Grant nos.
524460-22, 79772-21) and National Research Foundation of Korea (No.
2021M2E3A3040092). The National High Magnetic Field Laboratory is supported by
the National Science Foundation through NSF/DMR-1644779 and the State of
Florida.

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# Causal Transformers Perform Below Chance on Recursive Nested Constructions,
Unlike Humans

Yair Lakretz
Cognitive Neuroimaging Unit
NeuroSpin center
91191 Gif-sur-Yvette, France
<EMAIL_ADDRESS>
&Théo Desbordes
Facebook AI Research
NeuroSpin center
France
<EMAIL_ADDRESS>
&Dieuwke Hupkes
Facebook AI Research
Paris, France
<EMAIL_ADDRESS>
&Stanislas Dehaene
Cognitive Neuroimaging Unit
NeuroSpin center
Collège de France
Paris, France

###### Abstract

Recursive processing is considered a hallmark of human linguistic abilities. A
recent study evaluated recursive processing in recurrent neural language
models (RNN-LMs) and showed that such models perform below chance level on
embedded dependencies within nested constructions – a prototypical example of
recursion in natural language. Here, we study if state-of-the-art Transformer
LMs do any better. We test four different Transformer LMs on two different
types of nested constructions, which differ in whether the embedded (inner)
dependency is short or long range. We find that Transformers achieve near-
perfect performance on short-range embedded dependencies, significantly better
than previous results reported for RNN-LMs and humans. However, on _long-
range_ embedded dependencies, Transformers’ performance sharply drops below
chance level. Remarkably, the addition of only three words to the embedded
dependency caused Transformers to fall from near-perfect to below-chance
performance. Taken together, our results reveal Transformers’ shortcoming when
it comes to recursive, structure-based, processing.

##  1 introduction

$..$

Figure 1: A tree-structure representation of a recursive structure with two
long-range dependencies, one nested within the other one.

One of the fundamental principles of contemporary linguistics states that
language processing requires the ability to deal with nested structures.
Recursion, a specific type of computation that involves repeatedly applying a
function to its own output, is suggested to be at the core of this ability
(Hauser et al., 2002). The strongest evidence for recursion in human language
processing arises from the tree-like nested structure of sentences in natural
language, in which phrases of a particular type (i.e., NPs) can be embedded in
other phrases of that same type (Figure 1). Humans, it is argued, are endowed
with a unique competence for recursive processing, which allows them to
represent and process such nested tree structures (Chomsky, 2000; Hauser et
al., 2002; Dehaene et al., 2015).

In recent years, neural language models (NLMs) have shown tremendous advances
on a variety of linguistic tasks, such as next-word prediction (also known as
"language modeling"), translation or semantic inference. Furthermore,
evaluations of their syntactic abilities have shown promising results, with
similar or even above-human performance on a variety of different tasks
(Marvin and Linzen, 2018; Goldberg, 2019; Jumelet et al., 2021; Giulianelli et
al., 2018). However, negative results were recently also presented (Warstadt
et al., 2020; Hu et al., 2020; Petty and Frank, 2021). In particular, when it
comes to recursive processing, Lakretz et al. (2021b) showed that while
recurrent neural network language models (RNN-LMs) perform well on long-range
dependencies, such as the relationship between keys and are in sentences like
“The keys that the man near the cabinet holds, are red” (Figure 2b), they
perform below chance on the shorter, embedded dependency (man-holds). Humans,
instead, perform significantly better on such dependencies, although
interestingly, for them too, the shorter inner dependency is more difficult
than the long outer one.

The study by Lakretz et al. illustrates how investigations of neural networks
can inspire experiments about human language processing. However, their study
focuses on only a single architecture, an RNN-LM with LSTM units Hochreiter
and Schmidhuber (1997), which is currently outperformed on many fronts by the
newer _Transformer_ models (Vaswani et al., 2017). In this short paper, our
main question is therefore whether Transformer models do any better when it
comes to processing recursive constructions. We then further explore
similarities and differences in performance patterns of RNN and Trasformer
language models.

Our main results show that when tested on nested constructions with a short-
range embedded dependency, Transformers outperform RNN-LM across all
conditions, with error rates close to zero. However, when the embedded
dependency is long-range, their performance dramatically drops to below
chance, similarly to the case of RNNs. The mere addition of a short
prepositional phrase (‘near the cabinet’ in the example shown in figure 1) to
the embedded dependency causes model performance to drop from near perfect to
below chance level. Thus, contrary to what might be expected based on their
much improved performance and the fact that they are trained on substantially
more data, Transformer models share RNNs’ shortcoming when it comes to
recursive, structure-sensitive, processing.

Last, all models made more errors when trying to carry a noun in the singular
across dependencies which involved a plural noun, than in the converse
situation. Interestingly, this bias towards greater interference by plural
than by singular is opposite to that reported in Italian RNN-LMs (Lakretz et
al., 2021b), and is akin to the Markedness Effect reported for humans.

##  2 Related Work

In psycholinguistics, grammatical agreement became a standard method to probe
online syntactic processing in humans (Bock and Miller, 1991; Franck et al.,
2002), since it is ruled by hierarchical structures rather than by the linear
order of words in a sentence. More recently, it has also become a standard way
to probe grammatical generalization in NLMs (Linzen et al., 2016; Bernardy and
Lappin, 2017; Giulianelli et al., 2018; Gulordava et al., 2018; Jumelet et
al., 2019; Kersten et al., 2021; Lakretz et al., 2019; Sinha et al., 2021),
pointing to both similarities and differences between human and model error
patterns.

Lakretz et al. (2019) showed that RNN-LMs trained on a large corpus with
English sentences develop a number-propagation mechanism for long-range
dependencies. The core circuit of this mechanism was found to be extremely
sparse, comprising of only a very small number of units. This sparsity of the
mechanism suggests that models are not able to process two long-distance
dependencies simultaneously, and indeed, this was later confirmed in
simulations (Lakretz et al., 2021b). Inspired by this finding, Lakretz et al.
(2021b) conducted a following experiment with humans, which showed that they,
too, make more errors on nested long-range dependencies. However, contrary to
LMs, their performance was above chance on these constructions. This finding
suggests that human recursive processing remains significantly better than
that of RNN-LMs.

Recursive processing of nested constructions in RNN-LMs was also studied using
artificial grammars (Cleeremans et al., 1989; Servan-Schreiber et al., 1991;
Gers and Schmidhuber, 2001; Christiansen and Chater, 1999; Hewitt et al.,
2020). Recently, Suzgun et al. (2019) showed that memory-augmented RNNs can
capture recursive regularities of Dyck languages (also known as "bracket
languages"). However, when tested on a simple extension of these languages,
RNN-LMs failed to generalize to unseen data with a greater nesting depth
(Lakretz et al., 2021a). Specifically, the models failed also in cases in
which the training data contained deep structures, up to five levels of
nesting. This suggests that the poor recursive processing of RNN-LMs is not
merely due to shallow nesting depth in natural data, which is typically not
more than two (Karlsson, 2007).

Taken together, previous work suggests that RNN-LMs struggle to capture
recursive regularities in either natural or artificial data. Inspired by this
line of work, we focus here on Transformer LMs: do they show different
patterns when it comes to processing recursive structures? Do they better
approximate human ability for recursion?

##  3 Experimental Setup

We largely follow the experimental setup of Lakretz et al. (2021b), but
consider a different language (English instead of Italian) and a different set
of models.

#### Data

We consider two number-agreement tasks (NA-tasks): Short-Nested and Long-
Nested. Both tasks contain two subject-verb dependencies; they differ in terms
of whether the embedded dependency is _short-_ or _long-range_. In short-
nested, the subject and verb in the nested dependency are adjacent (see Figure
2a). They are embedded in a sentence by inserting an object-relative clause to
modify the subject of a different sentence. The Long-Nested task (Figure 2b)
uses the same constructions, except that an additional three-word
prepositional phrase (“near the cabinet”) is added in the embedded dependency.
We recently made this test data set available in the BigBench collaborative
benchmark.111https://github.com/google/BIG-
bench/tree/main/bigbench/benchmark_tasks/subject_verb_agreement

#### Models

We run experiments with all causal transformer-based NLMs that are currently
compatible with the BigBench framework, available from
HuggingFace.222https://huggingface.co/transformers/ Specifically, we include
four GPT-2 models that differed in size: GPT2, GPT2-Medium, GPT2-Large and
GPT-XL (Radford et al., 2019). In addition, as a baseline, we conduct an
experiment with an English LSTM-LM, which was studied in numerous work in the
past (Gulordava et al., 2018).

The keys that the man holds are … (a) Short-Nested

The keys that the man near the cabinet holds are … (b) Long-Nested

Figure 2: Experimental Design: the two number-agreement tasks – Short-Nested
and Long-Nested. In Short-Nested, the embedded dependency (in bold) is short-
range (in bold); in Long-Nested, it is long-range, through the insertion of a
three-word prepositional phrase.

#### Model evaluation

Following previous work, we evaluated model performance on grammatical
agreement by comparing the output probabilities of the model for the correct
(e.g., ‘are’) vs. wrong (e.g., ‘is’) verb form. For both tasks (short- and
long-nested), we evaluated model performance on subject-verb agreement for
both the embedded and the inner verb, and separately for each task condition
(see SM for details).

##  4 Results

(a) Short-Nested.

(b) Long-Nested.

Figure 3: Error rates on nested constructions for all models, for both the
main (‘keys’-‘are’) and embedded (‘man’-‘holds’) agreements. Conditions are
marked by the value of the grammatical number of all nouns in the sentence.
For example, condition SP in Short-Nested means that the first (main) noun is
singular and the second (embedded) one is plural. While error-rates are near
zero for Short-Nested, they are worse than chance-level for one of the
incongruent conditions of Long-Nested, consistently across all models. In this
condition (PSP), grammatical agreement is with respect to the second noun,
which is singular.

###  4.1 Short-Nested task

In Figure 3a, we show model performance on the Short-Nested task for all
models. Overall, the English LSTM made more errors on the main (outer)
dependency compared to the embedded (inner) one, with more than 20% errors,
across all four conditions. In contrast, Transformers, and in particular
GPT2-XL, achieved close to perfect performance across all conditions, on both
the embedded and main dependency. For GPT2, GPT2-Medium and Large, the longer
main dependency was, however, overall more difficult than the embedded one,
but with no more than 20% errors in the incongruent conditions (SP and PS;
Table S2).

Interestingly, consistently across all models, both Transformers and the LSTM
model made more errors on conditions in which the agreement was with respect
to singular, compared to plural.

###  4.2 Long-Nested task

In Figure 3b, we further show the performance of all models for the Long-
Nested task. Overall, all models made more errors across all conditions
compared to Short-Nested, but with the same tendency of making more errors on
dependencies with respect to singular compared to plural. The most striking
difference between the two tasks was the performance of the models on the
embedded dependency. In particular, for Transformers, their error rate was
close to zero in Short-Nested, but dropped to below-chance on one of the
incongurent conditions (PSP) in Long-Nested. Similarly, For the LSTM, this was
the case for both incongruent cases (PSP and SPS).

In contrast to the embedded dependency, all models performed above chance on
the main, longer, dependency. This shows that for Long-Nested, the length of
the dependency affected model performance less than the presence of recursive
embedding.

##  5 Discussion

In this study, we evaluated the recursive abilities of Transformer LMs on two
number-agreement tasks that were previously shown to be exceptionally
challenging for LSTM language models. Our experiments showed that, overall,
Transformers outperformed LSTM-LMs, and in particular, achieved close to
perfect performance on short embedded dependencies. However, similarly to
LSTM-LMs, syntactic processing in the models was found to be ‘brittle’ – the
addition of only a short prepositional phrase to the embedded dependency
caused model performance to sharply drop to below chance level.

Furthermore, we found that all models showed a bias towards plural and
therefore err more when the subject of a verb is in the singular. A similar
bias was previously observed in Italian LSTM models (Lakretz et al., 2021b),
however, in the opposite direction, with more errors on plural dependencies.
We hypothesize that this difference might be due to marking of the verb form,
given that in English, the marked form of the verb is singular, whereas in
Italian, it is plural. Related biases were previous reported for humans in
both languages, a phenomenon known as the Markedness Effect (Bock and Miller,
1991; Vigliocco et al., 1995). The relation between emerging biases in NLMs
and humans is an interesting topic for future work.

In LSTM language models, the poor performance was predicted by the underlying
neural mechanism for grammatical agreement identified in the models (Lakretz
et al., 2019, 2021b). The fact that Transformer models perform similarly
poorly on these constructions, and on the same dependency (inner), raises
interesting questions. Do transformers use syntactic-processing strategies
akin to those emerged in RNN-LMs? And what does that tell us about the data
that those models are trained on and about the potential processes that humans
may use to process such constructions (Lakretz et al., 2020)?

However, currently, the neural mechanisms underlying syntactic processing in
transformers are poorly understood (Belinkov and Glass, 2019). Our findings of
below-chance performance by transformer models calls for a further
investigation in _how_ these models achieve their earlier found successes on
syntactic related tasks, and why they generalise so poorly on constructions
which only minimally differ (a single three-word prepositional phrase) from
the constructions they process well.

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111 Ben Gurion University of the Negev222 College of William and Mary, VA, USA

# Quasiconformal Whitney Partition

Vladimir Gol’dshtein1 and Nahum Zobin2 Ben-Gurion University of the Negev,
Department of Mathematics,\newline P.O. Box 653, 8410501 Beer Sheva, Israel.
<EMAIL_ADDRESS>College of William and Mary, Department of
Mathematics,\newline P.O. Box 8795, Williamsburg, VA, USA<EMAIL_ADDRESS>
###### Abstract.

Whitney partition is a very important concept in modern analysis. We discuss
here a quasiconformal version of the Whitney partition that can be usefull for
Sobolev spaces.

##  1\. Introduction

We start with a definition of a Whitney partition of domains in
$\mathrm{\mathbf{R}^{n}}$. Classical Whitney partition is a partition of a
bounded domain $\Omega$ into diadic cubes with disjoint interiors and edges
comparable to the distance to $\partial\Omega$. Its modern generalization that
is called a Whitney partition is a partition into convex polyhedra with
similar properties. Let us give a more accurate definition (see, for example
[1]).

Let $\Omega$ be a bounded domain of $\mathrm{\mathbf{R}^{n}}$ and
$\Lambda=\mathrm{\mathbf{R}^{n}}\setminus\Omega$. Let $E_{i}$ be a family of
convex closed $n$-dimensional polyhedra in $\mathrm{\mathbf{R}^{n}}$, disjoint
from $\Lambda$, covering $\Omega$ and with pairwise disjoint interiors. We
will also need these polyhedra to have uniformly bounded ratio $K_{A}$ of
their exterior to interior radii. By the interior radius of a set $A$ with
nonempty interior we mean the greatest radius $r_{A}$ of a ball contained in
$A$; similarly the exterior radius of a set $A$ is the smallest radius $R_{A}$
of a ball containing $A$. The ratio $K_{A}$ will be called the dilatation of
$A$.

Such families of polyhedra with uniformly bounded dilatation will be called
uniformly regular. We will also demand that the edges of these polyhedra are
long, i.e. they have lengths uniformly comparable to the diameter of the
polyhedron. As an example we can take a family of dyadic cubes disjoint from
$\Lambda$. The diameter of $E_{i}$ will be denoted by
$\delta\left(E_{i}\right)$. The set $\Lambda$ will be called the residual set
of the family $\left\\{E_{i}\right\\}$ and $K_{int}:=\sup_{i}K_{E_{i}}$ will
be called the interior dilatation of the family $\left\\{E_{i}\right\\}$.

There are two conditions which are usually imposed on such families:

  1. (1)

$\delta\left(E_{i}\right)\geq C\mathrm{\>dist}\left(E_{i},\Lambda\right)$,

  2. (2)

$\delta\left(E_{i}\right)\leq
C\mathrm{{}^{-1}\>dist}\left(E_{i},\Lambda\right)$.

If a family $E_{i}$ satisfies both of these conditions, it is called a Whitney
family.

We generalize this definition for a more general setting under additional
assumption that the residual set
$\mathrm{\mathbf{R}^{n}}\setminus\Omega^{\prime}$ is unbounded and connected.
We will call such domains as simple domains.

Definition.(Rough Whitney family). Let $\Omega$ be a bounded subset of
$\mathrm{\mathbf{R}^{n}}$, $\Lambda=\mathrm{\mathbf{R}^{n}}\setminus\Omega$
and $\left\\{E_{i}\right\\}$ be a family of closed sets in
$\mathrm{\mathbf{R}^{n}}$ with nonempty interiors $\mathrm{V_{i}=Int}(E_{i})$
and the closures of $V_{i}$ coinciding with $E_{i}$. The sets $E_{i}$ are
disjoint from $\Lambda$, covering $\Omega$ and their interiors $V_{i}$ are
pairwise disjoint. The family $\left\\{E_{i}\right\\}$ has bounded interior
dilatation.

The family $E_{i}$ should also satisfy two geometric regularity conditions for
some constant $C>0$:

1.$\delta\left(E_{i}\right)\geq C\>\mathrm{dist}\left(E_{i},\Lambda\right)$,

2.$\delta\left(E_{i}\right)\leq
C^{-1}\>\mathrm{dist}\left(E_{i},\Lambda\right)$.

If a family $E_{i}$ satisfies all these conditions, we will call it as a rough
Whitney family.

The smallest possible constant $C$ will be called the exterior dilatation
$K_{ext}$ of the rough Whitney family.

Our main result connects concepts of Whitney families and rough Whitney
families.

Theorem. For any simple domain $\Omega$ quasiconformal image of any its
Whitney family is a rough Whitney family.

More detailed formulation with corresponding estimates will be discussed
later.

We prove this result with the help of classical estimates of a corresponding
capacity.

A two dimensional version of this result can be found in [3].

##  2\. Quasiconformal Whitney family

A well-ordered triple $(F_{0},F_{1};\Omega)$ of nonempty sets, where $\Omega$
is an open set in $\mathrm{\mathbf{R^{n}}}$, and $F_{0}$, $F_{1}$ are closed
subsets of $\overline{\Omega}$, is called a condenser on the Euclidean space
$\mathrm{\mathbf{R^{n}}}$.

The value

$\mathrm{Cap_{p}}(E)=\mathrm{Cap_{p}}(F_{0},F_{1};\Omega)=\inf\int\limits_{\Omega}|\nabla
v|^{p}dx,$

where the infimum is taken over all Lipschitz non-negative functions
$v:\overline{\Omega}\to\mathrm{\mathbf{R}}$, such that $v=0$ on $F_{0}$, and
$v=1$ on $F_{1}$, is called $p$-capacity of the condenser
$E=(F_{0},F_{1};\Omega)$. For $p=n$ it is the classical conformal capacity. We
will use notation $\mathrm{Cap}(F_{0},F_{1};\Omega)$ for the conformal
capacity. Of course the set of admissible functions can be empty. In this case
$\mathrm{Cap_{p}}(F_{0},F_{1};\Omega)=\infty$.

For $1<p$ for a finite value of $p$-capacity
$0\leq\mathrm{Cap}_{p}(F_{0},F_{1};\Omega)<+\infty$ there exists a unique
continuous weakly differentiable function $u_{0}$ (an extremal function) such
that:

$\mathrm{Cap}_{p}(F_{0},F_{1};\Omega)=\int\limits_{\Omega}|\nabla
u_{0}|^{p}dx.$

Definition. Let $\varphi:\Omega\rightarrow\Omega\prime$ be a homeomorphism
between two domains in $\mathrm{\mathbf{R^{n}}}$, $n\geq 2$. Then $\varphi$ is
said to be $Q$-quasiconformal, $Q\geq 1$, if

$Q^{-1}\mathrm{Cap}(F_{0},F_{1};\Omega)\leq\mathrm{Cap}(\varphi\left(F_{0}\right),\varphi\left(F_{1}\right);\Omega^{\prime})\leq
Q\>\mathrm{Cap}(F_{0},F_{1};\Omega)$

for any condenser $E=(F_{0},F_{1};\Omega)$.

The minimal possible constant $Q$ will be called the (quasiconformal)
dilatation of $\varphi$.

This geometric definition is a global requirement that quickly yields many
important properties of quasiconformal mappings. For example, the inverse of a
quasiconformal mapping is automatically quasiconformal, quasiconformal
mappings are weakly differentiable and its weak derivatives are integrable in
degree $n$, etc…

For any $Q$-quasiconfromal homeomorphism
$\varphi:\mathrm{\mathbf{R^{n}}}\to\mathrm{\mathbf{R^{n}}}$ images of closed
balls have uniformly bounded interior dilatation $K_{int}$ that depends only
on $n$ and $Q$ [4]. Any $Q$-quasiconformal image of closed balls
$\bar{B}(0,r)$ will be called the closed $Q$-quasiconformal ball or
$Q$-quasiball. For example any convex polyhedra is a closed $Q$-quasiconfromal
ball. The constant $Q$ depends on $n$ and its interior dilatation. This
collection of known facts can be formalized as

Proposition 1. ([4]) For any $Q$-quasiconfromal homeomorphism
$\varphi:\mathrm{\mathbf{R^{n}}}\to\mathrm{\mathbf{R^{n}}}$ and any closed
ball $B:=B(x_{0},r)$ the interior dilatation $K_{B,\Omega}$ is less or equal
to a constant $C(Q,n)$ that depends only on the dilatation $Q$ and $n$.

This proposition will be generalized for balls in domains under an additional
conditions for balls.

We will use notations: $B(x_{0},r)$ is an open ball of radius $r$ and center
$x_{0}$ and $\bar{B}(x_{0},r)$ is its closure.

Definition. Let $\Omega$ be a domain in $\mathrm{\mathbf{R^{n}}}$ and a closed
ball $\overline{B}(x_{0},r)\subset\Omega$. Choose such concentric open ball
$B(x_{0},K_{B(x_{0},r),\Omega}r)\subset\Omega$ that its closure
$\bar{B}(x_{0},K_{B(x_{0},r),\Omega}r)$ intersects $\partial\Omega$. We will
call the constant $K_{B(x_{0},r),\Omega}\in(1,\infty]$ the embedding
coefficient of the ball $\bar{B}(x_{0},r)$ in $\Omega$.

The embedding coefficient of any ball in $\mathbf{R^{n}}$ is $\infty$. In
section 4 we will prove the following property of quasiconformal
homeomorphisms.

Proposition 2. Let $\varphi:\Omega\to\mathrm{\mathbf{R}^{n}}$ be a
$Q$-quasiconformal homeomorphism of a domain
$\Omega\subset\mathrm{\mathbf{R}^{n}}$ and $\overline{B}(x_{0},r)$ is a closed
ball with the embedding coefficient $K_{B(x_{0},r),\Omega}$. Then the interior
dilatation $K_{\varphi(B(x_{0},r))}$ of its image $\varphi(B(x_{0},r)$ depends
only on $Q,K_{B(x_{0},r),\Omega}$ and $n$.

We will call a closed set $E$ the closed relative $Q$-quasiball if there exist
a domain $\Omega\subset\mathrm{\mathbf{R}^{n}}$, a closed ball
$\overline{B}(x_{0},r)$ and a $Q$-quasiconformal homeomorphism
$\varphi:\Omega\to\mathrm{\mathbf{R}^{n}}$ such that
$E=\varphi\left(\overline{B}(x_{0},r)\right)$. We let $K_{E}$ denote the
embedding coefficient $K_{B(x_{0},r),\Omega}$.

Definition ($Q$-quasiconfromal Whitney family). Let $\Omega$ be a bounded
domain in $\mathrm{\mathbf{R}^{n}}$ and $E_{i}$ be a family of closed relative
$Q$-quasiballs in $\mathrm{\mathbf{R}^{n}}$. The sets $E_{i}$ are disjoint
from $\Lambda$, covering $\Omega$, with pairwise disjoint interiors $V_{i}$
and their embedding coefficients $K_{i}:=K_{E_{u}}$ are uniformly bounded.

If a family $E_{i}$ satisfies this conditions, it is called a
$Q$-quasiconformal Whitney family.

Because any convex polyhedra is a quasiconformal image of the unit ball, any
Whitney family is a $Q$-quasiconformal Whitney family. Of course a rough
Whitney family is not necessary a $Q$-quasiconformal Whitney family.

##  3\. Classical estimates of conformal capacity

We will need two estimates of the conformal capacity (see, for example [4,
2]). Because proofs are very simple we reproduce here slight modifications of
both estimates, using classical embedding theorems.

Choose an open ball $B(x_{0},r)$ an open ball of radius $r$ and center $x_{0}$
and a positive constant $C_{0}>1$. Denote $R:=C_{0}\>r$.

Lemma 1. $\mathrm{Cap}\left(\bar{B}(x_{0},r),\mathrm{\mathbf{R}^{n}}\setminus
B(x_{0},R);\mathrm{\mathbf{R}^{n}}\right)=\omega_{n-1}\left[\ln\left(C_{0}\right)\right]^{1-n}.$
Here $\omega_{n-1}$is the volume of the unit $\left(n-1\right)$-dimensional
sphere $S^{n-1}$.

Proof. Let $u:\mathrm{\mathbf{R}^{n}}\to\mathrm{\mathbf{R}}$ be an admissible
smooth function for condensor
$E=\left(\bar{B}(x_{0},r),\mathrm{\mathbf{R}^{n}}\setminus
B(x_{0},R);\mathrm{\mathbf{R}^{n}}\right)$. By definition of an admissible
function $u(x)\equiv 0$ on $\bar{B}(x_{0},r)$ and $u(x)\equiv 1$ on
$\mathrm{\mathbf{R}^{n}}\setminus B(x_{0},R)$. Using H$\ddot{\mathrm{o}}$lder
inequality for the spherical coordinate system $(\rho,\sigma)$ we have

$1\leq\int_{r}^{R}\nabla u(\rho,\sigma)d\rho\leq\left(\int_{r}^{R}\left|\nabla
u(\rho,\sigma)\right|^{n}\rho^{n-1}d\rho\right)^{1/n}\left(\int_{r}^{R}\rho^{-1}d\rho\right)^{\frac{n-1}{n}}.$

Here $\rho:=|x|$ and $\sigma$ are spherical coordinates on the unit sphere
$S^{n-1}:=S^{n-1}(0,1)$. By previos calculations

$\frac{1}{\left[\ln\left(C_{0}\right)\right]^{n-1}}=\frac{1}{\left[\ln\left(\frac{R}{r}\right)\right]^{n-1}}\leq\int_{r}^{R}\left|\nabla
u(\rho,\sigma)\right|^{n}\rho^{n-1}d\rho$

for any $\sigma\in S^{n-1}$.

Integrating this inequality over $S^{n-1}$ we obtain finally

$\frac{\omega_{n-1}}{\left[\ln\left(C_{0}\right)\right]^{n-1}}\leq\int_{S^{n-1}}\int_{r}^{R}\left|\nabla
u(\rho,\sigma)\right|^{n}\rho^{n-1}d\rho=\int_{\mathrm{\mathbf{R}}^{n}}\left|\nabla
u(x)\right|^{n}dx$

where $\omega_{n-1}$ is area of $S^{n-1}$.

The function
$u_{0}(x):=\ln\left(\frac{R}{\left|x\right|}\right)\left[\ln\left(C_{0}\right)\right]^{-1}$
for $r\leq\left|x\right|\leq R$ , $0$ for any $\left|x\right|\geq R$ and $1$
for any $\left|x\right|\leq r$ is the extremal function for condensor $E$. It
follows from direct calculations:

$\int_{\mathrm{\mathbf{R}}^{n}}\left|\nabla
u_{0}(x)\right|^{n}dx=\omega_{n-1}\left[\ln\left(C_{0}\right)\right]^{-n}\int_{r}^{R}\rho^{-1}d\rho=\omega_{n-1}\left[\ln\left(C_{0}\right)\right]^{1-n}.$

Therefore

$\mathrm{Cap}\left(\bar{B}(x_{0},r),\mathrm{\mathbf{R}^{n}}\setminus
B(x_{0},R);\mathrm{\mathbf{R}^{n}}\right)=\omega_{n-1}\left[\ln\left(C_{0}\right)\right]^{1-n}.$

We will call a connected closed set a continuum. Choose two concentric
$(n-1)$-dimensional spheres $S^{n-1}(0,r)$ and $S^{n-1}(0,R)$, $R\geq r$ and
two continua $F_{0}$ and $F_{1}$ that join spheres. We shall use notations
$R=C_{0}\>r$, $C_{0}>1$ and $D_{r,R}:=\bar{B(0,R)\setminus B(0,r)}$. In the
next lemma we will prove a below estimate of conformal capacity of condensor
$E=(F_{0},F_{1},\mathrm{\mathbf{R}^{n})}$ using embedding theorems for the
unit sphere.

Lemma 2. $\mathrm{Cap}\left(F_{0},F_{1};\mathrm{\mathbf{R}^{n}}\right)\geq
C(n)\>\ln\left(C_{0}\right)$, where a constant $C(n)$ depends on $n$ only.

Proof. Because continua $F_{0},F_{1}$ join spheres $S^{n-1}(0,r)$ and
$S^{n-1}(0,R)$ intersections $F_{0,\rho}:=S_{\rho}^{n-1}\cap F_{0}$ and
$F_{1,\rho}:=S_{\rho}^{n-1}\cap F_{1}$ are not empty for any sphere
$S_{\rho}^{n-1}:=S^{n-1}(0,\rho)$, $r\leq\rho\leq R$. Any function $u$
admissible for the conformal capacity of the condensor $E$ is also admissible
for conformal capacity of any condensor
$E_{\rho}=(F_{0,\rho},F_{1,\rho},\mathrm{S_{\rho}^{n-1})}$ in the sphere
$S_{\rho}^{n-1}$. By elementary calculations in the spherical coordinates
$\rho,\sigma$ we have

$\int_{r}^{R}\left(\int_{S_{\rho}^{n-1}}\left|\nabla
u(\sigma,\rho)\right|^{n}d\sigma\right)d\rho\leq\int_{D_{R,r}}\left|\nabla
u(x)\right|^{n}dx\leq\int_{\mathrm{\mathbf{R}^{n}}}\left|\nabla
u(x)\right|^{n}dx$

where $D_{R,r}$ is a closed ring between spheres $S^{n-1}(0,r)$ and
$S^{n-1}(0,R)$.

Using similarities $\varphi_{\rho}(x)=\rho x$ we get

$\int_{S_{\rho}^{n-1}}\left|\nabla
u(\sigma,\rho)\right|^{n}d\sigma=\frac{1}{\rho}\int_{S^{n-1}(0,1)}\left|\nabla\tilde{u}(\sigma)\right|^{n}d\sigma,$

where $\tilde{u}(x)=u(\varphi_{\rho}(x))$.

By the classical Sobolev inequality for the unit sphere $S^{n-1}$ we have

$1=\left\|\tilde{u}|L_{\infty}(S^{n-1})\right\|^{n}\leq
K(n)\int_{S^{n-1}(0,1)}\left|\nabla\widetilde{u}(\sigma)\right|^{n}d\sigma$

where constant $K(n)$ depends on $n$ only.

Combining these two estimates we obtain

$1=\left\|u|L_{\infty}(S_{\rho}^{n-1})\right\|^{n}=\left\|\tilde{u}|L_{\infty}(S^{n-1}(0,1))\right\|^{n}\leq\rho
K(n)\int_{S_{\rho}^{n-1}}\left|\nabla u(\sigma,\rho)\right|^{n}d\sigma.$

Dividing by $\rho$ and integrating we finally get

$\frac{\ln\left(C_{0}\right)}{K(n)}=\frac{1}{K(n)}\int_{r}^{R}\frac{d\rho}{\rho}\leq\int_{r}^{R}\left(\int_{S_{\rho}^{n-1}}\left|\nabla
u(\sigma,\rho)\right|^{n}d\sigma\right)d\rho$
$\leq\int_{\mathrm{\mathbf{R}^{n}}}\left|\nabla u(x)\right|^{n}dx$

for any admissible function $u$ of condensor $E$. By definition of conformal
capacity

$\frac{\ln\left(C_{0}\right)}{K(n)}\leq\mathrm{Cap}\left(F_{0},F_{1};\mathrm{\mathbf{R}^{n}}\right).$

##  4\. Local estimates of dilatations

Let $\varphi:\Omega\to\Omega^{\prime}$ be a $Q$-quasiconformal homeomorphism
of a domain $\Omega\subset\mathrm{\mathbf{R}^{n}}$ onto a domain
$\Omega^{\prime}\subset\mathrm{\mathbf{R}^{n}}$. Choose a closed ball
$\bar{B}(x_{0},r)\subset\Omega$ with the embedding coefficient
$C_{r}:=K_{B\\{x_{0},r),\Omega}$ .

Our goal is to prove for $F_{r}:=\varphi(\bar{B}(x_{0},r))$ the following two
inequalities:

  1. (1)

There exists a constant $C\mathrm{{}_{1}(Q,C_{r},n)}$ such that

$K_{int}(F_{r})\leq C\mathrm{{}_{1}(Q,C_{r},n)}$

  2. (2)

There exists a constant $C\mathrm{{}_{2}(Q,C_{r},n)}$ such that

$C^{-1}\>\mathrm{dist}\left(F_{r},\Omega^{\prime}\right)\leq\delta\left(F_{r}\right)\leq
C\>\mathrm{dist}\left(F_{r},\Omega^{\prime}\right)$

,

Constants $C_{1},C_{2}$ depend only on $Q,C_{r},n$ and do not depend on choice
of domains $\Omega,\Omega^{\prime}$ and the $Q$-quasiconformal homeomorphism
$\varphi$.

Proposition 3. Suppose that $\Omega$,$\Omega^{\prime}$ are domains in
$\mathbf{R}^{n}$ and the residual set $\Lambda=\mathbf{R}^{n}\setminus\Omega$
of $\Omega$ is unbounded and connected. Let $\varphi:\Omega\to\Omega^{\prime}$
be a $Q$-quasiconformal homeomorphism of a domain
$\Omega\subset\mathrm{\mathbf{R}^{n}}$ onto a domain
$\Omega^{\prime}\subset\mathrm{\mathbf{R}^{n}}$ and
$\bar{B}(x_{0},r)\subset\Omega$ be a closed ball with the embedding
coefficient $K_{B(x_{0},r),\Omega}$ . Then there exists such positive constant
$C_{1}(Q,C_{r},n)$ that $K_{int}(F_{r})\leq C_{1}(Q,K_{B(x_{0},r),\Omega},n)$.

The constant $C_{1}(Q,K_{B(x_{0},r),\Omega},n)$ depends on
$Q,K_{B(x_{0},r),\Omega}$ and $n$ only.

Proof. Denote by $\bar{B}_{r}:=\bar{B}(y_{0},\bar{r})\subset F_{r}$ a closed
ball of a maximal radius that belongs to $F_{r}:=\varphi(B(x_{o},r))$ and has
the center at a point $y_{0};=\varphi(x_{0})$, by $B_{R}:=B(y_{0},\bar{R})$ an
open ball of a minimal radius whose closure contains $F_{r}$, by
$CB_{R}:=\mathrm{\mathbf{R}^{n}}\setminus B_{R}$, by $K_{r}$ the interior
dilatation $K_{int}(F_{r})$ and by $C_{r}$ the embedding coefficient
$K_{B(x_{0},r),\Omega}$. By definition of the interior dilatation
$K_{r}\leq\frac{\bar{R}}{\bar{r}}$.

By definition of the conformal capacity and previous Lemma

(4.1) $\mathrm{Cap}(B_{R},\bar{\Omega}^{\prime}\setminus
B,\Omega^{\prime})\leq\mathrm{Cap}(B_{r},CB_{R},\mathrm{\mathbf{R}^{n}})=\omega_{n-1}\left[\ln\left(\frac{\bar{R}}{\bar{r}}\right)\right]^{1-n}$
(4.2) $\leq\omega_{n-1}\left[\ln\left(K_{r}\right)\right]^{1-n}.$

Because a homeomorphism $\varphi^{-1}$ is $Q$-quasiconformal we have for two
compact sets $F_{0}:=\varphi^{-1}(\bar{B}_{r})\subset\bar{B}(x_{0},r)$ and
$F_{1}=\varphi^{-1}(\bar{\Omega}^{\prime}\setminus
B_{R})\subset\Omega\setminus B(x_{0},r)$ the following inequality

(4.3) $\mathrm{Cap}(F_{0},F_{1},\Omega)\leq
Q\mathrm{\>Cap}(\bar{B}_{r},\bar{\Omega}^{\prime}\setminus
B_{R},\Omega^{\prime})\leq\omega_{n-1}\left[\ln\left(K_{r}\right)\right]^{1-n}.$

Let us estimate $\mathrm{Cap}(F_{0},F_{1},\Omega)$ with the help of Lemma 2.
By construction both sets $F_{0},F_{1}$ have nonempty intersections
$S_{0},S_{1}$ with the sphere $S(x_{0},r)$. We distinguish two different
cases:

$\mathrm{1.\;dist}(S_{0},S_{1})\leq\min\left(\left(C_{r}-1\right)\frac{r}{2},\frac{r}{2}\right)$;

$\mathrm{2.\;dist}(S_{0},S_{1})>\min\left(\left(C_{r}-1\right)\frac{r}{2},\frac{r}{2}\right)$.

Let us use a short notation
$\bar{r}:=\min\left(\left(C_{r}-1\right)\frac{r}{2},\frac{r}{2}\right).$

Choose points $y_{o}\in S_{0}$ and $\bar{y_{1}\in S_{1}}$ such that
$\mathrm{dist}(S_{0},S_{1})=|y_{1}-y_{0}|$. Let
$\tilde{y}:=y_{0}+\frac{y_{1}-y_{0}}{2}$ and
$B_{1}:=\bar{B(}\tilde{y},\bar{r})$ be a closed ball with center at
$\tilde{y}$.

In the first case $|y_{1}-y_{0}|\leq\bar{r}$ and continua $F_{0},F_{1}$
intersect any sphere $S(\tilde{y},\rho)$ for any $\bar{r}\leq\rho\leq
2\bar{r}$ . By Lemma 2

(4.4)
$\mathrm{\mathrm{Cap}\left(F_{0},F_{1};\Omega\right)\geq\mathrm{Cap}\left(F_{0}\cap
B_{1},F_{1}\cap B_{1};B_{1}\right)}\geq C(n)\>\ln 2.$

Combining this inequality and inequality 4.3 we obtain finally

(4.5) $C(n)\>\ln 2\leq\omega_{n-1}\left[\ln\left(K_{r}\right)\right]^{1-n}.$

The proposition is proved for the first case when

$\mathrm{\;dist}(S_{0},S_{1})\leq\min\left(\left(C_{r}-1\right)\frac{r}{2},\frac{r}{2}\right).$

In the second
case$\mathrm{\;dist}(S_{0},S_{1})\leq\min\left(\left(C_{r}-1\right)\frac{r}{2},\frac{r}{2}\right)$
we have $|y_{1}-y_{0}|>\bar{r}$. By definition of capacity

$\mathrm{Cap}\left(F_{0},F_{1};\Omega\right)\geq\mathrm{Cap}\left(F_{0}\cap
B(x_{0},C_{r}r),F_{1}\cap B(x_{0},C_{r}r);B(x_{0},C_{r}r)\right).$

There exists a quasiconformal homeomorphism $\psi$ that maps $B(x_{0},C_{r}r)$
onto itself, maps any sphere $S(x_{0},\rho)$, $0<\rho<C_{r}r$ onto itself and
satisfies conditions: $\psi(y_{1})=y_{1}$, and image of $y_{0}$ is the point
$y_{0}^{\perp}:=\psi(y_{0})$ opposite to $y_{0}$ on $S(x_{0},r)$. The
coefficient of quasiconformality $Q_{1}$ of $\psi$ can be easily estimated
$Q_{1}\leq\frac{\pi r}{\bar{r}}=\frac{2}{\min(C_{r}-1,1)}$.

By definition of quasiconformal homeomorphism we have

$\begin{array}[]{c}\mathrm{Cap}\left(F_{0}\cap B(x_{0},C_{r}r),F_{1}\cap
B(x_{0},C_{r}r);B(x_{0},C_{r}r)\right)\geq\\\
\frac{1}{Q_{1}}Cap\left(\psi(F_{0})\cap B(x_{0},C_{r}r),\psi(F_{1})\cap
B(x_{0},C_{r}r);B(x_{0},C_{r}r)\right).\end{array}$

Choose two closed balls

$B_{2}:=\bar{B}\left(x_{0}+\frac{y_{0}^{\perp}-x_{0}}{2},\frac{r}{2}\right)$

and

$B_{3}:=\bar{B}\left(x_{0}+\frac{y_{0}^{\perp}-x_{0}}{2},\frac{r}{2}+(C_{r}-1)r\right)\subset
B(x_{0},C_{r}r)$

. By construction continua $\psi(F_{0})$ and $\psi(F_{1})$ intersect any
sphere $S\left(x_{0}+\frac{y_{0}^{\perp}-x_{0}}{2},\rho\right)$ for
$\frac{r}{2}\leq\rho\leq\frac{r}{2}+(C_{r}-1)r$. Therefore by Lemma 2

$\mathrm{Cap}\left(\psi(F_{0})\cap B(x_{0},C_{r}r),\psi(F_{1})\cap
B(x_{0},C_{r}r);B(x_{0},C_{r}r)\right)\geq C(n)\ln\left(1+2(C_{r}-1)\right).$

Finally we have

(4.6) $\mathrm{Cap}\left(F_{0},F_{1};\Omega\right)\geq
C(n)\min(C_{r}-1,1)\ln\left(1+2(C_{r}-1)\right).$

Combining this inequality and inequality 4.3 we obtain finally

(4.7)
$C(n)\min(C_{r}-1,1)\ln\left(1+2(C_{r}-1)\right)\leq\omega_{n-1}\left[\ln\left(K_{r}\right)\right]^{1-n}.$

The proposition is proved for the second case:

$\mathrm{dist}(S_{0},S_{1})>\min\left(\left(C_{r}-1\right)\frac{r}{2},\frac{r}{2}\right)$

Recall that $K_{r}$ is the short notation for $K_{int}(F_{r})$. Combining
inequalities (4.3, 4.4, 4.6) we obtain finally the constant $C_{1}(Q,C_{r},n)$

$C(n)\min\left[\ln
2,\min(C_{r}-1,1)\ln\left(1+2(C_{r}-1)\right)\right]\leq\omega_{n-1}\left[\ln\left(K_{r}\right)\right]^{1-n}$

i.e.

$K_{r}\leq\exp\left[\left\\{\frac{\omega_{n-1}}{C(n)\min\left[\ln
2,\min(C_{r}-1,1)\ln\left(1+2(C_{r}-1)\right)\right]}\right\\}^{\frac{1}{n-1}}\right].$

Remark. We proved a stronger estimate:

$\frac{\bar{R}}{\bar{r}}\leq\exp\left[\left\\{\frac{\omega_{n-1}}{C(n)\min\left[\ln
2,\min(C_{r}-1,1)\ln\left(1+2(C_{r}-1)\right)\right]}\right\\}^{\frac{1}{n-1}}\right].$

Let us return to estimates of $\delta\left(F_{r}\right)$ with the help of
$\mathrm{dist}\left(F_{r},\partial\Omega^{\prime}\right)$ i.e. to estimates of
$K_{ext}(F_{r})$. We start with the upper estimate:

Proposition 4. Suppose that $\Omega$,$\Omega^{\prime}$ are domains in
$\mathbf{R}^{n}$ and the residual set
$\Lambda=\mathbf{R}^{n}\setminus\Omega^{\prime}$ of $\Omega^{\prime}$ is
unbounded and connected. Let $\varphi:\Omega\to\Omega^{\prime}$ be a
$Q$-quasiconformal homeomorphism of a bounded domain
$\Omega\subset\mathrm{\mathbf{R}^{n}}$ onto a bounded domain
$\Omega^{\prime}\subset\mathrm{\mathbf{R}^{n}}$,
$\bar{B}(x_{0},r)\subset\Omega$ be a closed ball with the embedding
coefficient $K_{B(x_{0},r),\Omega}$. Then

$\delta\left(F_{r}\right)\leq
C_{1}\mathrm{(Q,C_{r},n)dist}\left(F_{r},\Lambda^{\prime}\right)=C_{1}\mathrm{dist}\left(F_{r},\partial\Omega^{\prime}\right)$

where a constant $C_{2}:=C_{2}(Q,C_{r},n)$ depends only on $Q,C_{r},n$.

Proof. We will use notations: $F_{r}=\varphi(\bar{B}(x_{0},r))$$,K_{r}$ for
the interior dilatation $K_{int}(F_{r})$, $C_{r}$ for the embedding
coefficient $K_{B(x_{0},r),\Omega}$ and
$F_{R}:=\overline{\Omega}^{\prime}\setminus\varphi(B(x_{0},C_{r}r))$.

Because $\varphi:\Omega\to\Omega^{\prime}$ is $Q$-quasiconformal

$\mathrm{Cap}(F_{r},F_{R};\Omega^{\prime})\leq
Q\>\mathrm{Cap}(\bar{B}(x_{0},r),\overline{\Omega}\setminus
B(x_{0},C_{r}r));\Omega).$

By definition of the conformal capacity

$\>\mathrm{Cap}(\bar{B}(x_{0},r),\overline{\Omega}\setminus
B(x_{0},C_{r}r));\Omega)=\>\mathrm{Cap}(\bar{B}(x_{0},r),\mathrm{\mathbf{R}^{n}}\setminus
B(x_{0},Cr));\mathrm{\mathbf{R}^{n}}).$

By Lemma 1

$\mathrm{Cap}(\bar{B}(x_{0},r),\mathrm{\mathbf{R}^{n}}\setminus
B(x_{0},Cr));\mathrm{\mathbf{R}^{n}})=\omega_{n-1}\left[\ln\left(C_{r}\right)\right]^{1-n}.$

Hence

$\mathrm{Cap}(F_{r},F_{R};\Omega^{\prime})\leq\omega_{n-1}\left[\ln\left(C_{r}\right)\right]^{1-n}$

Choose points $y_{o}\in F_{r}$and $y_{1}\in\partial\Omega^{\prime}$ such that
$\mathrm{dist}(F_{r},\partial\Omega^{\prime})=|y_{1}-y_{0}|$. Let
$\tilde{y}:=y_{0}+\frac{y_{1}-y_{0}}{2}$ and $\bar{B(}\tilde{y},\rho_{1})$ be
a closed ball with center at $\tilde{y}$.

By Proposition 3 $F_{r}$ contains a closed ball of a radius
$\rho_{2}:=\frac{1}{2}K_{r}\delta(F_{r})$. Therefore both continuums $F_{r}$
and $F_{R}$ intersect any sphere $S(\tilde{y},\rho)$ for any
$\rho\in[\rho_{1},\rho_{1}+\rho_{2}]$. By Lemma 2 we have

$\begin{array}[]{c}\mathrm{Cap}(F_{r},F_{R};\Omega^{\prime})\geq\mathrm{Cap}(F_{r},F_{R};\bar{B(}\tilde{y},\rho_{2})\setminus
B(\tilde{y},\rho_{1}))\geq\\\
C(n)\>\ln\left(\frac{\mathrm{dist}(F_{r},\partial\Omega^{\prime})+K_{r}\delta(F_{r})}{\mathrm{dist}(F_{r},\partial\Omega^{\prime})}\right).\end{array}$

Combining both estimates for $\mathrm{Cap}(F_{r},F_{R};\Omega^{\prime})$ we
obtain finally

$\mathrm{C(n)\>\ln\left(\frac{\mathrm{dist}(F_{r},\partial\Omega^{\prime})+K_{r}\delta(F_{r})}{\mathrm{dist}(F_{r},\partial\Omega^{\prime})}\right)\leq
Cap}(F_{r},F_{R};\Omega^{\prime})\leq\omega_{n-1}\left[\ln\left(C_{r}\right)\right]^{1-n}.$

It means that

$\mathrm{dist}(F_{r},\partial\Omega^{\prime})+K_{r}\delta(F_{r})\leq\mathrm{dist}(F_{r},\partial\Omega^{\prime})\exp\left[\frac{\omega_{n-1}}{C(n)\left[\ln\left(C_{r}\right)\right]^{n-1}}\right].$

Because $C_{r}>1$ the number under exponent is positive and the exponent value
is bigger than $1$. Therefore

$\delta(F_{r})\leq\left\\{K_{r}^{-1}\exp\left[\frac{\omega_{n-1}}{C(n)\left[\ln\left(C_{r}\right)\right]^{n-1}}\right]\right\\}\mathrm{dist}(F_{r},\partial\Omega^{\prime}).$

The second estimate is an estimate from below with some additional restriction
on the domain $\Omega^{\prime}$. We suppose additionally that any component of
$\mathrm{\mathbf{R}^{n}}\setminus\Omega^{\prime}$ is unbounded. We call such
domains simple domains.

Proposition 5. Suppose that $\Omega$,$\Omega^{\prime}$ are domains in
$\mathbf{R}^{n}$ and the residual set $\Lambda=\mathbf{R}^{n}\setminus\Omega$
of $\Omega$ is unbounded and connected (i.e $\Omega$ is a simple domain). Let
$\varphi:\Omega\to\Omega^{\prime}$ be a $Q$-quasiconformal homeomorphism ,
$\bar{B}(x_{0},r)\subset\Omega$ be a closed ball with embedding coefficient
$K_{B(x_{0},r),\Omega}$ . Then

$\delta\left(F_{r}\right)\geq
C_{3}\mathrm{(Q,K_{B(x_{0},r),\Omega},n)dist}\left(F_{r},\Lambda^{\prime}\right)$

where a constant $C_{3}:=C_{3}(Q,K_{B(x_{0},r),\Omega},n)$ depends only on
$Q,K_{B(x_{0},r),\Omega},n$.

Proof. Denote by $\bar{B}_{r}:=\bar{B}(y_{0},\bar{r})\subset F_{r}$ a closed
ball of a maximal radius that belongs to $F_{r}:=\varphi(B(x_{o},r))$ and has
the center at a point $y_{0};=\varphi(x_{0})$, by $B_{R}:=B(y_{0},\bar{R})$ an
open ball of a minimal radius whose closure contains $F_{r}$, by
$CB^{R}:=\mathrm{\mathbf{R}^{n}}\setminus B^{R}$, by $K_{r}$ the ratio
$\overline{K}_{r}\leq\frac{\bar{R}}{\bar{r}}$ and by $C_{r}$ the embedding
coefficient $K_{B(x_{0},r),\Omega}$. By definition of the interior dilatation
$K_{r}\leq\overline{K}_{r}=\frac{\bar{R}}{\bar{r}}$.

By Proposition 3
$\overline{K}_{r}\leq\tilde{C}_{r}(Q,C_{r},n):=\tilde{C}_{r}$. It means that
$\delta\left(F_{r}\right)\leq 2\tilde{C}_{r}\overline{r}$. Let
$B_{r_{d}}:=B(y_{0},r_{d})$ be a greatest open ball such that
$S(y_{0},r_{d})\cap\partial\Omega^{\prime}\neq\emptyset$ . By construction
$r_{d}\geq\overline{r}+dist\left(F_{r},\partial\Omega^{\prime}\right)$. By
Lemma 1 and monotonicity of the conformal capacity

$\mathrm{Cap}(\bar{B}_{r},\Omega^{\prime}\setminus
B_{r_{d}};\mathrm{\Omega^{\prime}})=\mathrm{Cap}(\bar{B}_{r},\mathrm{\mathbf{R}^{n}}\setminus
B_{r_{d}};\mathrm{\mathbf{R}^{n}})=\omega_{n-1}\left[\ln\left(\frac{r_{d}}{\overline{r}}\right)\right]^{1-n}$
$\leq\omega_{n-1}\left[\ln\left(\frac{\overline{r}+dist\left(F_{r},\partial\Omega^{\prime}\right)}{\overline{r}}\right)\right]^{1-n}.$

Using monotonicity of conformal capacity and quasi-invariance of conformal
capacity under $Q$-quasiconformal homeomorphisms we obtain

$\mathrm{\mathrm{Q^{-1}Cap}(\overline{B}(x_{0},r),\Lambda;\mathrm{\Omega})\leq\mathrm{Cap}(\bar{B}_{r},\Lambda^{\prime};\mathrm{\Omega^{\prime}})\leq\mathrm{Cap}(\bar{B}_{r},\mathrm{\mathbf{R}^{n}}\setminus
B_{r_{d}};\mathrm{\Omega^{\prime}})\leq
Cap}(\bar{B}_{r},\overline{\Omega}^{\prime}\setminus
B_{r_{d}};\mathrm{\Omega^{\prime}}).$

Choose a point $x_{1}\in\partial\Omega$ closest to $\overline{B}(x_{0},r)$ and
two balls

$B_{1}:=B\left(x_{0}+\frac{x_{1}-x_{0}}{2},\frac{C_{r}-1}{2}\right)$

and

$\overline{B}_{2}:=\overline{B}\left(x_{0}+\frac{x_{1}-x_{0}}{2},\frac{C_{r}-1}{2}+2r\right)$

. Continuums $F_{0}:=\overline{B}(x_{0},r)$ and $F_{1}:=\Lambda\cap
B\left(x_{0}+\frac{x_{1}-x_{0}}{2},\frac{C_{r}-1}{2}r+2r\right)$ join spheres
$S^{n-1}\left(x_{0}+\frac{x_{1}-x_{0}}{2},\frac{C_{r}-1}{2}r+2r\right)$ and
$B\left(x_{0}+\frac{x_{1}-x_{0}}{2},\frac{C_{r}-1}{2}r\right)$. By
monotonicity of capacity and Lemma 2 we get

$\mathrm{C(n)\ln\left\\{1+\frac{1}{C_{r}-1}\right\\}\leq
Cap}(F_{0},F_{1};\overline{B}_{2}\setminus
B_{1})\leq\mathrm{Cap}(\overline{B}(x_{0},r),\Lambda;\mathrm{\Omega})$

Combining all previous estimates we get

$\ln\left(1+\frac{1}{C_{r}-1}\right)\leq\frac{Q\omega_{n-1}}{C(n)}\omega_{n-1}\left[\ln\left(\frac{\overline{r}+dist\left(F_{r},\partial\Omega^{\prime}\right)}{\overline{r}}\right)\right]^{1-n}.$

It means that

$\left[\ln\left(1+\frac{dist\left(F_{r},\partial\Omega^{\prime}\right)}{\overline{r}}\right)\right]^{n-1}\leq\frac{Q\omega_{n-1}}{C(n)}\omega_{n-1}\left[\ln\left(1+\frac{1}{C_{r}-1}\right)\right]^{-1}$

The constant

$C_{0}(Q,C_{r},n):=\frac{Q\omega_{n-1}}{C(n)}\omega_{n-1}\left[\ln\left(1+\frac{1}{C_{r}-1}\right)\right]^{-1}$

is positive.

After elementary calculations we have

$\frac{dist\left(F_{r},\partial\Omega^{\prime}\right)}{\overline{r}}\leq\exp\left(C_{0}(Q,C_{r},n)^{\frac{1}{n-1}}\right)-1.$

Recall that $C_{r}$ is a short notation for $K_{B(x_{0},r),\Omega}$ and denote

$C_{3}(Q,K_{B(x_{0},r),\Omega},n):=\left[\exp\left(C_{0}(Q,C_{r},n)^{\frac{1}{n-1}}\right)-1\right]^{-}.$

Because $\overline{r}\leq\delta(F_{r})$ we rewrite the last inequality as

$C_{3}(Q,K_{B(x_{0},r),\Omega},n)\>dist\left(F_{r},\partial\Omega^{\prime}\right)\leq\overline{r}\leq\delta(F_{r})$

The main result of this paper can be formulated in more general way.

Theorem. For any simple domain $\Omega$ quasiconformal image of any its
Whitney family is a rough Whitney family.

Follows directly from Propositions 3,4,5.

## References

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