#  Kagomé quantum oscillations in graphene superlattices

Folkert K. de Vries  Laboratory for Solid State Physics, ETH Zürich, CH-8093
Zürich, Switzerland  Sergey Slizovskiy  National Graphene Institute,
University of Manchester, Manchester M13 9PL, United Kingdom  Department of
Physics & Astronomy, University of Manchester, Manchester M13 9PL, United
Kingdom  Second author contributed equally Petar Tomić  Laboratory for Solid
State Physics, ETH Zürich, CH-8093 Zürich, Switzerland  Roshan Krishna Kumar
National Graphene Institute, University of Manchester, Manchester M13 9PL,
United Kingdom  Department of Physics & Astronomy, University of Manchester,
Manchester M13 9PL, United Kingdom  ICFO-Institut de Ciencies Fotoniques, The
Barcelona Institute of Science and Technology, Barcelona, Spain  Aitor Garcia-
Ruiz  National Graphene Institute, University of Manchester, Manchester M13
9PL, United Kingdom  Department of Physics & Astronomy, University of
Manchester, Manchester M13 9PL, United Kingdom  Giulia Zheng  Elías Portolés
Laboratory for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland
Leonid A. Ponomarenko  Department of Physics, University of Lancaster,
Lancaster LA1 4YW, United Kingdom  Andre K. Geim  National Graphene Institute,
University of Manchester, Manchester M13 9PL, United Kingdom  Department of
Physics & Astronomy, University of Manchester, Manchester M13 9PL, United
Kingdom  Kenji Watanabe  National Institute for Materials Science, 1-1 Namiki,
Tsukuba 305-0044, Japan  Takashi Taniguchi  National Institute for Materials
Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Vladimir Fal’ko
<EMAIL_ADDRESS>National Graphene Institute, University of
Manchester, Manchester M13 9PL, United Kingdom  Department of Physics &
Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom  Henry
Royce Institute for Advanced Materials, M13 9PL, Manchester, United Kingdom
Klaus Ensslin<EMAIL_ADDRESS>Laboratory for Solid State Physics, ETH
Zürich, CH-8093 Zürich, Switzerland  Thomas Ihn  Laboratory for Solid State
Physics, ETH Zürich, CH-8093 Zürich, Switzerland  Peter Rickhaus  Laboratory
for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland

Periodic systems feature the Hofstadter butterfly spectrum produced by
Brown–Zak minibands of electrons formed when magnetic field flux through the
lattice unit cell is commensurate with flux quantum and manifested by magneto-
transport oscillations. Quantum oscillations, such as Shubnikov – de Haas
effect and Aharonov–Bohm effect, are also characteristic for electronic
systems with closed orbits in real space and reciprocal space. Here we show
the intricate relation between these two phenomena by tracing quantum magneto-
oscillations to Lifshitz transitions in graphene superlattices, where they
persist even at relatively low fields and very much above liquid-helium
temperatures. The oscillations originate from Aharonov–Bohm interference on
cyclotron trajectories that form a kagomé-shaped network characteristic for
Lifshitz transitions. In contrast to Shubnikov - de Haas oscillations, the
kagomé oscillations are robust against thermal smearing and they can be
detected even when the Hofstadter butterfly spectrum is undermined by
electron’s scattering. We expect that kagomé quantum oscillations are generic
to rotationally-symmetric two-dimensional crystals close to Lifshitz
transitions.

Lifshitz transitions Lifshitz (1960) (LTs) are generic for the electronic
bands in solids. They mark the sign change for the effective mass of
electrons, accompanied by saddle-point features in electrons’ dispersion and
van Hove singularities in their density of states. In two-dimensional (2D)
crystals, a LT also singles out a band energy for which disconnected closed-
loop contours, $\varepsilon({\bf p})=\rm const$, merge into a multiply-
connected network. For most generic LTs where the ’electron-like’ dispersion
transforms into ’hole-like’ near its top, these constant-energy contours,
$\varepsilon({\bf p})=E_{\text{LT}}$, form networks, spanning across the
entire reciprocal space of a crystal (see an example in Fig. 1a).

Constant energy maps are important for magnetotransport phenomena, as they
fully determine the shape of ballistic electron trajectories in a 2D metal
subjected to magnetic field, ${\bf B}=B\hat{\bf z}$. Because the electron’s
dynamics is set by $\dot{\bf p}=eB\hat{\bf z}\times\dot{\bf r}$ and $\dot{\bf
r}\equiv{\bf v}=\nabla_{{\bf p}}\epsilon({\bf p})$, the real-space
trajectories can be obtained from constant-energy contours by a $90^{\circ}$
rotation and re-scaling using a $(eB)^{-1}$ factor, Fig. 1b. For closed-loop
energy contours, this transformation results in closed cyclotron trajectories.
Aharonov–Bohm interference along such trajectories is known to lead to
semiclassical Shubnikov–de Haas oscillations (SdHO) that are forerunners of
the fully quantized electronic spectrum and the appearance of, for example,
the quantum Hall effect. In this report, we show that multiply connected
trajectories near Lifshitz transitions ($\epsilon\approx E_{\text{LT}}$), lead
to a peculiar interference contribution to conductivity which turns out to be
insensitive to thermal broadening of the Fermi step (unlike SdHO) due to the
trihexagonal geometry of these trajectories (Fig. 1b) ), whereas the magnetic
breakdown spans the oscillations over a substantial interval of carrier
densities around LTs. We refer this behaviour as kagomé oscillations, as the
low-field forerunners of Brown–Zak minibands Brown (1964); Zak (1964);
Hofstadter (1976) which determine the transport properties of the latter.
Manifestations of the latter in quantum oscillations of capacitance
Ponomarenko  _et al._ (2013) and transport Dean  _et al._ (2013); Kumar  _et
al._ (2017); Saito  _et al._ (2021) have been observed in several graphene
superlattices at high magnetic fields, and they were even seen to persist at
high-temperatures Kumar  _et al._ (2017, 2018). Here, we show that the high-
temperature magneto-transport oscillations emerge from kagomé oscillations
near Lifshitz transitions in graphene superlattices.

Figure 1: (a) Fermi contours for LT and dispersion map for the 1st miniband in
the K-valley conduction band of a AB/BA tDBLG (modelled using Hamiltonian
described in Ref. Garcia-Ruiz  _et al._ (2021) for $\theta=1.9^{\circ}$ twist
angle); $\kappa$, $\kappa^{\prime}$ and $\gamma$ mark the high-symmetry points
in the mini Brillouin zone of mSL. Red/blue indicate miniband energies
above/below $E_{\text{LT}}$. (b) At the LT, ballistic trajectories of
electrons in a magnetic field form a trihexagonal kagomé network. Green and
blue lines in (b) exemplify the shortest paths responsible for quantum
magneto-oscillations at the LT, zoomed-in for $E=E_{\text{LT}}+\epsilon$. Note
that both having the same length, ${\mathcal{L}}$. (c) Electrons scatter at
the intersections of paths linked to the saddle points in the band dispersion.
Figure 2: Magneto-transport oscillations in a tDBLG device with
$\theta=1.9^{\circ}$, sketched in (d). (a) Landau fan diagram
$\sigma_{\mathrm{xx}}(n,B)$ at $T=$1.2\text{\,}\mathrm{K}$$. Arrows mark
Landau levels formed around $\kappa$’s (blue) and $\gamma$ (red) edges of the
1st miniband on the conduction band side of graphene, shown in the inset. (b)
The LTs are identified from the changing sign of the low-field
$\sigma_{\mathrm{xy}}$. (c) $\sigma_{\mathrm{xx}}(n,B)$ at
$20\text{\,}\mathrm{K}$. Oscillations, periodic in $\phi_{0}/\phi$, are
pronounced in the vicinity of the LTs. (d) Schematic representation of the
tDBLG device; its double-gating enables tuning the displacement field to $D=0$
for each electron density $n$. (e) Oscillating part of conductivity,
$\Delta\sigma_{\mathrm{xx}}$ at
$n=$3.2\text{\times}{10}^{12}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$, and (f) the
amplitude of these oscillations, plotted as a function of $\phi_{0}/\phi$ for
various temperatures using colour coding from (e) (see SM SM  for details).

In Fig. 1a and Fig. S5 we illustrate a characteristic LT contour for lattices
which possess a C3 rotational symmetry. The illustrated geometry of the LT
contour is generic for graphene minibands formed by moiré superlattices (mSL)
in graphene-hBN heterostructures (G/hBN) Yankowitz  _et al._ (2012);
Ponomarenko  _et al._ (2013); Kumar  _et al._ (2017, 2018), twisted graphene
bilayers Lin  _et al._ (2020); Shen  _et al._ (2021), trilayers Brown  _et
al._ (2012); Chen  _et al._ (2016); Mogera and Kulkarni (2020), and double-
bilayers Burg  _et al._ (2019); Liu  _et al._ (2020); Shen  _et al._ (2020);
He  _et al._ (2021) (tDBLG). It has the distinct form of a trihexagonal kagomé
network (which would be inverted for graphene’s K and K’ valleys), with a
hexagonal part centered at $\gamma$ and two triangular shapes around $\kappa$
and $\kappa^{\prime}$ points of the mini Brillouin zone of mSL. The
corresponding hexagonal and triangular areas in Fig. 1a are painted in red and
blue, indicating which parts of a particular miniband dispersion are at the
energies above and below $E_{\text{LT}}$, respectively. The example shown in
Fig. 1 corresponds to the 1${}^{\text{st}}$ mSL miniband on the conduction
band side of tDBLG spectrum (to represent the LT map for the 1${}^{\text{st}}$
mSL miniband on the valence band side, we would have to swap red and blue
colours).

Below, we relate the kagomé-shaped LT contours to magneto-transport
oscillations observed in different graphene-based superlattices. We start by
discussing the oscillations observed in a tDBLG, Fig. 2. These oscillations
were measured in a Hall bar device SM ; Tomić  _et al._ (2022), Fig. 2d, at
low magnetic fields and in the $10\text{\,}\mathrm{K}$–$50\text{\,}\mathrm{K}$
temperature range, where they replace SdHO dominant at low temperatures, Figs.
2a. The latter, reflecting Landau levels (LL) formed by electrons revolving
along closed cyclotron loops (Fig.S1), have an amplitude,
$\Delta\sigma_{\text{SdHO}}\propto\operatorname{sech}\frac{2\pi^{2}k_{B}T\rho}{B/\phi_{0}}$,
suppressed at temperatures larger than the LL splitting Lifshitz and Kosevich
(1956), in particular, around LTs, where the density of states of electrons,
$\rho$, is high. The limitations on the appearance of SdHO and the data on the
measured transport mean free path, $\ell_{\text{mfp}}$ in the studied tDBLG
are discussed in Figs. S7 and S8. The remaining high-temperatures oscillations
have $\sigma_{xx}$ maxima when the magnetic flux through the mSL unit cell,
$\phi=B\mathcal{A}_{\varhexagon}$, is commensurate, $\phi=\phi_{0}/q$ with an
integer $q$, with the flux quantum, $\phi_{0}=h/e$. While this formally
coincides with the conditions for Brown–Zak minibands formation Brown (1964);
Zak (1964); Hofstadter (1976), the oscillations in Fig. 2c are clearly visible
at temperatures largely exceeding the miniband widths and gaps in-between
them. Below, we argue that these oscillations are produced by the interference
contribution to the electronic transport which would be present (though, in a
weaker form) even if the length of the electron mean free path,
$\ell_{\text{mfp}}$ would be comparable to the perimeter,
${\mathcal{L}}_{q}\sim
4\sqrt{\frac{2}{\sqrt{3}}}q\sqrt{\mathcal{A}_{\varhexagon}}$, of a magnetic
supercell Brown (1964); Zak (1964); Chen  _et al._ (2014).

Let us consider an unbound propagation of an electron along the kagomé network
at the LT in Fig. 1b. Due to the quantum nature of electrons, their
propagation along such a network has a stochastic element. The meander-like
paths bifurcate at their intersections, Fig. 1c, associated with the saddle-
point features in the electron dispersion, captured by transmission
amplitudes, ${\overset{\curvearrowright}{S}}$ and
${\overset{\curvearrowleft}{S}}$ Alexandradinata and Glazman (2018); Berry and
Mount (1972); Wilkinson (1984); Davis and Liu (1967),

$\begin{split}&\frac{|{\overset{\curvearrowright}{S}}|}{|{\overset{\curvearrowleft}{S}}|}=e^{\pi\mu};\,\mu=\frac{\hbar\,(\epsilon-
E_{\text{LT}})}{eB{\mathfrak{r}}};\,|{\overset{\curvearrowright}{S}}|^{2}+|{\overset{\curvearrowleft}{S}}|^{2}=1;\\\
&\arg{\overset{\curvearrowleft}{S}}=\arg{\overset{\curvearrowright}{S}}=\arg\left[\Gamma\left(\frac{1}{2}-i\mu\right)\right]+\mu(\ln|\mu|-1).\end{split}$
(1)

These amplitudes are comparable within the ’magnetic breakdown’ Cohen and
Falicov (1961) energy interval $\sim eB{\mathfrak{r}}/h$, dependent on the
Gaussian curvature of the dispersion saddle-point,
${\mathfrak{r}}=\hbar\sqrt{|\det\frac{\partial^{2}\epsilon({\bf
p})}{\partial{p_{i}}\partial{p_{j}}}|}$, which determines the energy window
around $E_{\text{LT}}$ where the LT kagomé network remains relevant for
electron transport. This makes the LT network similar to the networks of
topologically protected channels at the AB/BA domain boundaries in marginally
twisted bilayer graphene Huang  _et al._ (2018); Xu  _et al._ (2019). However,
the LT networks of trajectories — such as in Fig. 1b — are not set in stone,
as each of those is linked to the initial position and velocity, $|in\rangle$,
of an electron propagating ballistically.

Having in mind that the lengths of electronic trajectories,
${\mathcal{L}}\propto B^{-1}$, scale up and can exceed the electron mean free
path at $B\rightarrow 0$, in Fig. 1b we exemplify the shortest paths visiting
several LT network nodes on the way from point ’in’ to ’out’. Interference of
electron waves propagating along those paths contributes towards the
probability for an electron to reach ’out’ from ’in’. The phases, acquired by
an electron along this pair of paths, have the same property, as noted in an
AB/BA domain wall network De Beule  _et al._ (2020): as these two paths are
composed of pairs of equivalent ballistic segments, the electron-energy-
dependent dynamical part of the phase related to the length of the path is the
same for both of them, making their interference condition (constructive or
destructive) set only by the encircled magnetic field flux, independent of the
electron’s energy deviation from $E_{\text{LT}}$. Note that the above-
mentioned cancellation of dynamical phases is also similar to the phase
cancellations for selected pairs of paths in the weak localisation effect in
electronic systems. It leads to the resilience of the resulting interference
effects against thermal broadening of the Fermi step.

Figure 3:  Comparison of $\sigma_{xx}$ oscillations around LTs observed in
moire superlattice minibands in tDBLG and G/hBN. Dots mark the shift of the
maxima of oscillations’ amplitudes with increasing magnetic field. (a)
Oscillations of $\sigma_{xx}(q,n)$ around LT in the first miniband on the
conduction band side of tDBLG spectrum, shown in (b). The experimental data
are plotted after subtracting a smooth background from data in Fig. 2 (see SM
and Figs. S2,S3 for details); T = 20K. (c) Electronic spectrum of G/hBN
superlattices Yu  _et al._ (2014); Lee  _et al._ (2016)) which features a
clear LT in the 1st miniband on the valence band side of graphene’s spectrum
(white contour). In agreement with Eq. (4), similarity between the highlighted
minibands in (b) and (c) prescribes the same direction of a shift (upon adding
electrons to the system) of the oscillations maxima from the LT in tDBLG and
(d) in a highly aligned G/hBN device, where mSL has a $15$ nm period (here,
$\sigma_{xx}$ was measured at $T=$50\text{\,}\mathrm{K}$$ and analysed as in
tDBLG device.

To take into account interference of these two highlighted partial waves, we
write an amplitude for an electron to reach ’out’ from ’in’, as

$\langle{\rm
out|in}\rangle=\alpha_{\text{diff}}+e^{i\varphi}\left[{\overset{\curvearrowleft}{S}}^{3}{\overset{\curvearrowright}{S}}^{2}+{\overset{\curvearrowleft}{S}}^{3}{\overset{\curvearrowright}{S}}^{2}e^{i\frac{eB\mathcal{A}}{\hbar}}\right]e^{-\frac{{\mathcal{L}}}{\ell}}.$
(2)

Here, $\varphi=\frac{1}{\hbar}\int_{\text{in}}^{\text{out}}\bf p\cdot dr$ is
the phase acquired by an electron propagating along the green path in Fig. 1b,
and

$\mathcal{A}\equiv\mathcal{A}_{1}+\mathcal{A}_{2}+\mathcal{A}_{3}=\frac{A_{\text{BZ}}}{(eB)^{2}}\equiv\frac{\phi_{0}^{2}}{\mathcal{A}_{\varhexagon}B^{2}},$

is the area between the green and the blue paths that is equal to the area of
the kagomé network unit cell ($A_{\text{BZ}}$ and $\mathcal{A}_{\varhexagon}$
are the mSL Brillouin zone area and the unit cell area, respectively). We also
account for Maslov’s phase multipliers $\mp i$ for each
clockwise/anticlockwise turning point SM ; Creagh  _et al._ (1990);
Alexandradinata and Glazman (2018) and list all the shortest interfering paths
in Fig. S4. An amplitude $\alpha_{\text{diff}}$ accounts for diffusive paths
with variable lengths linking "in" to "out" upon scattering from graphene’s
disorder, or phonons, which may even be the dominant transport mechanism. Such
scattering processes are also accounted for by a factor
$e^{-{\mathcal{L}}/\ell}<1$, and we neglect contributions of longer inter-
crossing ballistic paths, as their contributions would be additionally
suppressed by disorder scattering.

For the probability, $|\langle{\rm out|in}\rangle|^{2}$, of an electron to
reach ’out’ from ’in’, the interference of partial waves arriving along such
pairs of paths is set by the Aharonov–Bohm phase, $eB\mathcal{A}/\hbar$. This
produces a contribution,

$\Delta\sigma_{\text{LT}}\sim\frac{e^{2}}{h}\cos\frac{2\pi\phi_{0}}{\mathcal{A}_{\varhexagon}B}\times
e^{-\frac{2{\mathcal{L}}}{\ell}}\int d\epsilon\,\frac{\partial
n_{F}}{\partial\epsilon}|{\overset{\curvearrowleft}{S}}^{6}{\overset{\curvearrowright}{S}}^{4}|,$
(3)

to the conductivity $\sigma_{xx}=\sigma_{0}+\Delta\sigma_{\text{LT}}$, which
oscillates periodically as a function of $B^{-1}$. This contribution comes on
top of a diffusive conductivity background, $\sigma_{0}$, in which the
interference between waves following paths of variable long lengths, with
large energy-dependent phases, is washed out by averaging over disorder and
over energies, due to the thermal broadening of the Fermi step (here,
accounted for by the factor $\frac{\partial n_{F}}{\partial\epsilon}$).

Equation (3) describes the high-temperature low-field $1/B$-periodic
oscillations, Fig. 2c, observed in tDBLG around $n_{\text{LT}}$ (here, LT was
identified by the sign change of the lowest-field Hall conductivity, Fig. 2b).
In contrast to SdHO, shown in Fig. 2a for 1.2 K but disappearing above liquid-
helium temperature, kagomé oscillations sustain elevated temperatures (up to
50 K, Fig. 2e) and have a period set by the unit cell area of mSL, independent
of the gate-induced carrier density, $n$.

We note that the observed kagomé oscillations not only appear at temperatures
largely exceeding Brown–Zak miniband widths and gaps Brown (1964); Zak (1964);
Hofstadter (1976); Kumar  _et al._ (2017, 2018), but, also, are noticeable
even in the regimes of a substantial disorder and phonon scattering,
$\ell_{\text{mfp}}\leq 2{\mathcal{L}}_{q}$, where Brown–Zak minibands cannot
form. To discuss the influence of scattering on kagomé oscillations, accounted
for by the factor $e^{-2{\mathcal{L}}/\ell}$ in Eq. (3), we analysed the
envelopes of the oscillations in Fig. 2e, which decay as $e^{-\beta/B}$ over
two orders of magnitude, Fig. 2f and Fig. S6. Such exponential decay is a
natural consequence of scaling, ${\mathcal{L}}(B)\propto B^{-1}$, of the
kagomé network segments, and the exponential decay of the measure amplitudes
of kagomé oscillations shown in Fig. 2 suggest that their observation is made
in the scattering-dominated transport regime.

The high-temperature oscillations in Fig. 2c are the largest near the LTs
(here, in the 1${}^{\text{st}}$ miniband on both conduction and valence side
of tDBLG spectrum, but their maxima shift away from $n_{LT}$ upon increasing
magnetic field (this shift largely exceeds small deviations of the zero of
$\sigma_{xy}$ from the vertical direction in Fig. 2b, which can be attributed
to anomalous contributions to the Hall conductivity induced by the Berry
curvature). We relate this shift, highlighted in Fig. 3a to the kagomé network
topology and a typical — for LTs in all C3-symmetric crystals — energy
dependence of transmission amplitudes in Eq. (3). Indeed, the interference
term in Eq. (3) contains an unbalanced product of
$|{\overset{\curvearrowright}{S}}|^{2}$ and
$|{\overset{\curvearrowleft}{S}}|^{2}$ factors,
$\left|{\overset{\curvearrowleft}{S}}^{6}{\overset{\curvearrowright}{S}}^{4}\right|=\frac{e^{-\pi\mu}}{32\cosh^{5}\pi\mu}$,
which has the energy dependence skewed towards lower energies. A saddle-point
evaluation of the integral in Eq. (3), recalculated from the Fermi energy into
carrier density dependence (using density of states in the band, $\rho$,
averaged over $k_{B}T$ interval around LT), results in the following
expression for the low-field tail of oscillations,

$\begin{split}\Delta\sigma_{\text{LT}}\propto\frac{B}{\delta
n}e^{-\frac{(n-n_{\max})^{2}}{\delta
n^{2}}}e^{-\frac{2{\mathcal{L}}(B)}{\ell(T)}}\,\cos\frac{2\pi\phi_{0}}{\mathcal{A}_{\varhexagon}B};\\\
n_{\max}=n_{\text{LT}}\pm\frac{2\rho B{\mathfrak{r}}}{5\phi_{0}};\,\,\delta
n=2\rho\sqrt{(k_{B}T)^{2}+\frac{2}{5}\left(\frac{B{\mathfrak{r}}}{\phi_{0}}\right)^{2}}.\end{split}$
(4)

Here, $\pm$ accounts for the type of dispersion inside the hexagonal part of
mSL Brilloun zone around $\gamma$: it is $+$ for hole-like
($\epsilon(\gamma)>E_{\text{LT}}$, as shown in Fig. 3b) and $-$ for electron-
like ($\epsilon(\gamma)<E_{\text{LT}}$) part of the miniband. As a result, the
inverted form of minibands on the valence and conduction band side of tDBLG
spectrum (see Fig. 3b) sets the opposite shift directions,
$n_{\text{LT}}\pm\frac{2\rho{\mathfrak{r}}B}{5\phi_{0}}$, for the oscillations
amplitude maxima near the LT in n- and p-doped tDBLG.

To corroborate the generality of the trend described by Eq. (4) for graphene
and other C3-symmetric superlattices, we compare tDBLG to monolayer graphene
placed and aligned on hBN (G/hBN), where mSL is formed with a 15 nm period
Kumar  _et al._ (2017). The mSL minibands for such a system are shown in Fig.
3c. Here, we stress that, for G/hBN, the hole-like dispersions around $\gamma$
are separated by the LT from electron-like pockets around $\kappa$’s in the
1st miniband on the valence band side in the same way as in the 1st miniband
on the conduction band side in tDBLG. This similarity determines the same
direction of the shift of the oscillations maxima from the respective LTs upon
adding electrons in both systems, Fig. 3a,d. The $\Delta\sigma_{xx}$ data for
G/hBN were taken at $T=$50\text{\,}\mathrm{K}$$ and analysed for the low
magnetic field end in the same way as for tDBLG (Fig. 3d), whereas at high
magnetic fields they agree with the earlier observed magneto-oscillations
Kumar  _et al._ (2017) which reflect recurrent appearance of faster-
propagating Kumar  _et al._ (2018) Bloch states at superlattice-commensurate
flux values, $\phi_{0}/q$.

Also, from the computed mSL minibands, we estimate ${\mathfrak{r}}\approx
1.2\,{\,\mathrm{eV}}{\,\mathrm{nm}}^{2}$ for tDBLG and ${\mathfrak{r}}\approx
6\,{\,\mathrm{eV}}{\,\mathrm{nm}}^{2}$ for G/hBN SM ; as a result, we find
that $\delta n$ is temperature-limited up to $B\sim 10$ T for tDBLG at $T=20$
K and up to $B\sim 5$ T for G/hBN at $T=50$ K, explaining the weak
$B$-dependence of the width of the regions where the oscillations occur in
Fig. 3a,d. Fitting the measured magnetic field dependence of oscillations
amplitudes with Eq.(4) and length ${\mathcal{L}}$, Fig.S6 , we find
temperature dependence of scattering parameter, $\ell^{-1}(T)$, and compare
it, in Fig.S7 , to the inverse of the measured mean free path,
$\ell^{-1}_{\text{mfp}}(B=0,\varepsilon=E_{LT})$. This comparison shows that
(a) momentum relaxation (which requires larger momentum transfer) is dominated
by electron-phonon scattering in the Bloch-Grüneisen regime, and (b) that it
is slower than decay of a ballistic wave packet, which could be expected as
the latter is sensitive to the large momentum transfer upon scattering,
whereas $\ell^{-1}$ is caused by both large and small angle scattering. After
this, we note that formation of Brown–Zak minibands Brown (1964); Zak (1964)
and Hofstadter butterfly are determined by Bragg scattering from the
superlattice with a period inflated $q$ times as compared to the underlying
moiré pattern Chen  _et al._ (2014). For forming minibands the interfering
electron waves should propagate ballistically over - at least - two magnetic
supercells (to explore periodicity of magnetic superlattice), which requires
that $2{\mathcal{L}}_{q}(B)\leq\ell_{\text{mfp}}(T)$, hence, higher magnetic
fields. Figure S8 shows that the kagomé oscillations were observed in the
opposite regime of low magnetic fields, heralding the formation of minibands
and anticipating the oscillations of kinetic parameters (such as
characteristic group velocities) simultaneously occuring in several
consecutive Brown–Zak minibands) in the mSL spectrum.

While the studies of kagomé oscillations in this paper were focused on moiré
superlattices with trigonal (C3) symmetry, we expect similar forerunner of
Brown–Zak oscillations to exist in crystals with a C4 symmetry Azbel (1964);
Hofstadter (1976), too. However, we note that they would be suppressed in
systems with a lower symmetry. This is because LT contours in low-symmetry
crystals have the form of quasi-1D block-chains of intertwining meanders
Kaganov and Lifshits (1979); Kaganov and Slutskin (1983) which do not provide
pairs of paths needed for the energy-independent Aharonov–Bohm interference.
Therefore, breaking the C3 rotational symmetry of the mSL by straining one of
the 2D crystals in a stack Cosma  _et al._ (2014) and violating the kagomé
topology of the LT network would suppress these novel low-magnetic-field high-
temperature quantum oscillations.

##  I Acknowledgements

We acknowledge support from the European Graphene Flagship Core3 Project,
EPSRC grants EP/W006502/1, EP/V007033/1, EP/S030719/1, Swiss National Science
Foundation via NCCR Quantum Science, EU H2020 grant No 862660/QUANTUM E LEAPS,
and H2020 European Research Council (ERC) Synergy Grant under Grant Agreement
951541.

##  II Author contributions

K.E.,T.I. P.R. and V.F. conceived the work and designed the research strategy.
F.K.deV. and P.T. performed conductance measurements and data analysis under
K.E.’s supervision. G.Z. and E.P. fabricated the tDBLG samples. R.K.K., L.A.P.
and A.K.G. provided data on low-field magnetotransport in G/hBN superlattices.
K.W. and T.T. provided the hBN crystals. S.S. and V.F. developed the kagomé
network model, and S.S. performed the data analysis under V.F. supervision. A.
G.-R. and V.F. modelled miniband spectra in tDBLG and G/hBN heterostructure.
V.F., F.K.deV. and S.S. wrote the paper. All authors discussed the paper and
commented on the manuscript. All authors declare no competing interests. All
data is available from the authors upon reasonable request.

##  III Supplementary Materials include:

Materials and Methods:

  * •

tDBLG device characterization and experimental data analysis

  * •

G/hBN device characterization and data analysis

  * •

Maslov and Berry phases

  * •

Shortest interfering paths

  * •

Details of saddle-point calculation, leading to Eq. (4)

  * •

Calculation of dispersion for tDBLG and G/hBN

  * •

Temperature dependence of kagomé oscillation amplitude and estimate of
coherence length

Figs. S1 –– S8

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## Supplemental Material for "Kagomé quantum oscillations in graphene
superlattices"

##  S4 tDBLG device characterization and experimental data analysis

For this work we used the same device as reported on in Ref. Tomić  _et al._
(2022). Details on the fabrication process, as well as optical images of the
Van der Waals stack can be found there. We measured $V_{\mathrm{xx}}$ and
$V_{\mathrm{xy}}$, convert this to $\rho_{\mathrm{xx}}$ and
$\rho_{\mathrm{xy}}$ using the applied current $I$ and the width of the mesa
$W=$2374\text{\,}\mathrm{n}\mathrm{m}$$ and the average distance between the
contacts of $L=$910\text{\,}\mathrm{n}\mathrm{m}$$, and obtain
$\sigma_{\mathrm{xx}}$ and $\sigma_{\mathrm{xy}}$ through tensor inversion. We
control the temperature in a pumped Helium-4 cryostat by a heater and a
feedback loop, allowing us to reach stable values from
$1.2\text{\,}\mathrm{K}$ up to $60\text{\,}\mathrm{K}$.

Figure S1: (a) Shubnikov–de Haas map, $\sigma_{\mathrm{xx}}(n,B)$ at zero
displacement field, as plotted before in Fig.1(a). Here the SdH oscillations
are fit (red and black dashed lines) to extract the degeneracies and full
filling density $n(\gamma)$ as indicated. (b) $\sigma_{\mathrm{xx}}$ versus
both bottom gate ($V_{\mathrm{bg}}$) and top gate ($V_{\mathrm{tg}}$) voltages
at a constant magnetic field of $B=$2\text{\,}\mathrm{T}$$. Shubnikov–de Haas
oscillations indicate constant density lines for the two bilayers.

We calculate the density (in cm-2) and displacement field (in V/nm) using the
following equations:

$n=\frac{1}{e}(C_{\mathrm{bg}}V_{\mathrm{bg}}+C_{\mathrm{tg}}V_{\mathrm{tg}})\cdot
10^{-4}-n_{\mathrm{offset}},$ (S1)
$D=-\frac{0.5}{\epsilon_{0}}(C_{\mathrm{tg}}V_{\mathrm{tg}}-C_{\mathrm{bg}}V_{\mathrm{bg}})\cdot
10^{-9}$ (S2)

where $C_{\mathrm{xx}}$ are the capacitance to the respective gates, $e$ is
the elementary charge, $\epsilon_{0}$ the vacuum permittivity, and
$n_{\text{offset}}$ an offset in the density. The capacitances are obtained in
two steps. First we estimate the capacitance per area between the gate and the
TDBG using the parallel plate capacitor model. Input parameters are the
thickness of the hBN and AlOx layers as well as their dielectric constants.
Then we finetune the capacitance found by measuring the Landau levels as a
function of the estimated density, and fitting the Landau levels using
$n=h/q\cdot B/\nu$ where q is either +e for holes or -e for electrons (Fig.
S1(a)). To crosscheck the ratio of the capacitances to top and bottom gate we
then perform a measurements of the SdHO at constant magnetic field and as a
function of both gates ( Fig. S1(b)). The slope of the line where the total
density equals zero gives us the ratio of the capacitances.

Furthermore, the fit of the Shubnikov–de Haas oscillations in Fig. S1 allows
us to check the degeneracies. The red dashed lines correspond to SdHO with a
degeneracy of $8$. The degeneracy corresponds to spin, valley and minivalley
(or layer) degree of freedom. The data reveals that this degeneracy is broken
at higher magnetic fields where three additional lines appear between each
pair of red dashed lines. This corresponds to the situation where the spin-
and valley degeneracy is lifted, while the mini-valley degeneracy is
maintained even at $7\text{\,}\mathrm{T}$.

Finally, the oscillations emerging from large densities (i.e.
$8.95\text{\times}{10}^{12}\text{\,}\mathrm{c}\mathrm{m}^{-2}$) are four-fold
degenerate, as indicated by the black dashed lines. The change from eight-fold
degenerate electrons (at $n>0$) to four-fold degenerate holes occurs at the
Lifshitz transition. These fits allow us to estimate the density at full band
filling and from that calculate a twist angle of
$1.9\text{\,}\mathrm{\SIUnitSymbolDegree}$. We cross-checked this with the
Brown–Zak oscillations and were able to refine it to
$1.94\text{\,}\mathrm{\SIUnitSymbolDegree}$.

The background subtraction procedure used to plot Fig. 3 (a) in the main text
is shown in Fig. S2. The kagom’e oscillations are visible in Fig. S2, where we
show $\sigma_{\mathrm{xx}}(n,q)$ at $20\text{\,}\mathrm{K}$. For better
visibility, we apply a background subtraction procedure. We use a smoothing
filter (Savitzky-Golay) along the $n$-axis (27 points, $1^{st}$ order) and
obtain the background $\left<\sigma_{\mathrm{xx}}\right>$ plotted in Fig.
S2(b). By subtracting the two maps from each other, we obtain Fig. S2(c), i.e.
$\Delta\sigma_{\mathrm{xx}}=\sigma_{\mathrm{xx}}-\left<\sigma_{\mathrm{xx}}\right>$.
The exact procedure is used for different temperatures for Fig. 3(e) and to
obtain Fig. 3(d) smoothing is done over 19 points instead.

Figure S2: Background subtraction used for Fig. 2(c). (a) obtained
$\sigma_{\mathrm{xx}}(n,B)$, (b) smoothed background
$\langle\sigma_{\mathrm{xx}}(n,B)\rangle$, (c)
$\Delta\sigma_{\mathrm{xx}}(n,B)$, the difference between (a) and (b).

Next, we obtain the kagomé oscillation amplitude $A_{\text{osc}}(n)$, used for
Fig. 2(f). We first extract cuts $\sigma_{\mathrm{xx}}(n)$ at integer $q$ and
half-integer $q$, see Fig. S3(a). Then from these, we calculate
$A_{\text{osc}}=\sigma_{\mathrm{xx}}(q)-\nicefrac{{1}}{{2}}(\sigma_{\mathrm{xx}}(q+\nicefrac{{1}}{{2}})+\sigma_{\mathrm{xx}}(q-\nicefrac{{1}}{{2}}))$,
i.e. $A_{\text{osc}}$ is given by constructive minus destructive interference,
see Fig. S3(b). To obtain $A_{\text{osc}}$ for many different temperatures for
Fig. 2(f) we have taken linetraces at different temperatures at the density
indicated in Fig. S3(b).

Figure S3: (a) Examples of $\sigma_{\mathrm{xx}}$ traces at half integer and
integer $q$. (b) The extracted amplitude of the oscillations $A_{\text{osc}}$
for $q$ ranging from 14 (red) to 32 (blue). The dashed line indicates the
density at which the amplitude data for the temperature dependent measurements
is taken. Both panels are for $T=$15\text{\,}\mathrm{K}$$.

##  S5 G/hBN device characterization and data analysis

The data for graphene aligned on hBN at $T=50$ K (Fig. 3d of main text) is a
previously unpublished data, obtained in the experiment reported in Ref.Kumar
_et al._ (2017). The device details can be found in Supplementary Section 1 of
Ref.Kumar  _et al._ (2017). The oscillatory part of $\sigma_{xx}$ has been
extracted by the same method as for tDBLG sample discussed above.

##  S6 Maslov and Berry phases

The phases of the amplitudes ${\overset{\curvearrowright}{S}}$ and
${\overset{\curvearrowleft}{S}}$, presented in Eq.(1), do not include the
Maslov phase, which should be included in the amplitude as $-i$ multiplier for
each clock-wise $p_{y}$ turning point and as $i$ for each anti-clock-wise
$p_{y}$ turning point Creagh  _et al._ (1990). Such separation of Maslov
phases allows us to describe all the differently-oriented saddle points in a
gauge-independent way.

A more precise version of semiclassical formalism includes also the Berry
curvature and Berry magnetic moment corrections to the semiclassical phase,
see Wilkinson (1984); Davis and Liu (1967); Alexandradinata and Glazman
(2018); Alexandradinata  _et al._ (2018). Effectively, the Aharonov–Bohm
interference correction that we describe corresponds to a phase gained on a
contour encircling the Brillouin zone. The Berry magnetic moment can be
accounted for by a magnetic field dependent shift of the dispersion, while the
Berry curvature term leads to a Chern number of the band, $C_{n}$, which is
integer and hence does not affect the phase of the kagomé oscillations.

##  S7 Shortest interfering paths

In this section, we describe all the shortest interference paths contributing
to kagomé oscillations. We start with a random phase space point "in", as
shown in Fig.S4, and show all 4 of the possible shortest interfering
contributions. For "out" points chosen on any other segments (not shown in
Fig.S4), the Aharonov–Bohm interference contribution will contain longer path
and will not be dominant in the limit of short phase coherence.

Figure S4: Schematic illustration of 4 shortest-path amplitudes producing
kagomé oscillations in conductance. Black dot is a starting point on a
randomly selected segment, red dots indicate final segments for which
probabilities get a $+\cos(2\pi\phi_{0}/\phi)$ interference correction (due to
the real Maslov phase) and blue dots correspond to $-\cos(2\pi\phi_{0}/\phi)$
terms (due to the imaginary Maslov phase). Interfering paths are shown in
green and blue (for "out1" and "out2" on the left and for "out3" and "out4" on
the right). Adding up the 4 leading contributions shows that “+” terms
dominate in the conductance, leading to $+\cos(2\pi\phi_{0}/\phi)$
oscillations in the conductance.

Explicitly, the amplitudes are

$\displaystyle\langle{\rm out1|in}\rangle$ $\displaystyle=$
$\displaystyle\alpha_{\text{diff}}+e^{i\phi}\left[{\overset{\curvearrowleft}{S}}^{3}{\overset{\curvearrowright}{S}}^{2}+{\overset{\curvearrowleft}{S}}^{3}{\overset{\curvearrowright}{S}}^{2}e^{i\frac{eB(\mathcal{A}_{1}+\mathcal{A}_{2}+\mathcal{A}_{3})}{\hbar}}\right]e^{-\frac{{\mathcal{L}}_{\text{in-
out1}}}{2\ell}}$ (S3) $\displaystyle\langle{\rm out2|in}\rangle$
$\displaystyle=$
$\displaystyle\alpha_{\text{diff}}+e^{i\phi}\left[-i{\overset{\curvearrowleft}{S}}^{2}{\overset{\curvearrowright}{S}}^{3}+i{\overset{\curvearrowleft}{S}}^{4}{\overset{\curvearrowright}{S}}e^{i\frac{eB(\mathcal{A}_{1}+\mathcal{A}_{2}+\mathcal{A}_{3})}{\hbar}}\right]e^{-\frac{{\mathcal{L}}_{\text{in-
out2}}}{2\ell}}$ (S4) $\displaystyle\langle{\rm out3|in}\rangle$
$\displaystyle=$
$\displaystyle\alpha_{\text{diff}}-e^{i\phi}\left[{\overset{\curvearrowleft}{S}}{\overset{\curvearrowright}{S}}^{4}+{\overset{\curvearrowleft}{S}}^{5}e^{i\frac{eB(\mathcal{A}_{1}+\mathcal{A}_{2}+\mathcal{A}_{3})}{\hbar}}\right]e^{-\frac{{\mathcal{L}}_{\text{in-
out3}}}{2\ell}}$ (S5) $\displaystyle\langle{\rm out4|in}\rangle$
$\displaystyle=$
$\displaystyle\alpha_{\text{diff}}+e^{i\phi}\left[-i{\overset{\curvearrowleft}{S}}^{2}{\overset{\curvearrowright}{S}}^{3}+i{\overset{\curvearrowleft}{S}}^{4}{\overset{\curvearrowright}{S}}e^{i\frac{eB(\mathcal{A}_{1}+\mathcal{A}_{2}+\mathcal{A}_{3})}{\hbar}}\right]e^{-\frac{{\mathcal{L}}_{\text{in-
out4}}}{2\ell}}$ (S6)

where we have taken into account the Maslov phases. When module-squared to get
the probability, all the 4 amplitudes presented above produce an oscillatory
term
$\sim\pm{\overset{\curvearrowleft}{S}}^{6}{\overset{\curvearrowright}{S}}^{4}\cos(2\pi\phi/\phi_{0})$
and the sign is $+$ for “out1” and “out4” and $-$ for “out2” and “out3”.

There are two possible approaches to extract conductivity: first is the
Einstein relation between conductivity and diffusion, which implies that a
contribution to the conductivity is proportional to the squared distance,
$\langle x^{2}\rangle$, covered between the two scatterings. Since the
contributions with $+$ sign are seen to have higher values of $\langle
x^{2}\rangle$, the overall sign of oscillating term is
$+\cos(2\pi\phi_{0}/\phi)$. Another approach to conductivity is based on Kubo
formula and involves the velocity-velocity correlator. Since the positive
(negative) contributions have velocities with positive (negative) projection
onto the initial velocity, the total contribution is, clearly, positive.

##  S8 Details of saddle-point calculation, leading to Eq.(4)

The amplitude of kagomé oscillations is clearly peaked near the LT since it
involves a product of both ${\overset{\curvearrowright}{S}}$ and
${\overset{\curvearrowleft}{S}}$ terms, the typical energy width of
oscillation region is $\sim\hbar eB{\mathfrak{r}}$. A distinguishing feature
of hexagonal network is that the maximum amplitude of oscillations is shifted
from the Lifshitz transition point ($\mu=0$) in the direction of a higher
doping (as shown in Fig. 2c and Fig. 3 a,d of the Main Text). This occurs
because higher power of ${\overset{\curvearrowleft}{S}}$ compared to
${\overset{\curvearrowright}{S}}$ is involved in the oscillation amplitude,
Eq.(2),
$\left|{\overset{\curvearrowleft}{S}}^{6}{\overset{\curvearrowright}{S}}^{4}\right|=\frac{e^{-\pi\mu}}{32(\cosh\pi\mu)^{5}}$.
The maximum of this expression is shifted by

$\epsilon_{\text{max}}=-\frac{eB{\mathfrak{r}}\ln(3/2)}{2\pi\hbar}=-\frac{{\mathfrak{r}}B}{\phi_{0}}\ln\frac{3}{2}$
(S7)

from the energy of Lifshitz transition. The peak in
$\left|{\overset{\curvearrowleft}{S}}^{6}{\overset{\curvearrowright}{S}}^{4}\right|$
is further broadened by accounting for the finite temperature. To evaluate the
integral analytically, we can expand the exponents in Taylor series either
around the LT ($\epsilon=0$), or around the maximum ($\epsilon_{\text{max}}$),
or, one can approximate the integrand with a Gaussian according to a mean
value and dispersion. All these approaches lead to very similar results that
differ numerically by a few percent. Taylor-expanding the exponent around the
LT point and also approximating the $n_{F}^{\prime}$ with a Gaussian gives

$\displaystyle\int
d\epsilon\,n_{F}^{\prime}(\epsilon)|{\overset{\curvearrowleft}{S}}^{6}{\overset{\curvearrowright}{S}}^{4}|\propto$
$\displaystyle\int\frac{d\epsilon}{T}\,\exp\left[-\frac{5\pi^{2}\hbar^{2}}{2e^{2}{\mathfrak{r}}^{2}B^{2}}\left(\epsilon+\frac{2{\mathfrak{r}}B}{5\phi_{0}}\right)^{2}\right]\exp\left[\frac{-(\epsilon-\epsilon_{F})^{2}}{4T^{2}}\right],$

leading to Eq.(4) of the Main Text. Note that the resulting position of the
maximum in Eq.(4), $\epsilon_{\text{max
(saddle)}}=-\frac{2}{5}\frac{{\mathfrak{r}}B}{\phi_{0}}$ is numerically very
close to the value in Eq.(S7), because $\ln(3/2)=0.4055\approx 2/5$.

##  S9 Calculation of dispersion for tDBLG and G/hBN

Although our results do not depend on the details of dispersion, we used the
dispersion relations to plot the network and estimate the Gaussian curvature
near the saddle-points. The dispersion of tDBLG was calculated according to
Refs.García-Ruiz  _et al._ (2020); Garcia-Ruiz  _et al._ (2021), and the
details can be found in Supplementary to Ref. Tomić  _et al._ (2022)
(parameters of Slonczewski-Weiss-McClure model used are
$\gamma_{0}=3.16{\,\mathrm{eV}}$, $\gamma_{1}=0.381{\,\mathrm{eV}}$,
$\gamma_{3}=-0.38{\,\mathrm{eV}}$, $\gamma_{4}=0.14{\,\mathrm{eV}}$). For
aligned G/hBN system, we used Refs. Wallbank  _et al._ (2019); Lee  _et al._
(2016) (parameter values $U_{0}=8.5{\mathrm{\,meV}},\
U_{1}=-17{\mathrm{\,meV}},\ U_{3}=-14.7{\mathrm{\,meV}}$).

The kagomé networks of saddle-point trajectories in magnetic field look very
similar for tGBLG and G/hBN and shown in Fig.S5.

Figure S5: Comparison of Fermi contours at LT between (a) $1.9^{\circ}$ tDBLG
(conductance band), (b) valence band of G/hBN, (c) graphene monolayer doped up
to LT In all the cases we see the same LT topology where one FS turns into two
complimentary ones.

##  S10 Temperature dependence of kagomé oscillation amplitude and estimate of
coherence length

In Fig. S6 we show the full dataset for oscillation amplitude at doping where
it is maximal, $n\approx n_{max}$. The minimal real-space path required to see
the oscillations can be estimated by taking the length of a green path in
Fig.1 (same as the length of a blue path) in momentum space, and rescaling
that into the real space length using a factor $1/(eB)$, resulting in
${\mathcal{L}}\approx\frac{\hbar}{eB}\,2{\,\mathrm{nm}}^{-1}$. Note that
$2{\mathcal{L}}$ is comparable to the perimeter of an extended magnetic
supercell,
${\mathcal{L}}_{q}=4\sqrt{\frac{2}{\sqrt{3}}}q\sqrt{\mathcal{A}_{\varhexagon}}=\frac{\hbar}{eB}\,3.98{\,\mathrm{nm}}^{-1}$,
where $\mathcal{A}_{\varhexagon}\approx 46{\,\mathrm{nm}}^{2}$ for the studied
tDBLG.

Fitting the measured magnetic field dependence of oscillations amplitudes with
Eq.(4) and length ${\mathcal{L}}$, we extract the value of the scattering
parameter, $\ell^{-1}(T)$, which determines the loss of electrons from
ballistic propagation. The result of such fitting are displayed as black
circles in Fig.S7, where we also compare it with the inverse of the mean free
path, $\ell^{-1}_{\text{mfp}}$ (blue curve), determined from the conductivity
measured at the Lifshitz transition density and $B=0$ (here, we use average
velocity estimated from the computed dispersion, shown in Fig. 3 of the main
text). This comparison shows that momentum relaxation is slower than decay of
ballistic beam, which could be expected, based on that
$\ell^{-1}_{\text{mfp}}$ is more sensitive to the large momentum transfer upon
scattering, whereas $\ell^{-1}$ is caused by both large and small momentum
transfers. Both of these two quantities are temperature-dependent, indicating
the contribution of inelastic scattering processes, most likely, generated by
phonons. In contrast to that, their difference, shown in Fig.S7 for the
overlapping temperature interval for the available data using empty circles,
is almost temperature-independent, suggesting that the low-angle scattering in
the system is mostly elastic, rather than inelastic.

Having compared the lengths $\ell_{\text{mfp}}$ and $\ell$, we come back to
the discussion of their influence on conditions for the formation of Brown–Zak
magnetic minibands. We note that the latter are determined by Bragg scattering
from the superlattice with a period inflated $q$ times as compared to the
underlying moiré pattern Brown (1964); Zak (1964); Chen  _et al._ (2014). Note
that such magnetic supercell has a perimeter ${\mathcal{L}}_{q}\propto
B^{-1}$. For establishing periodicity, required for Bragg scattering, the
interfering electron waves should propagate ballistically over at least two
unit cells, exploring their areas for establishing the magnetic flux
commensurability with the flux quantum, so that the minimal condition for the
formation of magnetic miniband spectrum would require that

$\ell_{\text{mfp}}(T)>2{\mathcal{L}}_{q}(B).$ (S8)

This requirement is reflected on the $B-T$ parametric diagram in Fig. S8, as a
borderline between the the ’higher’ magnetic field regime, where magnetic
minibands would have a chance to form (green area), and a low-field region
(red area), where only kagomé oscillations appear, as a precursor of Brown–Zak
minibands formation. By inspection, we notice that the data displayed in Fig.
2 and discussed in Fig. 3a belong to the latter parametric interval.

Figure S6: Full data for $\ln A_{\text{osc}}$ as a function of $q$ for
different temperatures, the lines correspond to $\ln
A_{\text{osc}}=a_{0}-\frac{2{\mathcal{L}}}{\ell}-\ln(q)$ fit according to
Eq.(4) of the main text (note that
${\mathcal{L}}(q)\approx\frac{\hbar}{eB}\,2{\,\mathrm{nm}}^{-1}\propto q$,
producing the main contribution to slope of the lines).  Figure S7: Length
$\ell$ vs the mean free path, $l_{\text{mfp}}$ extracted from conductivity. We
plot $\ell^{-1}$, $l_{\text{mfp}}^{-1}$ and the difference
$\ell^{-1}-l_{\text{mfp}}^{-1}$, showing weak temperature dependence of
dephasing contributions. Figure S8: Regions where different type of phenomena
appear. For $B<B^{*}$ the Brown–Zak minibands are not yet formed, still the
kagomé oscillations are visible. For $B>B^{*}$, there are sizeable
contributions from path encircling multiple magnetic unit cells, leading to
formation of Brown–Zak minibands.


# Number of $A+B\neq C$ solutions in abelian groups and application to
counting independent sets in hypergraphs††This work was supported by the
Russian Federation Government (Grant number 075-15-2019-1926).

Aliaksei Semchankau††Lomonosov Moscow State University, Faculty of Mechanics
and Mathematics, Department of Dynamical Systems Theory, Leninskie Gory, 1,
Moscow, Russia; Steklov Mathematical Institute of Russian Academy of Sciences,
Gubkina 8, Moscow, Russia<EMAIL_ADDRESS>Dmitry
Shabanov††Moscow Institute of Physics and Technology, Laboratory of
Combinatorial and Geometric Structures, Institutskiy per. 9, Dolgoprudny,
Moscow Region, Russia; HSE University, Faculty of Computer Science,
Myasnitskaya Str. 20, Moscow, Russia<EMAIL_ADDRESS>Ilya Shkredov††Steklov Mathematical Institute of Russian Academy of Sciences,
Gubkina 8, Moscow, Russia; IITP RAS, Bolshoy Karetny per. 19, Moscow, Russia;
Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny,
Moscow Region, Russia<EMAIL_ADDRESS>
Abstract

The paper deals with a problem of Additive Combinatorics. Let ${\mathbf{G}}$
be a finite abelian group of order $N$. We prove that the number of subset
triples $A,B,C\subset{\mathbf{G}}$ such that for any $x\in A$, $y\in B$ and
$z\in C$ one has $x+y\neq z$ equals

$3\cdot 4^{N}+N3^{N+1}+O((3-c_{*})^{N})$

for some absolute constant $c_{*}>0$. This provides a tight estimate for the
number of independent sets in a special 3-uniform linear hypergraph and gives
a support for the natural conjecture concerning the maximal possible number of
independent sets in such hypergraphs on $n$ vertices.

##  1 Introduction

The paper deals with estimating the number of special subset triples in
abelian groups. Let us start with some motivation of the main result and
discuss the problem concerning counting the number of independent sets in
hypergraphs. Recall that if $H=(V,E)$ is a hypergraph, then its vertex subset
$W\subset V$ is _independent_ if it does not contain complete edges from $E$,
i.e. for any $A\in E$, $A\nsubseteq W$. Let $i(H)$ denote the total number of
independent sets in $H$.

###  1.1 Related work: counting independent sets in hypergraphs

The classical problem of extremal graph theory asks what is the maximal number
of independent sets in a $d$-regular graph $G$ on $n$ vertices. In 1991 Alon
[1, Section 5] conjectured that if $n$ is divisible by $2d$ then the number of
independent sets in this class of graphs is maximized when $G$ is a union of
$n/(2d)$ disjoint copies of $K_{d,d}$, the complete bipartite graph with equal
size part $d$. The conjecture was proved by Kahn [9] for bipartite graphs and,
finally, by Zhao [16] for all $d$-regular graphs on $n$ vertices. So, if $G$
is a $d$-regular graph on $n$ vertices, then

$i(G)\leqslant\left(i(K_{d,d})\right)^{\frac{n}{2d}}=\left(2^{d+1}-1\right)^{\frac{n}{2d}}.$

Note that the inequality holds even when $n$ is not divisible by $2d$.

It is quite natural to consider the same problem for hypergraphs, especially
for the class of linear hypergraphs. Recall that a hypergraph $H=(V,E)$ is
said to be _linear_ if every two of its distinct edges do not share more than
one common vertex, i.e. for any $A,B\in E$, $A\neq B$, we have

$|A\cap B|\leqslant 1.$

In recent paper [3] Cohen, Perkins, Sarantis and Tetali posed the following
general question: _which $d$-regular, $k$-uniform, linear hypergraph on a
given number of vertices has the most number of independent sets_?

Suppose that $H=(V,E)$ is a $d$-regular $k$-uniform linear hypergraph on $n$
vertices. The general container method (see, e.g., [12]) implies the following
upper bound on the number of independent sets:

$\frac{\log_{2}i(H)}{n}\leqslant\frac{k-1}{k}+O_{k}\left(\frac{\log_{2}^{2(k-1)/k}d}{d^{1/k}}\right).$
(1)

It is known that the first order term in the exponent in (1) is correct,
however the second is expected to be improved. Authors of [3] formulated the
following conjecture:

######  Conjecture 1.

For any $d$-regular $k$-uniform linear hypergraph $H$ on $n$ vertices the
following holds

$\frac{\log_{2}i(H)}{n}\leqslant\frac{k-1}{k}+\frac{\log_{2}k}{kd}.$ (2)

As in Alon’s conjecture for graphs, authors of [3] provide some example of a
small hypergraph, whose appropriate number of copies will give bound (2). In
the current paper we concentrate on the case of 3-uniform hypergraphs, so we
will describe the construction from [3] only for $k=3$.

Consider the following _mod_ hypergraph $H^{mod}_{d}=(V,E)$, where
$V=V_{1}\sqcup V_{2}\sqcup V_{3}$, every $V_{i}$ is equal to the cyclic group
$\mathbb{Z}/d\mathbb{Z}$ and

$E=\left\\{(x,y,z):\;x\in V_{1},\;y\in V_{2},\;z\in
V_{3},\;x+y=z\pmod{d}\right\\}.$

Clearly, this hypergraph is 3-uniform, 3-partite, $d$-regular and linear (for
each $x\in V_{1}$, $y\in V_{2}$ there is a unique $z\in V_{3}$ such that
$(x,y,z)\in E$). It was noticed in [3] that

$i(H^{mod}_{d})\geqslant 3\cdot 4^{d}-3\cdot 2^{d}+1.$

This lower bound estimates only the independent sets that do not touch some of
the parts $V_{1}$, $V_{2}$, $V_{3}$. But can we say that this bound is almost
tight? One the main results of our paper answers this question positively.

######  Theorem 2.

We have

$i(H^{mod}_{d})=3\cdot 4^{d}+O((4-c_{*})^{d})$ (3)

and moreover

$i(H^{mod}_{d})=3\cdot 4^{d}+d3^{d+1}+O((3-c_{*})^{d})$ (4)

for some absolute constant $c_{*}>0$.

As a consequence, we immediately obtain that if $H$ is a union of $n/(3d)$
disjoint copies of $H^{mod}_{d}$, then in view of (3)

$\frac{\log_{2}i(H)}{n}\leqslant\frac{1}{3d}\log_{2}\left(3\cdot
4^{d}+O((4-4\varepsilon)^{d})\right)=\frac{2}{3}+\frac{\log_{2}3}{3d}+O((1-\varepsilon)^{d})$

for some positive $\varepsilon$. This estimate strongly supports Conjecture 1,
see inequality (2). Surprisingly, that on the other hand, from our more
refined asymptotic formula (4) it follows that

$\frac{\log_{2}i(H)}{n}\geqslant\frac{2}{3}+\frac{\log_{2}3}{3d}+\Omega\left(\frac{3^{d}}{d4^{d}}\right)$

and hence one must add exponentially small error terms to make Conjecture 1
correct.

We also would like to mention that the problem of estimating the number of
independent sets has some natural extensions. One can count the strong
independent sets (see [3], [10], [5]) or general $j$-independent sets (see
[2]).

###  1.2 Problem statement for abelian groups

By analogy with $\mathbb{Z}/d\mathbb{Z}$ we can consider any finite abelian
group ${\mathbf{G}}$ and ask the following question: _what is the number of
triples $A,B,C\subset{\mathbf{G}}$ such that there is no solution $x+y=z$,
where $x\in A$, $y\in B$ and $z\in C$?_ If $|{\mathbf{G}}|=N$, then clearly,
this number is at least

$3\cdot 4^{N}-3\cdot 2^{N}+1,$

because any triple with one empty subset always fits. If we take one of $A$,
$B$ or $C$ equals a one–element set, that would add $N\cdot 3^{N+1}$ more
triples, and therefore the lower bound is

$3\cdot 4^{N}+N\cdot 3^{N+1}+O(2^{N}).$ (5)

Our first aim is to show that this lower bound is tight up to some summand
exponentially smaller than $4^{N}$. Then we further elaborate our argument to
find the correct exponent from (5).

The organization of this paper is the following. In section 2 we consider the
case when ${\mathbf{G}}$ is the prime field. In the next section 3 we deduce
the main result in the general case. The arguments in the prime case are
simpler but nevertheless, the scheme of the proof is similar to the case of
general abelian group. Finally, in sections 4 and 5 we will repeat the same
scheme but obtain a stronger estimate for the error term, using a new result
on structure of dense subsets $A,B$ of $\mathbb{F}_{p}$ with
$A+B\neq\mathbb{F}_{p}$, see Proposition 13 below. A similar instrument was
introduced for the first time in [13], where it has found already some
applications to the structure of sets with small Wiener norm. Also, it allows
to estimate size of $A(A+A)$ for any $A\subseteq\mathbb{F}_{p}$, see [13].
Thus this part of the paper has an independent interest for Additive
Combinatorics.

##  2 The case of the prime field

Let ${\mathbf{G}}$ be a finite abelian group, $\widehat{{\mathbf{G}}}$ be the
dual group. It is well–known that $\widehat{{\mathbf{G}}}$ is isomorphic to
${\mathbf{G}}$. In this paper we use the same letter to denote a set
$A\subseteq{\mathbf{G}}$ and its characteristic function
$A:{\mathbf{G}}\to\\{0,1\\}$. If $f,g:{\mathbf{G}}\to\mathbb{C}$ are
functions, then we write

$(f*g)(x)=\sum_{y\in{\mathbf{G}}}f(y)g(x-y)\,.$

The sumset of sets $A,B\subseteq{\mathbf{G}}$ is

$A+B=\\{a+b\leavevmode\nobreak\ :\leavevmode\nobreak\ a\in A,\,b\in B\\}\,.$

Similarly, one can define the difference set of $A$ and $B$. Given a prime $p$
we write $\mathbb{F}_{p}$ for the prime field. All logarithms are to base $2.$
The signs $\ll$ and $\gg$ are the usual Vinogradov symbols.

Now suppose that $A,B\subseteq{\mathbf{G}}$ are two sets and put
$\mathcal{C}=\mathcal{C}(A,B)={\mathbf{G}}\setminus(A+B)$. By the
Cauchy–Davenport inequality, see, e.g., [15, Theorem 5.4] if
${\mathbf{G}}=\mathbb{F}_{p}$ and $A+B\neq\mathbb{F}_{p}$, that is,
$\mathcal{C}\neq\emptyset$, we have

$|A+B|\geqslant|A|+|B|-1\,.$ (6)

We need one of the main results from [6]. Recall that for a given subset
$\Gamma\subseteq\widehat{{\mathbf{G}}}$ and a number $\varepsilon\in(0,1)$ the
set ${\rm Bohr}(\Gamma,\varepsilon)$ is called a Bohr neighbourhood (or a Bohr
set), see, e.g., [15, Section 4.4] if

${\rm Bohr}(\Gamma,\varepsilon)=\\{x\in{\mathbf{G}}\leavevmode\nobreak\
:\leavevmode\nobreak\
|\chi(x)-1|\leqslant\varepsilon\,,\forall\chi\in\Gamma\\}\,.$

Size of $\Gamma$ is called the dimension of ${\rm Bohr}(\Gamma,\varepsilon)$,
$\varepsilon$ is the radius of ${\rm Bohr}(\Gamma,\varepsilon)$ and it is
well–known the connection of size of Bohr sets with its dimension and radius,
e.g., see [15, Lemma 4.20]

$|{\rm
Bohr}(\Gamma,\varepsilon)|\geqslant(\varepsilon/2\pi)^{|\Gamma|}|{\mathbf{G}}|\,.$
(7)

Now we are ready to formulate the required extracted result from [6] about
shifts of Bohr sets in sumsets (actually one can check that the arguments work
for composite $p$ and even for an arbitrary finite abelian group as well). The
quantity $\log^{1/4}p$ below is not so important and it has chosen just for
convenience.

######  Theorem 3.

Let $p$ be a sufficiently large prime number and $A,B$ be sets from
$\mathbb{F}_{p}$, $|A|\geqslant\alpha p$, $|B|\geqslant\beta p$,
$\kappa=\sqrt{\alpha\beta}\geqslant\log^{-1/4}p$. Also, let
$|\mathcal{C}(A,B)|=\gamma p$ and $\omega\leqslant\gamma$ be a parameter,
$\omega\geqslant\exp(-(\log^{1/4}p))$. Then there is a shift
$x\in\mathbb{F}_{p}$ and a Bohr set $\mathcal{B}={\rm
Bohr}(\Gamma,\frac{\kappa}{64d})$, $|\Gamma|=d\leqslant
1000\kappa^{-2}\log(1/\omega)$, such that

$\left|(A+B)\cap(\mathcal{B}+x)\right|\geqslant\left(1-\frac{32\omega}{\kappa}\right)|\mathcal{B}|\,.$

A similar result on almost periodicity of convolutions was obtained in [4].
For the sake of the completeness we formulate a consequence of it. The
dependencies on the parameters in Theorems 3, 8 are slightly different but in
our regime ($\alpha,\beta\gg 1$) this is absolutely not important.

######  Theorem 4.

Let ${\mathbf{G}}$ be a finite abelian group, $N=|{\mathbf{G}}|$ and $A,B$ be
sets from ${\mathbf{G}}$, $|A|\geqslant\alpha N$, $|B|\geqslant\beta N$,
$\kappa=\sqrt{\alpha\beta}$. Also, let $q\geqslant 2$ and
$\varepsilon\in(0,1)$ be parameters. Then there is a shift
$x\in\mathbb{Z}/N\mathbb{Z}$ and a Bohr set $\mathcal{B}={\rm
Bohr}(\Gamma,c\varepsilon)$, $c>0$ is an absolute constant, $|\Gamma|\ll
q/\varepsilon^{2}$, such that

$\left|(A+B)\cap(\mathcal{B}+x)\right|\geqslant\left(1-\left(\frac{\varepsilon}{\kappa}\right)^{q}\right)|\mathcal{B}|\,.$
(8)

Indeed, by the main result of [4, Theorem 1.2] for almost periodicity of
$f(x):=(A*B)(x)$, we have

$\sum_{x\in{\mathbf{G}}}\sum_{t\in\mathcal{B}}\left((A*B)(x+t)-(A*B)(x)\right)^{q}\leqslant(\varepsilon\kappa
N)^{q}|\mathcal{B}|N$

and hence if (8) does not hold for any $x\in{\mathbf{G}}$, then we obtain a
contradiction in view of the simple bound

$\kappa^{2q}N^{q+1}\leqslant\sum_{x\in{\mathbf{G}}}(A*B)^{q}(x)\,.$

Let us write $\\#\\{R\ |\ P(R)\\}$ for the number of objects $R$, satisfying a
property $P$. We simply write $\\#\\{R\\}$, if the required property is clear
from the context. Also, let us introduce the following nonstandard notation.
For sets $X,Y\subseteq\mathbb{F}_{p}$ we write $X\neq Y$ if the equation $x=y$
has no solutions in $x\in X,y\in Y$.

Now we are ready to prove the main result of this section.

######  Theorem 5.

Let $p$ be a sufficiently large prime number. Then the number of sets $A,B,C$
from $\mathbb{F}_{p}$ such that $x+y\neq z$, $x\in A$, $y\in B$, $z\in C$
equals

$3\cdot 4^{p}+O((4-c_{*})^{p})\,,$ (9)

where $c_{*}>0$ is an absolute constant.

P r o o f. As we have noted above, if one takes $A,B$ or $C$ equals the empty
set and the rest is an arbitrary, then we obtain $3\cdot 4^{p}+O(2^{p})$ of
the required sets. The quantity $3\cdot 4^{p}$ is the main term and our task
is to estimate the rest.

Suppose that all sets $A,B,C$ are non–empty. Put $a=|A|$, $b=|B|$, $c=|C|$.
Clearly, $|\mathcal{C}(A,B)|\geqslant c$ hence by inequality (6), we get
$c\leqslant p-|A+B|=p-a-b+1$ and, similar bounds hold for $a$ and $b$. We
begin with a crude upper bound for the number of triples $A,B,C$ with small
$a,b$ or $c$. Namely, for $M\leqslant p/16$, say, one has

$\sum_{a\leqslant
M}\binom{p}{a}\sum_{b=1}^{p}\binom{p}{b}\sum_{c=1}^{p-a-b+1}\binom{p-a-b+1}{c}\leqslant
2^{p+1}\sum_{a\leqslant
M}\binom{p}{a}2^{-a}\sum_{b=1}^{p}\binom{p}{b}2^{-b}\leqslant$ (10) $\leqslant
4\cdot 3^{p}\binom{p}{M}2^{-M}\leqslant 4\cdot
3^{p}\left(\frac{ep}{2M}\right)^{M}<4(3.75)^{p}\,.$ (11)

Thus $a,b,c\geqslant p/16$.

Now we apply Theorem 3 with the parameters $\alpha=\beta=\gamma=\kappa=1/16$
and let $\omega\leqslant 2^{-10}$ be a sufficiently small number, which we
will choose later (an alternative way is to use Theorem 8). Let $d$ and
$\mathcal{B}$ be as in Theorem 3. The number of all possible shifted Bohr sets
is $p^{d+1}$. We know that $C$ intersects $\mathcal{B}$ by at most
$2^{9}\omega|\mathcal{B}|:=E$ points. Let us set
$C^{\prime}:=C\cap\mathcal{B}$. The number of all sets of size at most $E$
does not exceed $E\binom{p}{E}$. Put
$q=|\mathcal{B}|-E\geqslant|\mathcal{B}|/2>0$. Without loss of generality
(since it will not affect the exponent) we assume that $0\in B$, and therefore
$A\neq C$. For fixed $a,b,c$ we obtain

$\\#\\{A,B,C\ |\ A+B\neq C\\}\leqslant$ $\leqslant\\#\\{\mathcal{B}\\}\
\\#\\{C^{\prime}\\}\ \\#\\{C\setminus
C^{\prime}\subseteq\mathbb{F}_{p}\setminus\mathcal{B}\\}\ \\#\\{A\ |\ A\neq
C\\}\ \\#\\{B\ |\ B\neq C-A\\}\leqslant$ $\leqslant
p^{d+1}E\binom{p}{E}\binom{p-q}{c}\binom{p}{a}\binom{p-c-a+1}{b}.$

Summing it for all possible triples $a,b,c$ and considering obvious
inequalities $c\leqslant p-q,a\leqslant p-c,b\leqslant p-c-a+1$ we see that
the number of all possible triples $A,B,C$ is at most

$\sigma:=p^{d+1}E\binom{p}{E}\sum_{c=1}^{p-q}\binom{p-q}{c}\sum_{a=1}^{p-c}\binom{p}{a}\sum_{b=1}^{p-c-a+1}\binom{p-c-a+1}{b}\leqslant$
$\leqslant
2Ep^{d+1}\binom{p}{E}\sum_{c=1}^{p-q}\binom{p-q}{c}2^{p-c}\sum_{a=1}^{p-c}\binom{p}{a}2^{-a}\leqslant
2Ep^{d+1}\binom{p}{E}3^{p}\sum_{c=1}^{p-q}\binom{p-q}{c}3^{-c}\leqslant$
$\leqslant
p^{d+2}\left(\frac{ep}{E}\right)^{E}\left(\frac{3}{4}\right)^{q}\cdot
4^{p}\,.$ (12)

Put $l=\log(1/\omega)$. By estimate (7), we know that $|\mathcal{B}|\geqslant
q\gg p\exp(-Cl\cdot\log l)$, where $C>0$ is an absolute constant. Taking
$\omega$ sufficiently small constant, we can attain

$\left(\frac{ep}{E}\right)^{E}\ll\left(\frac{C_{1}\exp(Cl\log
l)}{\omega}\right)^{2^{10}\omega q}\leqslant\left(\frac{8}{7}\right)^{q}\,,$
(13)

where $C_{1}>0$ is another absolute constant. Thus the number $d\leqslant
1000\kappa^{-2}\log(1/\omega)$ is a constant and hence the multiple $p^{d+2}$
in (12) is negligible. Whence for sufficiently large $p$, we have, say,

$\sigma\ll\left(\frac{9}{10}\right)^{q}\cdot
4^{p}\ll\left(\frac{9}{10}\right)^{p\exp(-Cl\cdot\log l)}\cdot
4^{p}=(4-c_{*})^{p}\,.$

This completes the proof. $\hfill\Box$

##  3 The general case

The case of an arbitrary finite abelian group ${\mathbf{G}}$ requires more
refined arguments and generalizations. For example, inequality (6) is a
particular case of Kneser’s Theorem, see, e.g., [15, Theorem 5.5], which takes
place in any abelian group.

######  Theorem 6.

Let ${\mathbf{G}}$ be an abelian group, and $A,B\subseteq{\mathbf{G}}$ be
sets. Then

$|A+B|\geqslant|A+H|+|B+H|-|H|\,,$

where $H:=\\{x\in{\mathbf{G}}\leavevmode\nobreak\ :\leavevmode\nobreak\
A+B+x=A+B\\}$.

Now we are ready to prove an analogue of Theorem 5 and we will appeal to the
proof of this result.

######  Theorem 7.

Let ${\mathbf{G}}$ be a finite abelian group, $N=|{\mathbf{G}}|$. Then the
number of sets $A,B,C$ from ${\mathbf{G}}$ such that $x+y\neq z$, $x\in A$,
$y\in B$, $z\in C$ equals

$3\cdot 4^{N}+O((4-c_{*})^{N})\,,$ (14)

where $c_{*}>0$ is an absolute constant.

P r o o f. As in the proof of Theorem 5 (see calculations in (10), (11)) we
can assume that two sets from $A,B,C$ are large, say, $\Omega(N)$. Indeed,
using the notation of the proof of this Theorem, we see the rest can be
estimated as $2^{N}\binom{N}{M}^{2}\leqslant(3.75)^{N}$, say. Without losing
of the generality, suppose that $A$ and $B$ are large, i.e., $|A|,|B|\geqslant
c_{0}N$ with an absolute constant $c_{0}>0$. Put
$H:=\\{x\in{\mathbf{G}}\leavevmode\nobreak\ :\leavevmode\nobreak\
A+C+x=A+C\\}\leqslant{\mathbf{G}}$, and suppose firstly that $h:=|H|\leqslant
c_{1}N$, where $c_{1}>0$ is a sufficiently small absolute constant. Then
Kneser’s inequality gives us $|A+C|\geqslant|A|+|C|-|H|$. Hence combining this
with Theorem 3 or Theorem 8 and acting as in the proof of Theorem 5, we get

$Ep^{d+1}\binom{p}{E}\sum_{c=1}^{N-q}\binom{N-q}{c}\sum_{a=1}^{N-c}\binom{N}{a}\sum_{b=1}^{N-c-a+h}\binom{N-c-a+h}{b}\leqslant
N^{d+2}2^{h}\left(\frac{eN}{E}\right)^{E}\left(\frac{3}{4}\right)^{N}\cdot
4^{N}$

(again $d$ is as in Theorem 3). Hence as in inequality (13) we obtain the
required asymptotic formula (14) taking sufficiently small $c_{1}$. Thus
$h>c_{1}N$ and we can assume that $h<N$ because otherwise $A+C={\mathbf{G}}$.
Hence $h\leqslant N/2$. Put $n=N/h\geqslant 2$, $n\in\mathbb{Z}$. Let $k_{A}$
be the number of different cosets ${\mathbf{G}}/H$, which intersects our
random set $A$ (and, similarly, define $k_{B}$, $k_{C}$ for $B$ and $C$). From
Kneser’s Theorem we see that $k_{A},k_{B},k_{C}<n$ otherwise the correspondent
sumset coincides with the whole ${\mathbf{G}}$. Applying Theorem 6 again, we
obtain that $b\leqslant(n-k_{A}-k_{C}+1)h$. Using the arguments as in (10),
(11) one more time, combining with Theorem 6, we derive a crude upper bound
for the number of possible $A,B$ and $C$

$\sum_{k_{A}=1}^{n-1}\binom{n}{k_{A}}2^{k_{A}h}\sum_{k_{C}=1}^{n-k_{A}}\binom{n}{k_{C}}2^{k_{C}h}\sum_{k_{B}=1}^{n-k_{A}-k_{C}+1}\binom{n-k_{A}-k_{C}+1}{k_{B}}2^{k_{B}h}\leqslant$
$\leqslant
n2^{3n}\sum_{k_{A}=1}^{n-1}2^{k_{A}h}\sum_{k_{C}=1}^{n-k_{A}}2^{k_{C}h}\cdot
2^{(n-k_{A}-k_{C}+1)h}\leqslant n^{3}2^{3n}2^{N+h}\leqslant
n^{3}2^{3n}2^{3N/2}\,.$

Since $h\geqslant c_{1}N$, it follows that that $n\leqslant c_{1}^{-1}=O(1)$
and hence the multiple $n^{3}2^{3n}$ in the formula above is negligible. Also,
it is easy to see that the number of subgroups in ${\mathbf{G}}$ of index at
most $n=O(1)$ is $N^{O(1)}$ (e.g., consider the canonical homomorphism of left
cosets of $H$ in ${\mathbf{G}}$ to the symmetric group on $n$ letters) and
this latter number is also negligible. This completes the proof. $\hfill\Box$

##  4 An improvement

In this section we obtain an improvement of Theorem 5, using other new tools,
e.g., the Fourier transform on ${\mathbf{G}}$. Our main argument works in the
case of the prime field only although some statements hold to be true for a
general finite abelian group ${\mathbf{G}}$.

We denote the Fourier transform of a function $f:{\mathbf{G}}\to\mathbb{C}$ by
$\widehat{f},$ namely,

$\widehat{f}(\chi)=\sum_{x\in{\mathbf{G}}}f(x)\overline{\chi(x)}\,,$ (15)

where $\chi\in\widehat{{\mathbf{G}}}$ is an additive character on
${\mathbf{G}}$. We rely on the following basic identities. The first one is
called the Plancherel formula and its particular case $f=g$ is called the
Parseval identity

$\sum_{x\in{\mathbf{G}}}f(x)\overline{g(x)}=\frac{1}{|{\mathbf{G}}|}\sum_{\chi\in\widehat{{\mathbf{G}}}}\widehat{f}(\chi)\overline{\widehat{g}(\chi)}\,.$
(16)

Another particular case of (16) is

$\sum_{x\in{\mathbf{G}}}|(f*g)(x)|^{2}=\sum_{x\in{\mathbf{G}}}\Big{|}\sum_{y\in{\mathbf{G}}}f(y)g(x-y)\Big{|}^{2}=\frac{1}{|{\mathbf{G}}|}\sum_{\chi\in\widehat{{\mathbf{G}}}}\big{|}\widehat{f}(\chi)\big{|}^{2}\big{|}\widehat{g}(\chi)\big{|}^{2}\,.$
(17)

and the identity

$f(x)=\frac{1}{|{\mathbf{G}}|}\sum_{\chi\in\widehat{{\mathbf{G}}}}\widehat{f}(\chi)\chi(x)\,.$
(18)

is called the inversion formula.

Our proof is based on several auxiliary statements. First of all, we need a
consequence of the following result of J.M. Pollard [11] (also, see [14,
Corollary 1.2]). Given sets $A,B\subseteq\mathbb{F}_{p}$ and
$\varepsilon\in(0,1)$ we write $A+_{\varepsilon}B$ for the set of
$x\in\mathbb{F}_{p}$, having at least $\varepsilon p$ representations as sum
of $a+b$, $a\in A$, $b\in B$.

######  Theorem 8.

Let $A,B\subseteq\mathbb{F}_{p}$ be sets and $\varepsilon\in(0,1)$ be a real
number, such that $\sqrt{\varepsilon}p<|A|,|B|$. Then

$|A+_{\varepsilon}B|\geqslant\min\\{p,|A|+|B|\\}-2p\sqrt{\varepsilon}\,.$ (19)

We have the following combinatorial observation: if sets
$X,Y,Z\subseteq\mathbb{F}_{p}$ satisfy $X+Y\neq Z$, then they satisfy $X\neq
Z-Y$ as well. Let us prove the following robust version of this truism.

######  Lemma 9.

Let $X,Y,Z$ be nonempty subsets of $\mathbb{F}_{p}$ and $\delta\in(0,1)$ be a
parameter such that $Z\neq X+_{\delta}Y$. Also, let us assume that
$|X|\geqslant\varepsilon p$ for a certain $\varepsilon\in(0,1)$. Let $T>1$ be
a parameter. Then there exists $X^{\prime}\subseteq
X,|X^{\prime}|\leqslant|X|/T$ such that

$X\setminus{X^{\prime}}\neq Z-_{\delta T/\varepsilon}Y\,.$

P r o o f. Let $\eta:=\delta T/\varepsilon$. Suppose that for a certain
$X^{\prime}\subseteq X$ one has $X^{\prime}\subseteq Z-_{\eta}Y$. Then we have
at least $|X^{\prime}|\eta p$ triples $(x,y,z)$ such that $x+y=z$. By the
pigeonhole principle we see that there is $z_{0}$ with at least
$|X^{\prime}|\eta p/|Z|$ representations $z_{0}=x+y$. By assumption $Z\neq
X+_{\delta}Y$ and hence we derive the inequality $|X^{\prime}|\eta
p/|Z|\leqslant\delta p$, which gives
$|X^{\prime}|\leqslant|Z|\delta/\eta\leqslant p\delta/\eta\leqslant|X|/T$.
$\hfill\Box$

Secondly, we need the well–known Chang’s Theorem see, e.g., [15]. Recall that
for a set $A\subseteq{\mathbf{G}}$ and $\varepsilon\in(0,1]$ the spectrum
${\rm Spec\,}_{\varepsilon}(A)$ is defined as

${\rm
Spec\,}_{\varepsilon}(A):=\\{\chi\in\widehat{{\mathbf{G}}}\leavevmode\nobreak\
:\leavevmode\nobreak\ |\widehat{A}(\chi)|\geqslant\varepsilon|A|\\}\,.$

Also, the additive dimension ${\rm dim}(A)$ of a set $A\subseteq{\mathbf{G}}$
is size of the maximal dissociated subset of $A$, i.e., size of maximal
$\Lambda\subseteq A$ such that any equation
$\sum_{\lambda\in\Lambda}\varepsilon_{\lambda}\lambda=0$ with
$\varepsilon_{\lambda}\in\\{0,1,-1\\}$ implies $\varepsilon_{\lambda}=0$,
$\forall\lambda\in\Lambda$.

######  Theorem 10.

Let ${\mathbf{G}}$ be a finite abelian group, $A\subseteq{\mathbf{G}}$, and
$\varepsilon\in(0,1]$ be a real number. Then

${\rm dim}({\rm Spec\,}_{\varepsilon}(A))\leqslant
2\varepsilon^{-2}\log(|{\mathbf{G}}|/|A|)\,.$

Now let us formulate a rather general and simple result on level sets of an
arbitrary function.

######  Lemma 11.

Let ${\mathbf{G}}$ be an abelian group, $\delta$, $\alpha$, $\beta$ be real
numbers, $\alpha\leqslant\beta$, $Y$ be a set, and
$f,g:{\mathbf{G}}\to\mathbb{C}$ be functions. Put

$X=\\{x\in{\mathbf{G}}\leavevmode\nobreak\ :\leavevmode\nobreak\
f(x)\in[\alpha,\beta]\\}\,,$

and

$E^{-}=\\{x\in{\mathbf{G}}\leavevmode\nobreak\ :\leavevmode\nobreak\
f(x)\in[\alpha-\delta,\alpha)\\}\,,\quad
E^{+}=\\{x\in{\mathbf{G}}\leavevmode\nobreak\ :\leavevmode\nobreak\
f(x)\in(\beta,\beta+\delta]\\}\,.$

Suppose that for all $x\in{\mathbf{G}}\setminus Y$ one has
$\|f-g\|_{\infty}\leqslant\delta$ and write $g(x)=\sum_{j}g(x)S_{j}(x)$, where
$g$ differs by at most $\delta$ on some disjoint sets $S_{j}$. Then

$\bigsqcup_{j\leavevmode\nobreak\ :\leavevmode\nobreak\ S_{j}\cap(X\setminus
Y)\neq\emptyset}S_{j}=((X\setminus Y)\cup Y^{\prime})\cup E^{\prime}\cup
E^{{}^{\prime\prime}}\,,$ (20)

where $E^{\prime}\subseteq E^{-}$, $E^{\prime\prime}\subseteq E^{+}$,
$Y^{\prime}\subseteq Y$ are some sets.

P r o o f. Put $X^{\prime}=X\setminus Y$,
$X^{\prime\prime}=\bigsqcup_{j\leavevmode\nobreak\ :\leavevmode\nobreak\
S_{j}\cap X^{\prime}\neq\emptyset}S_{j}$. Clearly, $X^{\prime}\subseteq
X^{\prime\prime}$ because we can assume that
$\bigsqcup_{j}S_{j}={\mathbf{G}}$. Our task is to prove that
$X^{\prime\prime}$ coincides with the right–hand side of (20). If $x\in
X^{\prime\prime}$, then $f(x)\in[\alpha-\delta,\beta+\delta]$ and hence
$X^{\prime\prime}\subseteq X\sqcup E^{{}^{\prime}}\sqcup
E^{{}^{\prime\prime}}$, where $E^{\prime}:=E^{-}\cap X^{\prime\prime}$ and
$E^{\prime\prime}:=E^{+}\cap X^{\prime\prime}$. Let $Y^{\prime}:=Y\cap
X^{\prime\prime}$. Then $X^{\prime\prime}\subseteq((X\setminus Y)\cup
Y^{\prime})\cup E^{{}^{\prime}}\cup E^{{}^{\prime\prime}}$ but the reverse
inclusion is obvious (recall that $X^{\prime}=X\setminus Y\subseteq
X^{\prime\prime}$). This completes the proof. $\hfill\Box$

######  Remark 12.

In follows from the proof of Lemma 11 that the sets $S_{j}$ do not need to be
disjoint and one can consider any covering of ${\mathbf{G}}$ by a collection
of sets $S_{j}$. Nevertheless, then we lose disjointedness in (20) and it is
important sometimes to keep it, see [13].

Theorem 19, combining with Lemma 11 give us a useful result.

######  Proposition 13.

Let $A,B\subseteq\mathbb{F}_{p}$ be sets, $|A|=\alpha p$, $|B|=\beta p$,
$\delta,\varepsilon\in(0,1]$ be real parameters, and
$C=\mathcal{C}(A,B)\neq\emptyset$. Then there are sets $Y$, $W$, $C\setminus
Y\subseteq W$ such that the following holds

$|Y|\leqslant\varepsilon^{2}\alpha^{2}\beta\delta^{-2}p\,,\quad\quad
W=\bigcup_{j=1}^{t}S_{j}\,,$

where $S_{j}$ are shifts of a Bohr set $\mathcal{B}$ of dimension
$2\varepsilon^{-2}\log\alpha^{-1}$ and radius $\delta(\alpha\beta)^{-1/2}$,

$t\leqslant\exp(O(\varepsilon^{-2}\log\alpha^{-1}\cdot\log(\varepsilon\delta)^{-1}))\,,$
(21) $W\neq A+_{\delta}B,$

and

$|W|\leqslant p-|A|-|B|+2p\sqrt{\delta}\,.$

P r o o f. Let $f(x)=(A*B)(x)$ and $g(x)=p^{-1}\sum_{r\in{\rm
Spec\,}_{\varepsilon}(A)}\widehat{f}(r)e(rx)$. Due to formulae (16), (17), we
have

$\sum_{x}(f(x)-g(x))^{2}\leqslant p^{-1}\sum_{r\notin{\rm
Spec\,}_{\varepsilon}(A)}|\widehat{A}(r)|^{2}|\widehat{B}(r)|^{2}\leqslant\varepsilon^{2}|A|^{2}|B|$

and hence $\|f(x)-g(x)\|_{\infty}\leqslant\delta p$ outside of a set $Y$ of
size $|Y|\leqslant\varepsilon^{2}\alpha^{2}\beta\delta^{-2}p$. Apply Lemma 11
with the parameters $\alpha=\beta=0$ to the constructed functions $f$, $g$ and
to the set $Y$. Then $E^{\prime}$ is empty and $|E^{\prime\prime}|\leqslant
p-|A|-|B|+2\sqrt{\delta}p$ thanks to Theorem 19. Thus we put
$W:=E^{\prime\prime}=\\{x\leavevmode\nobreak\ :\leavevmode\nobreak\
f(x)\leqslant\delta\\}$. In other words, $W\neq A+_{\delta}B$. Further by
Chang’s Theorem 10 there exists a set $\Lambda$, $|\Lambda|\leqslant
2\varepsilon^{-2}\log\alpha^{-1}$ such that any element of ${\rm
Spec\,}_{\varepsilon}(A)$ can be expressed as
$\sum_{\lambda\in\Lambda}\xi_{\lambda}\lambda$,
$\xi_{\lambda}\in\\{0,1,-1\\}$. Put $\mathcal{B}={\rm
Bohr}(\Lambda,\zeta/|\Lambda|)$ with $\zeta=\delta(\alpha\beta)^{-1/2}$. Then
for any $b\in\mathcal{B}$ and for an arbitrary $x\in\mathbb{F}_{p}$, one has

$|g(x+b)-g(x)|\leqslant p^{-1}\sum_{r\in{\rm
Spec\,}_{\varepsilon}(A)}|\widehat{A}(r)\widehat{\mathcal{B}}(r)||e(rb)-1|\leqslant\zeta
p^{-1}\sum_{r}|\widehat{A}(r)||\widehat{\mathcal{B}}(r)|\leqslant\sqrt{\alpha\beta}\zeta
p=\delta p\,,$

where we have used the triangle inequality twice, the Cauchy–Schwarz
inequality and finally, formula (16). It means that the function $g$ differs
at most $\delta p$ on any shift of $\mathcal{B}$.

Now we want to find some $s_{1},s_{2},\ldots,s_{t}$ so that shifts
$s_{i}+\mathcal{B}$ cover the whole group $\mathbb{F}_{p}$. Let us consider a
Bohr set $\mathcal{B}^{\prime}$, identical to $\mathcal{B}$, but with twice
smaller radius. By the definition of Bohr sets it is clear that
$\mathcal{B}^{\prime}-\mathcal{B}^{\prime}\subseteq\mathcal{B}$. Let
$S:=\\{s_{1},s_{2},\ldots,s_{t}\\}$ be a maximal set such that shifts
$s_{i}+\mathcal{B}^{\prime}$ do not overlap. Due to its maximality, for any
$s\in\mathbb{F}_{p}$ there exists such $s_{i}$ so that
$s+\mathcal{B}^{\prime}$ intersects with $s_{i}+\mathcal{B}^{\prime}$, and
therefore $s\in s_{i}+\mathcal{B}^{\prime}-\mathcal{B}^{\prime}\subseteq
s_{i}+\mathcal{B}$. Therefore, shifts $s_{i}+\mathcal{B}$ cover the whole
group $\mathbb{F}_{p}$. Since $s_{i}+\mathcal{B}^{\prime}$ do not overlap, we
have $p\geqslant|S+\mathcal{B}^{\prime}|=|S||\mathcal{B}^{\prime}|$, and
therefore, by estimate (7), we see that the number $t$ of such shifts is at
most $p/|\mathcal{B}^{\prime}|\leqslant(4\pi|\Lambda|\zeta^{-1})^{|\Lambda|}$.
This completes the proof. $\hfill\Box$

Repeating the proof above, one can obtain the following slightly different

######  Proposition 14.

Let $A,B\subseteq\mathbb{F}_{p}$ be sets, $|A|=\alpha p$, $|B|=\beta p$,
$\eta,\delta,\varepsilon\in(0,1]$ be real parameters, and $C$ be a set, $C\neq
A+_{\eta}B$.

Then there are sets $Y$, $W$, $C\setminus Y\subseteq W$ such that the
following holds

$|Y|\leqslant\varepsilon^{2}\alpha^{2}\beta\delta^{-2}p\,,\quad\quad
W=\bigcup_{j=1}^{t}S_{j}\,,$

where $S_{j}$ are shifts of a Bohr set $\mathcal{B}$ of dimension
$2\varepsilon^{-2}\log\alpha^{-1}$ and radius $\delta(\alpha\beta)^{-1/2}$,

$t\leqslant\exp(O(\varepsilon^{-2}\log\alpha^{-1}\cdot\log(\varepsilon\delta)^{-1}))\,,$
(22) $W\neq A+_{(\eta+\delta)}B,$

and

$|W|\leqslant p-|A|-|B|+2p\sqrt{\eta+\delta}\,.$

We write Chernoff Bound, according to [15]:

######  Theorem 15 (Chernoff’s inequality).

Assume that $X_{1},\ldots,X_{n}$ are jointly independent random variables
where $|X_{i}-\mathbb{E}X_{i}|\leqslant 1$ for all $i$. Set
$X:=X_{1}+\ldots+X_{n}$ and let $\sigma:=\sqrt{\mathrm{Var}(X)}$ be the
standard deviation of $X$. Then for any $\lambda>0$

$\mathbb{P}(|X-\mathbb{E}X|\geqslant\lambda\sigma)\leqslant
2\max(e^{-\lambda^{2}/4},e^{-\lambda\sigma/2})$

Now we are ready to improve Theorem 5.

######  Theorem 16.

Let $p$ be a sufficiently large prime number and $\varepsilon\in(0,1]$ be any
real parameter. Then the number of sets $A,B,C$ from $\mathbb{F}_{p}$ such
that $x+y\neq z$, $x\in A$, $y\in B$, $z\in C$ equals

$3\cdot 4^{p}+3p\cdot 3^{p}+O((3-c_{*})^{p})\,,$ (23)

for some absolute constant $c_{*}>0$.

P r o o f. When one of the sets $A,B,C$ is empty we obtain the term $3\cdot
4^{p}$, and when one of them contains just one element, we get the term
$3p\cdot 3^{p}$. From now let us assume that all sets contain at least two
elements. We will consider two cases: when one of the sets is small, and when
all $A,B,C$ satisfy $|A|,|B|,|C|\gg\varepsilon p$, where $\varepsilon$ is a
sufficiently small constant.

Let us assume that $2\leqslant|B|\ll\varepsilon p$. Without loss of
generality, we can assume that $B$ contains elements $0$ and $-1$ (by
multiplying and shifting $A,B,C$ by the same constants). Therefore, $A\neq
C\cup(C+1)$. The number of pairs $A,C$ is at most

$\sum_{c=1}^{p}\sum_{C:\ |C|=c}\\#\\{A:A\neq C\cup(C+1)\\}=\
\sum_{c=1}^{p}\sum_{C:\ |C|=c}2^{p-|C\cup(C+1)|}\,.$ (24)

If we write $2^{p-|C|}$ instead of $2^{p-|C\cup(C+1)|}$, then the above sums
up to $3^{p}$. However,
$|C\cup(C+1)|=|C|+|C+1|-|C\cap(C+1)|=2|C|-|C\cap(C+1)|$, and we expect
$|C\cap(C+1)|$ to be equal roughly $\gamma^{2}p$ in average, for $|C|=\gamma
p$. Using Chernoff bound (15) we can estimate the number of cases when
$|C\cap(C+1)|\approx\gamma^{2}p$ (the latter has exponentially small
probability), and we obtain estimate $(3-c_{*})^{p}$ for sum (24) in this
case.

Summing it for all $B,2\leqslant\min(|A|,|B|,|C|)\ll\varepsilon p$ we obtain

$\\#\\{A,B,C:A+B\neq
C\\}\leqslant\\#\\{B\\}(3-c_{*})^{p}\leqslant\exp(O(\varepsilon
p))(3-c_{*})^{p}\leqslant(3-c_{*}+O(\varepsilon))^{p}.$

Now let us consider the case, when all $A,B,C$ satisfy
$|A|,|B|,|C|\gg\varepsilon p$.

Let us apply Proposition 13 to sets $A,B,C$ with the parameters
$\delta:=\delta_{C}$ and $\varepsilon:=\varepsilon_{C}$ ($\varepsilon_{C}$ and
$\delta_{C}$ will be defined later). We find sets $W_{C}$, $Y_{C}$, so that

$C\setminus Y_{C}\subseteq W_{C},\ \ W_{C}\neq A+_{\delta_{C}}B.$

Let us choose some $T>1$. By Lemma 9 we have $A\setminus A^{\prime}\neq
W_{C}-_{\delta_{C}T/\varepsilon}B$ for some $A^{\prime}$ of size at most
$|A|/T$. Let $\delta_{A}:=\delta_{C}T/\varepsilon$. Put $A_{1}:=A\setminus
A^{\prime}$. Apply Proposition 14 to the sets $A_{1},W_{C},B$ with the
parameters $\varepsilon:=\varepsilon_{A},\eta:=\delta:=\delta_{A}$ (again
$\varepsilon_{A}$ will be defined later). We see that for some $W_{A},Y_{A}$
one has

$A_{1}\setminus Y_{A}\subseteq W_{A},\ \ W_{A}\neq W_{C}-_{2\delta_{A}}B.$

In a similar way by Lemma 9 we have $B\setminus B^{\prime}\neq
W_{C}-_{2\delta_{A}T/\varepsilon}W_{A}$ for some $B^{\prime}$ of size at most
$|B|/T$. Let us put $\delta_{B}:=2\delta_{A}T/\varepsilon$. Let
$B_{1}:=B\setminus B^{\prime}$. By Proposition 14 applied to the sets
$B_{1},W_{A}$, and $W_{C}$ with parameters $\varepsilon:=\varepsilon_{B}$ and
$\eta:=\delta:=\delta_{B}$ ($\varepsilon_{B}$ will be defined later) we obtain
some $W_{B}$ and $Y_{B}$, so that

$B_{1}\setminus Y_{B}\subseteq W_{B},\ \ W_{B}\neq W_{C}-_{2\delta_{B}}W_{A}.$

We know that all $Y_{X}$ (with $X=A,B$ or $C$) satisfy the rough inequality
$|Y_{X}|\leqslant(\varepsilon_{X}/\delta_{X})^{2}p$. Let us put each
$\varepsilon_{X}$ to be equal $\varepsilon\delta_{X}$. Now we have bounds
$|Y_{X}|\leqslant\varepsilon^{2}p$ for $X=A,B,C$. It is easy to see that

$\\#\\{Y_{X}\\}\ll\exp(O(p\varepsilon)).$

Each $W_{X}$ is defined by a collection of at most
$t:=\exp(O(\varepsilon_{X}^{-2}\log\varepsilon^{-1}\log(\varepsilon\delta_{X})^{-1}))$
shifts of Bohr sets $S_{j}:=\mathcal{B}_{X}+s_{j}$, $s_{j}\in\mathbb{F}_{p}$.
We now guarantee that all $\delta_{X}$ (and therefore $\varepsilon_{X}$) do
satisfy $\delta_{X}\gg\varepsilon^{\Omega(1)}$, by setting
$T:=1/\varepsilon^{2}$ and $\delta_{C}=\varepsilon^{10}$ (notice that
$\delta_{A}=\delta_{C}T/\varepsilon=\varepsilon^{7}$, and
$\delta_{B}=2\delta_{A}T/\varepsilon=2\varepsilon^{4}$). From this we obtain
$t\ll\exp(O(\varepsilon^{-\Omega(1)}))$. We know that the dimension of our
Bohr set $\mathcal{B}_{X}$ is at most ${\rm dim}(\mathcal{B}_{X}),{\rm
dim}(\mathcal{B}_{X})\ll\varepsilon_{X}^{-2}\log\varepsilon^{-1}\ll\varepsilon^{-\Omega(1)}$.
Hence the number of possible sets $S_{j}=\mathcal{B}_{X}+s_{j}$ is at most

$\\#\\{\mathcal{B}_{X}\\}\\#\\{s_{1},\dots,s_{t}\\}\ll\binom{p}{{\rm
dim}(\mathcal{B}_{X})}\binom{p}{t}\ll p^{{\rm
dim}(\mathcal{B}_{X})+t}\ll\exp(O(\varepsilon p)).$

From here we obtain

$\\#\\{W_{X}\\}\ll\\#\\{\mathcal{B}_{X}\\}\\#\\{s_{1},\ldots,s_{t}\\}\ll\exp(O(\varepsilon
p))$

for all $X=A,B,C$.

Clearly, since
$|A_{1}|\leqslant|A|/T\ll\varepsilon^{2}p,|B_{1}|\ll\varepsilon^{2}p$, it
follows that

$\\#\\{A_{1}\\},\\#\\{B_{1}\\}\ll\binom{p}{\varepsilon^{2}p}\ll\exp(O(\varepsilon
p))\,.$

Finally, from the formula $W_{B}\neq W_{C}-_{2\delta_{B}}W_{A}$ and estimate
(19) from Theorem 19 we obtain that

$|W_{A}|+|W_{B}|+|W_{C}|\leqslant
p+2\sqrt{2\delta_{B}}p=p+O(\varepsilon^{2}p)$

For given sets $W_{A},W_{B},W_{C}$, satisfying
$|W_{A}|+|W_{B}|+|W_{C}|\leqslant p+O(\varepsilon^{2}p)$, and given
$Y_{A},Y_{B},Y_{C}$ the number of triples $A,B,C$ is bounded by

$\\#\\{A^{\prime}\\}\\#\\{A_{1}:A_{1}\setminus Y_{A}\subseteq W_{A}\\}\
\\#\\{B^{\prime}\\}\\#\\{B_{1}:B_{1}\setminus Y_{B}\subseteq W_{B}\\}\
\\#\\{C:C\setminus Y_{C}\subseteq W_{C}\\},$

which is at most

$\exp(O(\varepsilon p))2^{|W_{A}|}2^{|W_{B}|}2^{|W_{C}|}\ll 2^{p+O(\varepsilon
p)}.$

The number of sets $W_{A},W_{B},W_{C},Y_{A},Y_{B},Y_{C}$ can be estimated as

$\\#\\{W_{A}\\}\\#\\{Y_{A}\\}\ \\#\\{W_{B}\\}\\#\\{Y_{B}\\}\
\\#\\{W_{C}\\}\\#\\{Y_{C}\\}\ll\exp(O(\varepsilon p)),$

and the final result is $2^{p+O(\varepsilon p)}$. This completes the proof.
$\hfill\Box$

######  Remark 17.

It is highly likely, that when $2\leqslant|B|\ll\varepsilon p$, one can
estimate $\\#\\{A+B\neq C\\}$ as

$\exp(O(\varepsilon p))\max_{\gamma\in(0,1)}\binom{p}{\gamma p}2^{p-2\gamma
p+\gamma^{2}p}=\exp(O(\varepsilon
p))\max_{\gamma\in(0,1)}e^{(-\gamma\ln{\gamma}-(1-\gamma)\ln{(1-\gamma)}+(1-\gamma)^{2}\ln{2})p},$

which is optimized for $\gamma=0.2653\ldots$ and gives
$O(2.5926\ldots+O(\varepsilon))^{p}$.

######  Remark 18.

It is also highly likely, that consider separately the cases
$\min{(|A|,|B|,|C|)}$ $=2,3,4,\ldots,k$, we can obtain bound on
$\\#\\{A,B,C:A+B\neq C\\}$ of the form

$3\cdot 4^{p}+3p\cdot
3^{p}+Q_{2}(p)\lambda_{2}^{p}+Q_{3}(p)\lambda_{3}^{p}+\ldots+Q_{k}(p)\lambda_{k}^{p}+O((\lambda_{k}-c_{*})^{p}),$

where $4>3>\lambda_{2}>\lambda_{3}>\ldots>\lambda_{k}>c_{*}>0$ are absolute
constants, and $Q_{i}$ are some polynomials on $p$.

##  5 An improvement in the general case

The case of an arbitrary finite abelian group ${\mathbf{G}}$ requires more
refined arguments and generalizations. For example, we have more general
version of inequality (19) (see [7, Corollary 6.2]) which follows by the same
argument from the generalization of Pollard Theorem (see, e.g., [7,
Proposition 6.1] or [8, Theorem 1]).

######  Theorem 19.

Let ${\mathbf{G}}$ be an abelian group, and $A,B\subseteq{\mathbf{G}}$ be
sets. Let $\varepsilon\in(0,1)$ be such that
$\sqrt{\varepsilon}|{\mathbf{G}}|<|A|,|B|$. Let $H$ be a maximal proper
subgroup of ${\mathbf{G}}$. Then

$|A+_{\varepsilon}B|\geqslant\min(|{\mathbf{G}}|,|A|+|B|-|H|)-3\sqrt{\varepsilon}|{\mathbf{G}}|\,,$

######  Remark 20.

We will use this Theorem just to conclude the following observation. Let
$W_{a},W_{b},W_{c}$ be some subsets of ${\mathbf{G}}$ of size at least
$\varepsilon p$. Let us also assume they satisfy $W_{a}\neq
W_{b}+_{\delta}W_{c}$ for some $\delta\ll\varepsilon^{2}$. Then we have
$|W_{a}|\leqslant|{\mathbf{G}}|-|W_{b}+_{\delta}W_{c}|$. Thanks to Theorem 19
the last quantity is at most
$|{\mathbf{G}}|+|H|+3\sqrt{\delta}|{\mathbf{G}}|-|W_{b}|-|W_{c}|$ and this
implies

$|W_{a}|+|W_{b}|+|W_{c}|\leqslant|{\mathbf{G}}|+|H|+3\sqrt{\delta}|{\mathbf{G}}|\leqslant(3/2+O(\varepsilon))|{\mathbf{G}}|.$

Now we are ready to prove an analogue of Theorem 5 and we will appeal to the
proof of this result.

######  Theorem 21.

Let ${\mathbf{G}}$ be a finite abelian group, $N=|{\mathbf{G}}|$. Then the
number of sets $A,B,C$ from ${\mathbf{G}}$ such that $x+y\neq z$, $x\in A$,
$y\in B$, $z\in C$ equals

$3\cdot 4^{N}+3N\cdot 3^{N}+O((3-c_{*})^{N})\,,$ (25)

where $c_{*}>0$ is an absolute constant.

P r o o f. As in the proof of Theorem 16 we split the problem into two cases,
namely when one of the sets $A,B,C$ is small, and when all the sets have sizes
at least $\varepsilon N$ for a certain $\varepsilon\in(0,1)$. As in the proof
of Theorem 16 we see that the first case gives us at most $3\cdot
4^{N}+3N\cdot 3^{N}+O((3-c_{*})^{N})$ of such triples.

In the second case we repeat the same argumentation as for
${\mathbf{G}}=\mathbb{F}_{p}$ but we need to change it slightly, since we
cannot use the inequality from Theorem 19 in the general case. Replacing usage
of Theorem 19 with the more general Theorem 19 and taking into account Remark
20 we obtain for the sets $W_{a},W_{b},W_{c}$ that
$|W_{a}|+|W_{b}|+|W_{c}|\leqslant(1.5+O(\varepsilon))N$, which changes the
bound in the second case from $2^{N+O(\varepsilon)N}$ to
$2^{1.5N+O(\varepsilon)N}=(2\sqrt{2}+O(\varepsilon))^{N}$, and the proof is
complete.

$\hfill\Box$

######  Remark 22.

Let us take a finite abelian group ${\mathbf{G}}$, $N=|{\mathbf{G}}|$ which
has a subgroup $H$ of index $2$. Then for any $A,B\subseteq
H,C\subseteq{\mathbf{G}}\setminus H$ we have $A+B\neq C$ and
$\\#\\{A,B,C:A+B\neq C\\}\gg 2^{|H|}2^{|H|}2^{|H|}=2^{3N/2}=(2\sqrt{2})^{N}$.
Therefore, one cannot obtain a smaller exponent at least in the second case of
the proof.

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