# Exploring dynamics of $A_{4}$ flavour symmetry using low scale seesaw
mechanisms

Kalpana Bora1<EMAIL_ADDRESS>Maibam Ricky Devi1
<EMAIL_ADDRESS>Department of Physics, Gauhati University, Assam,
India1

###### Abstract

Low scale seesaw models, like low scale type II, inverse (ISS), and linear
seesaw (LSS) models provide an interesting mechanism to obtain light neutrino
masses and mixings, as they can be tested in future TeV scale experiments.
Discrete flavour symmetry groups like $A_{4}$ can be incorporated to explain
the flavour structure of particles. But, so far, nothing is known about
dynamics of flavour symmetry - scale of its breaking, or VEV alignment of the
flavon fields. In a recent study Ricky , we have investigated and shed light
on how to pinpoint the favoured VEV alignment of the flavon field using light
neutrino oscillation data. In this work, for the first time, we present an
analysis on dependence of light neutrino masses on scale of flavon VEV in
these three seesaw models, and comment on how this information can be used to
discriminate among them. We also discuss about the estimated value of the
constant $F$ which can constrain various coupling constants of the model, cut-
off scale of the theory and scale of flavour symmetry breaking. This study can
provide useful insight into the hitherto unknown dynamics of flavour symmetry
and hence can contribute as an important ingredient in the model building for
future studies.

##  I Introduction

Fundamental particles are found to follow certain pattern among them, with
reference to their masses and coupling strengths among themselves. Since in
standard model (SM), there are three families (generations) of particles, the
flavour group that can be used to describe them must have a triplet in it.
Many flavour groups have been used to explain this flavour structure of
particles - $A_{4}$, $S_{3}$, $S_{4}$ to name a few. Among these, $A_{4}$ is
the simplest group that can accommodate all the particles of the SM, and hence
in this work we have chosen this group for our analysis. Flavour symmetry is
not observed at low scales and hence it must be broken, and this can be done
by desiring that the flavon fields acquire VEV (vacuum expectation value).
Flavon fields couple to the particles of the theory to give desired coupling
or masses to them. Though many scenarios have been proposed, but there are no
experimental proofs in support of scale of this flavour symmetry breaking or
the direction in which the vacuum stabilizes.

With this motivation, recently we proposed a scenario Ricky  in which the
favoured direction of the VEV of these flavon fields can be pinpointed in low
scale seesaw models with $A_{4}$ flavour symmetry, with the help of some other
discrete symmetries like $Z_{3}$, $Z_{4}$ etc. We constructed the Lagrangian
such that the light neutrino mass matrix produces the neutrino oscillation
parameters within the $3\sigma$ range of their global best fit values. These
models have the advantage that they can be tested in near future experiments,
as the seesaw scales involved are of the order of few TeV. We chose a
benchmark value of $\dfrac{(\textrm{flavon VEV})}{\Lambda}$=0.1, for three low
scale seesaw models, where $\Lambda$=100 TeV is the cut-off scale of the
theory. So, the scale of the flavour symmetry breaking was considered to be
$\sim 10$ TeV, and we successfully obtained the light neutrino masses and
mixing pattern lying within the range of their global best fit values.

In this work, extending the work of our earlier work Ricky , we present an
interesting analysis on dynamics of the flavour symmetry breaking, i.e., how
the light neutrino mass depend on the ratio $\dfrac{(\textrm{flavon
VEV})}{\Lambda}$ in the three seesaw models. If in future experiments, we are
able to obtain this dependence of light neutrino mass on flavon VEV, then
through this study one can discriminate among the models. It may be noted that
so far, no information is available either on favoured VEV alignment of the
flavon VEV, or on the scale of flavour symmetry breaking, and hence the
findings of this work are important and may contribute in model building with
flavour symmetries. For some of the $A_{4}$ symmetry related works, the
interested reader can refer to Dinh:2016tu -Cai:2018upp .

##  II Dynamics of flavour symmetry

In Ricky , we discussed in detail, the structure of the Lagrangians in low
scale type-II seesaw Mohapatra:1980yp , ISS PhysRevLett.56.561
-PhysRevD.34.1642 , and LSS Malinsky:2005bi -Deppisch:2015cua  models with
$A_{4}$ flavour symmetry. We first constructed the Lagrangians in three
respective models, so that no unwanted terms appear in it, then obtained the
light neutrino mass matrices in all of them. Finally, we compared them with
the phenomenological neutrino mass matrix and obtained the equations for
unknown flavon VEVs, solving them using computation we obtained correlation
plots among the unknown neutrino oscillation parameters like - light neutrino
mass, atmospheric mixing angle $\theta_{23}$ (as its Octant is not yet fixed),
and leptonic CP violating phases. For more details, interested reader may
refer to Ricky . There we found that the light neutrino mass can be expressed
as:

$m_{\nu}=F\times(\textrm{flavon VEV}).$ (1)

The expressions of the dimensionless coupling constant F for the three seesaw
models can be summarized as follows Ricky :

$\displaystyle F=\dfrac{y\upsilon_{\Delta_{L}}}{\Lambda},\textrm{for type-II
seesaw model,}$ (2)

$\displaystyle
F=\dfrac{Y^{2}_{D}Y_{\mu}}{Y^{2}_{M}}[\dfrac{\upsilon^{2}_{h}\upsilon_{\rho^{\prime}}\upsilon^{\dagger}_{\rho^{\prime\prime}}}{\Lambda^{4}}],\textrm{for
ISS model,}$ (3)
$\displaystyle
F=\dfrac{2Y_{L}Y_{D}\upsilon^{2}_{h}\upsilon^{\dagger}_{\varepsilon}}{\Lambda^{2}Y_{M}\upsilon_{\rho}},\textrm{for
LSS model.}$ (4)

Here, $\upsilon_{\Delta_{L}}$ is the VEV of the additional (beyond SM) SU(2)
triplet scalar field that is responsible for generating light neutrino mass in
type-II see saw model, $y_{i}$($Y_{i}$) are dimensionless coupling constants
of various flavon fields with other fields of the SM in low scale type-II
seesaw(ISS, LSS) models, and $\upsilon_{\rho}$ etc are the VEVs of various
flavon fields. In ref.[1], we had used same values of $\dfrac{\langle
flavon\rangle}{\Lambda}$, lightest neutrino mass and heavy mass scale, in the
three models, just for the sake of comparison among them. The value of
constant F, as computed while solving simultaneous equations, was found to be
$\sim 10^{-2}$. Now the value of light neutrino mass scale can be fine-tuned
(to suit the experimentally observed values), if we adjust the values of
$\dfrac{\langle flavon\rangle}{\Lambda}$ and the seesaw scale, since we don’t
know any of them. Hence the couplings of the theory, and these two scales can
be adjusted, with the constraint that

$F\sim 10^{-2}\textrm{ (dimensionless constant)}$ (5)

as is evident from Equations (2),(3), and (4). For example, as mentioned
earlier, in say ISS,

$\dfrac{Y^{2}_{D}Y_{\mu}}{Y^{2}_{M}}=\Lambda\times 1.52\times
10^{-21}\textrm{eV}^{-1}$ (6)

for $\dfrac{\langle flavon\rangle}{\Lambda}=10^{-1}$, But same F can also be
obtained for $\dfrac{\langle flavon\rangle}{\Lambda}=10^{-2}$, and then the
couplings would satisfy

$\dfrac{Y^{2}_{D}Y_{\mu}}{Y^{2}_{M}}=\Lambda\times 1.52\times
10^{-18}\textrm{eV}^{-1}$ (7)

Another interesting feature observed from Equation (2),(3), and (4) is the
dependence of light neutrino mass, $m_{\nu}$ on $\left(\dfrac{\langle
flavon\rangle}{\Lambda}\right)^{x}$, where $x=1$ in type II and LSS, and $x=3$
in ISS model.

$m_{\nu}\propto\left(\dfrac{\langle flavon\rangle}{\Lambda}\right)^{x}$ (8)

Seesaw model $m_{\nu}$ (eV) for $\Lambda=100$ TeV, $\dfrac{\langle
flavon\rangle}{\Lambda}=0.1$ $\Lambda=1000$ TeV, $\dfrac{\langle
flavon\rangle}{\Lambda}=0.1$ Low-scale type-II seesaw
$y\upsilon_{\Delta}\dfrac{flavon}{\Lambda}$ y=0.1 and $\upsilon_{\Delta}=10$
eV y=0.1 and $\upsilon_{\Delta}=10$ eV Inverse seesaw
$\dfrac{Y^{2}_{D}Y_{\mu}}{Y^{2}_{M}}(\upsilon^{2}_{h}\upsilon_{\rho^{\prime}}\upsilon^{\dagger}_{\rho^{\prime\prime}})\dfrac{flavon}{\Lambda^{4}}$
$\dfrac{Y^{2}_{D}Y_{\mu}}{Y^{2}_{M}}=1.52\times 10^{-7}$
$\dfrac{Y^{2}_{D}Y_{\mu}}{Y^{2}_{M}}=1.52\times 10^{-6}$ Linear seesaw
$\dfrac{2Y_{L}Y_{D}\upsilon_{h}^{2}\upsilon_{\epsilon}^{\dagger}}{Y_{M}\upsilon_{\rho}}\dfrac{flavon}{\Lambda^{2}}$
$\dfrac{Y_{L}Y_{D}}{Y_{M}}=7.63\times 10^{-10}$
$\dfrac{Y_{L}Y_{D}}{Y_{M}}=7.63\times 10^{-9}$

Table 1: Values of light neutrino mass $m_{\nu}$ for some chosen values of
couplings and flavon VEVs for three low-scale seesaw models. It shows the
correlations among them, and that how $m_{\nu}$ scales with flavon VEV. Here,
$\upsilon_{\rho}$, $\upsilon_{\epsilon}$ etc are VEVs of various flavon fields
Ricky

.

In Table 1, we show the implications of Eq. (8), for the purpose of
demonstration - for similar decrease in light neutrino mass $m_{\nu}$ (say
$\dfrac{m_{\nu}}{y}$), keeping coupling constant, the $\dfrac{\langle
flavon\rangle}{\Lambda}$ decreases by a factor of $y$, in type II and LSS,
while in ISS, it decreases by a factor of $\dfrac{1}{y^{1/3}}$. This implies
that if experimentally observed value of light neutrino mass decreases by a
certain factor y, the scale of flavour symmetry decreases less in ISS
($\dfrac{1}{y^{1/3}}$).

Eqs. (6-8) and Table 1 are the novel results of our analysis, that
demonstrates new correlations among scale of symmetry and various coupling
constants, and are testable in future measurements. They give us useful
insight into the dynamics of flavour symmetry - that how the scale of flavour
symmetry breaking and various coupling constants can be adjusted, and hence
pinpointed so as to match these predictions with experimental measurements.

##  III Summary

To summarise, in this work, we presented a novel analysis on the dynamics of
$A_{4}$ flavour symmetry and new correlations among scale of symmetry and
various coupling constants, in context of low scale type-II, inverse seesaw
(ISS) and linear seesaw (LSS) models. Our new results are summarised in Eqs.
(6-8) and Table 1. We found interesting correlation among light neutrino mass,
scale of flavour symmetry breaking and various coupling constants of the
theory, which can be tested in future measurements. Thus, this study can
provide insightful guidelines on the important issue of dynamics of flavour
symmetry, which is still not resolved in flavour physics (in particular the
VEV scale of flavon and their coupling constants to other particles), and we
hope that future experiments would be able to support or refute the ideas
presented here.

## Acknowledgements

We thank RUSA grant (Govt. of India) to Gauhati University for the High
Performance Computing Facility in our computer Lab where this work was carried
out..

##  IV References

## References

  * (1) Maibam Ricky Devi and Kalpana Bora, arXiv:hep-ph/2103.10065 [hep-ph], arXiv:hep-ph/2112. 13004 [hep-ph], Communicated.
  * (2) D. N. Dinh, N. Anh Ky, P. Q. Văn and N. T. H. Vân, arXiv:1602.07437 [hep-ph].
  * (3) M. C. Chen, J. Huang, J. M. O’Bryan, A. M. Wijangco and F. Yu, JHEP 02 (2013), 021 doi:10.1007/JHEP02(2013)021 arXiv:1210.6982 [hep-ph].
  * (4) R. Kalita and D. Borah, Phys. Rev. D 92 (2015) no.5, 055012 doi:10.1103/PhysRevD.92.055012 [arXiv:1508.05466 [hep-ph]].
  * (5) N. Sarma, K. Bora and D. Borah, Eur. Phys. J. C 79 (2019) no.2, 129 doi:10.1140/epjc/s10052-019-6584-z [arXiv:1810.05826 [hep-ph]].
  * (6) S. Antusch, O. Fischer, A. Hammad and C. Scherb, JHEP 02 (2019), 157 doi:10.1007/JHEP02(2019)157 [arXiv:1811.03476 [hep-ph]].
  * (7) D. Borah, S. Mahapatra and N. Sahu, Nucl. Phys. B 968 (2021) 115407 [arXiv:2009.06294 [hep-ph]].
  * (8) M. Hirsch, S. Morisi and J. W. F. Valle, Phys. Lett. B 679 (2009), 454-459 doi:10.1016/j.physletb.2009.08.003 [arXiv:0905.3056 [hep-ph]].
  * (9) M. Sruthilaya, R. Mohanta and S. Patra, Eur. Phys. J. C 78 (2018) no.9, 719 doi:10.1140/epjc/s10052-018-6181-6 [arXiv:1709.01737 [hep-ph]].
  * (10) D. Borah and B. Karmakar, Phys. Lett. B 780 (2018), 461-470 doi:10.1016/j.physletb.2018.03.047 [arXiv:1712.06407 [hep-ph]].
  * (11) P. Sahu, S. Patra and P. Pritimita, [arXiv:2002.06846 [hep-ph]].
  * (12) V. Cirigliano, A. Kurylov, M. J. Ramsey-Musolf and P. Vogel, Phys. Rev. D 70 (2004), 075007 doi:10.1103/PhysRevD.70.075007 [arXiv:hep-ph/0404233 [hep-ph]].
  * (13) F. F. Deppisch, Fortsch. Phys. 61 (2013), 622-644 doi:10.1002/prop.201200126 [arXiv:1206.5212 [hep-ph]].
  * (14) S. Verma, M. Kashav and S. Bhardwaj, Nucl. Phys. B 946 (2019), 114704 doi:10.1016/j.nuclphysb.2019.114704 [arXiv:1811.06249 [hep-ph]].
  * (15) I. Girardi, S. T. Petcov, A. J. Stuart and A. V. Titov, Nucl. Phys. B 902 (2016), 1-57 doi:10.1016/j.nuclphysb.2015.10.020 [arXiv:1509.02502 [hep-ph]].
  * (16) S. T. Petcov and A. V. Titov, Phys. Rev. D 97 (2018) no.11, 115045 doi:10.1103/PhysRevD.97.115045 [arXiv:1804.00182 [hep-ph]].
  * (17) S. T. Petcov and A. V. Titov, Int. J. Mod. Phys. A 33 (2018) no.31, 1844024 doi:10.1142/S0217751X18440244
  * (18) D. Borah and B. Karmakar, Phys. Lett. B 789 (2019), 59-70 doi:10.1016/j.physletb.2018.12.006 [arXiv:1806.10685 [hep-ph]].
  * (19) I. Sethi and S. Patra, J. Phys. G 48 (2021) 10, 105003. [arXiv:1909.01560 [hep-ph]].
  * (20) H. Cai, T. Nomura and H. Okada, Nucl. Phys. B 949 (2019), 114802 doi:10.1016/j.nuclphysb.2019.114802 [arXiv:1812.01240 [hep-ph]].
  * (21) R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 23 (1981), 165 doi:10.1103/PhysRevD.23.165
  * (22) R. N. Mohapatra, Phys. Rev. Lett. 56 (1986), 561-563 doi:10.1103/PhysRevLett.56.561
  * (23) R. N. Mohapatra and J. W. F. Valle, Phys. Rev. D 34 (1986), 1642 doi:10.1103/PhysRevD.34.1642
  * (24) M. Malinsky, J. C. Romao and J. W. F. Valle, Phys. Rev. Lett. 95 (2005), 161801 doi:10.1103/PhysRevLett.95.161801 [arXiv:hep-ph/0506296 [hep-ph]].
  * (25) F. F. Deppisch, L. Graf, S. Kulkarni, S. Patra, W. Rodejohann, N. Sahu and U. Sarkar, Phys. Rev. D 93 (2016) no.1, 013011 doi:10.1103/PhysRevD.93.013011 [arXiv:1508.05940 [hep-ph]].
  * (26) H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto, Lect. Notes Phys. 858 (2012), 1-227 doi:10.1007/978-3-642-30805-5

##  Appendix A $A_{4}$ product rules

A discrete non-abelian group of even permutations of four objects called
$A_{4}$ flavour symmetry is considered here to describe the flavour structure
of fundamental particles. It is also the symmetry group of a tetrahedron. This
group consist of four irreducible representations: three one-dimensional and
one three dimensional whose conventional notations are
$\bf{1},\bf{1^{\prime}},\bf{1^{\prime\prime}}$ and $\bf{3}$ respectively
Ishimori:2012zz . Their product rules are given as

$\bf{1}\otimes\bf{1}=\bf{1}$
$\bf{1^{\prime}}\otimes\bf{1^{\prime}}=\bf{1^{\prime\prime}}$
$\bf{1^{\prime}}\otimes\bf{1^{\prime\prime}}=\bf{1}$
$\bf{1^{\prime\prime}}\otimes\bf{1^{\prime\prime}}=\bf{1^{\prime}}$
$\bf{3}\otimes\bf{3}=\bf{1}\otimes\bf{1^{\prime}}\otimes\bf{1^{\prime\prime}}\otimes\bf{3}_{a}\otimes\bf{3}_{s}$

where $a$ and $s$ in the subscript refers to anti-symmetric and symmetric
parts respectively. If $(a_{1},b_{1},c_{1})$ and $(a_{2},b_{2},c_{2})$ are two
triplets, then their direct product can be decomposed into the direct sum as
follows

$\bf{1}=a_{1}a_{2}+b_{1}c_{2}+c_{1}b_{2}$
$\bf{1^{\prime}}=c_{1}c_{2}+a_{1}b_{2}+b_{1}a_{2}$
$\bf{1^{\prime\prime}}=b_{1}b_{2}+c_{1}a_{2}+a_{1}c_{2}$
$\bf{3}_{s}=(2a_{1}a_{2}-b_{1}c_{2}-c_{1}b_{2},2c_{1}c_{2}-a_{1}b_{2}-b_{1}a_{2},$
$\bf{2b_{1}b_{2}-a_{1}c_{2}-c_{1}a_{2})}$
$\bf{3}_{a}=(b_{1}c_{2}-c_{1}b_{2},a_{1}b_{2}-b_{1}a_{2},c_{1}a_{2}-a_{1}c_{2})$


# On the Band Functions and Bloch Functions.

O. A. Veliev
Dogus University, Istanbul, Turkey.
e-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we consider the continuity of the band functions and Bloch
functions of the differential operators generated by the differential
expressions with periodic matrix coefficients.

Key Words: Differential operator, band functions, Bloch functions.

AMS Mathematics Subject Classification: 34L20, 47F05, 35 P15.

##  1 Introduction

In this paper, we consider the continuity of the Bloch eigenvalues, band
functions and Bloch functions with respect to the quasimomentum of the
differential operator $T$, generated in the space $L_{2}^{m}(\mathbb{R}^{d})$
by formally self-adjoint differential expression

$Tu={\textstyle\sum\limits_{\left|\alpha\right|=2s}}Q_{\alpha}D_{\alpha}u+{\textstyle\sum\limits_{\left|\alpha\right|\leq
2s-1}}Q_{\alpha}(x)D_{\alpha}u+Bu,$ (1)

where $d\geq 1$, $\alpha=(\alpha_{1},\alpha_{2},...,\alpha_{d})$ is a multi-
indeks,
$\left|\alpha\right|=\alpha_{1}+\alpha_{2}+\cdot\cdot\cdot+\alpha_{d}$,

$D_{\alpha}=\left(\frac{1}{i}\frac{\partial}{\partial
x_{1}}\right)^{\alpha_{1}}\left(\frac{1}{i}\frac{\partial}{\partial
x_{2}}\right)^{\alpha_{2}}...\left(\frac{1}{i}\frac{\partial}{\partial
x_{d}}\right)^{\alpha_{d}},$

$Q_{\alpha}$, for each $\alpha$, is an $m\times m$ matrix. Here the entries of
$Q_{\alpha}(x)$, for $\left|\alpha\right|\leq 2s-1$, are bounded functions
that are periodic with respect to a lattice $\Omega$, the entries of
$Q_{\alpha}$, for $\left|\alpha\right|=2s$, are real numbers and $B$ is a
bounded operator commuting with the shift operators
$S_{\omega}:u(x)\rightarrow u(x+\omega)$, for $\omega\in\Omega$. Note that
$L_{2}^{m}(\mathbb{R}^{d})$ is the space of the vector-valued functions
$f=\left(f_{1},f_{2},...,f_{m}\right)$ with $f_{k}\in L_{2}(\mathbb{R}^{d})$,
for $k=1,2,...,m$.

To describe briefly the scheme of this paper, let us introduce the following
notations. Let $\Gamma$ be the lattice dual to $\Omega$. Denote by $F$ and
$F^{\star}$ the fundamental domains of the lattices $\Omega$ and $\Gamma$,
respectively. Let$\ T_{t}$ be the operator generated in $L_{2}^{m}(F)$ by (1)
and the quasiperiodic conditions

$u(x+\omega)=e^{i\left\langle t,\omega\right\rangle}u(x),\
\forall\omega\in\Omega,$ (2)

where $t\in F^{\star}$ and $\left\langle\cdot,\cdot\right\rangle$ is the inner
product in $\mathbb{R}^{d}$. Denote by $SBC(H)$ the set of below-bounded self-
adjoint operators, with compact resolvents acting in the Hilbert space $H$. If

$T_{t}\in SBC(L_{2}^{m}(F)),$ (3)

then the spectrum of $T_{t}$ consists of the eigenvalues. Let
$\lambda_{1}(t),\lambda_{2}(t),...$ be the eigenvalues of $T_{t}$ numerated in
the nondecreasing order

$\lambda_{1}(t)\leq\lambda_{2}(t)\leq\cdot\cdot\cdot.$ (4)

The eigenvalues and eigenfunctions of $T_{t}$, for $t\in F^{\star}$, are
called the Bloch eigenvalues and Bloch functions of $T$, respectively. The
function $\lambda_{n}:F^{\star}\rightarrow\mathbb{R}$ is said to be the $n$-th
band function of $T$.

To investigate the continuity of the band functions and Bloch functions of
$T$, first we study the continuity of the eigenvalues and eigenfunctions for
the family $\left\\{A_{t}:t\in E\right\\}$ of the operators $A_{t}\in SBC(H)$,
where $E$ is a metric space and $A_{t}$ continuously depends on $t\in E$ in
the generalized sense defined in [2, Chap. 4, page 202]. For the simplicity of
the notations, the parameter of the family $\left\\{A_{t}:t\in E\right\\}$ is
also denoted by $t$, the eigenvalues of $A_{t}$ also are denoted by
$\lambda_{1}(t),\lambda_{2}(t),...$ and are numerated in the nondecreasing
order as in the case $T_{t}$ (see (4)). In Theorems 1-3, we prove that the
eigenvalue $\lambda_{n}(t)$ and corresponding eigenspace of $A_{t}$
continuously depend on $t\in E$. Moreover, we prove that, if
$\lambda_{n}(t_{0})$ is a simple eigenvalue, then the eigenvalues
$\lambda_{n}(t)$ are simple in some neighborhood of $t_{0}$ and the
corresponding normalized eigenfunctions $\Psi_{n,t}$ of $A_{t}$ can be chosen
so that $\left\|\Psi_{n,t}-\Psi_{n,t_{0}}\right\|\rightarrow 0$ as
$t\rightarrow t_{0}$, where $\left\|\cdot\right\|$ is the norm of $H$. Then,
we use these results for the family $\left\\{T_{t}:t\in F^{\star}\right\\}$ of
the operators $T_{t}.$ Namely, in Theorem 4 we prove that, if (3) and an
additional condition (10) hold, then $T_{t}$ continuously depends on $t\in
F^{\star}$ in the generalized sense and hence the results obtained for $A_{t}$
continue to hold for $T_{t}$. In order not to deviate from the purpose of this
paper, we do not discuss the conditions on (1) under which (3) and (10) hold,
since the consideration of these conditions is long technical. Nevertheless,
in the end of this paper (see Corollary 1), we give an example which shows
that the obtained results include the continuity of the band functions and
Bloch functions of a large class of the differential operators generated by
(1).

Note that, as was noted in the physical and mathematical literatures (see for
example the books [1, 3, 5] and paper [4]), the continuity of the band
functions of the Schrödinger operator with a periodic potential and some other
periodic operators is well-known or immediately follows from the perturbation
theory described in [2]. Here we do not discuss all the numerous
investigations about this, and give only the detailed proof of the continuity
of the band functions and Bloch functions for the large class of the systems
of differential operators by using the generalized convergence in the sense of
[2]. Finally, we note that, using the results of [Chapter 13 of [3]] in a
standard way, one can easily verify that $T$ can be expressed as the direct
integral of the operators $T_{t}$, for $t\in F^{\star}$ and that the spectrum
of the operator $T$ is the union of the spectra of the operators $T_{t}$ for
$t\in F^{\star}$.

##  2 Main Results

First, we consider the general operators by using the generalized convergence
of the closed operators defined in [2, Chap. 4, page 202]. Let us state the
definition of the generalized convergence in the notation which is suitable
for this investigation. Let $A$ and $B$ be self-adjoint operators in the
Hilbert space $H$. Since the self-adjoint operators are closed, the graphs
$G(A)=\left\\{\left(u,Au\right):u\in D(A)\right\\}$ and
$G(B)=\left\\{\left(u,Bu\right):u\in D(B)\right\\}$ of these operators are
closed linear manifolds of the Hilbert space $H^{2}$. Let $S_{A}$ and $S_{B}$
be respectively the unit spheres of $G(A)$ and $G(B)$. The gap $g(T,S)$
between the self-adjoint operators $A$ and $B$ is defined as follows:

$g(A,B)=\max\left\\{\sup_{v\in S_{A}}\left(\inf_{w\in
G(B)}\left\|w-v\right\|\right),\sup_{v\in S_{B}}\left(\inf_{w\in
G(A)}\left\|w-v\right\|\right)\right\\}.$ (5)

We use the following definitions and theorems of [2].

###### Definition 1

(See [2, Chap. 4, page 202]). We say that, the sequence of the closed
operators $A_{n}$ converges to the closed operator $A$ in the generalized
sense, if $g(A_{n},A)\rightarrow 0$ as $n\rightarrow\infty$.

This convergence determines the following definition of the continuity.

###### Definition 2

We say that, a family of the closed operators $A_{t}$ is continuous at $E$ in
the generalized sense, if for each $t_{0}\in E$ and for any sequence
$\left\\{t_{n}\right\\}\subset E$ converging to $t_{0}$,
$g(A_{t_{n}},A_{t_{0}})\rightarrow 0$ as $n\rightarrow\infty$.

Besides these definitions, we use Theorems 3.1 and 3.16 of [2] (see Chap. 4,
pages 208 and 212). Let us formulate these theorems in the suitable form by
using the above notations. Theorem 3.1 of [2] states that if the closed curve
$\gamma$ belongs to the resolvent set $\rho(A)$ of $A\in SBC(H),$ then there
exists $\delta>0$ such that $\gamma\in\rho(B)$, for any operator $B\in SBC(H)$
with $g(A,B)<\delta$. Theorem 3.16 of [2] states that, in this case the
numbers of the eigenvalues (counting the multiplicity) of $A$ and $B$ lying
inside $\gamma$ are the same. Moreover,

$\text{ }\left\|\int_{\gamma}(A-\lambda
I)^{-1}d\lambda-\int_{\gamma}(B-\lambda I)^{-1}d\lambda\right\|\rightarrow
0\text{,}$ (6)

as $g(A,B)\rightarrow 0$.

Now using these statements, we consider a continuous family $\left\\{A_{t}\in
SBC(H):t\in E\right\\}$ of the operators by using the following notations.

###### Notation 1

In (4), the eigenvalues of the operator $A_{t_{0}}$ are denoted by counting
the multiplicity. Let us denote by $\mu_{1}(t_{0}),\mu_{2}(t_{0}),...$ the
eigenvalues of $A_{t_{0}}$ without counting the multiplicity. In other words,
$\mu_{1}(t_{0})<\mu_{2}(t_{0})<\cdot\cdot\cdot$ are the eigenvalues of
$A_{t_{0}}$ with the multiplicities $k_{1},k_{2},...$, respectively. Since
$\lambda_{n}(t_{0})\rightarrow\infty$ as $n\rightarrow\infty$, for each $n$
there exists $p$ such that $n\leq k_{1}+k_{2}+\cdot\cdot\cdot+k_{p}$. This
notation, together with the notation (4) implies that

$\lambda_{1}(t_{0})=\lambda_{2}(t_{0})=\cdot\cdot\cdot=\lambda_{s_{1}}(t_{0})=\mu_{1}(t_{0}),$
$\lambda_{s_{1}+1}(t_{0})=\lambda_{s_{1}+2}(t_{0})=\cdot\cdot\cdot=\lambda_{s_{2}}(t_{0})=\mu_{2}(t_{0}),$

and

$\lambda_{s_{p-1}+1}(t_{0})=\lambda_{s_{p-1}+2}(t_{0})=\cdot\cdot\cdot=\lambda_{s_{p}}(t_{0})=\mu_{p}(t_{0}),$

where $s_{p}=k_{1}+k_{2}+\cdot\cdot\cdot+k_{p}$ and $n\leq s_{p}$.

Now we prove the following theorems for the continuous family
$\left\\{A_{t}\in SBC(H):t\in E\right\\}$.

###### Theorem 1

Let $\left\\{A_{t}:t\in E\right\\}$ be the continuous family of the operators
$A_{t}\in SBC(H)$, $t_{0}\in E$ and $\left\\{t_{k}\in
E:k\in\mathbb{N}\right\\}$ be a sequence converging to $t_{0}.$ Then, for
every $r>0$ satisfying the inequality

$r<\frac{1}{2}\min_{j=1,2,...p}\left(\mu_{j+1}(t_{0})-\mu_{j}(t_{0})\right),$
(7)

there exists $N>0$ such that, each of the operators $A_{t_{k}}$, for $k>N$,
has $k_{j}$ eigenvalues in the interval
$\left(\mu_{j}(t_{0})-r,\mu_{j}(t_{0})+r\right)$, where $j=1,2,...,p$, the
numbers $p$ and $\mu_{j}(t_{0})$ are defined in Notation 1. Moreover, the
eigenvalues lying in $\left(\mu_{j}(t_{0})-r,\mu_{j}(t_{0})+r\right)$ are
$\lambda_{s_{j-1}+1}(t_{k}),\lambda_{s_{j-1}+2}(t_{k}),...,\lambda_{s_{j}}(t_{k})$,
where $s_{0}=0$ and $s_{j}$’s, for $j\geq 1$, are defined in Notation 1.

Proof. By (7), Definition 2 and Theorem 3.1 of [2], there exists $N>0$ such
that the circle
$D_{j}(r)=\left\\{z\in\mathbb{C}:\left|z-\mu_{j}(t_{0})\right|=r\right\\}$
belongs to the resolvent set of the operators $A_{t_{k}}$, for $k>N$.
Therefore, by Theorem 3.16 of [2], $A_{t_{k}}$ has $k_{j}$ eigenvalues inside
$D_{j}$. In the same way, we prove that $A_{t_{k}}$, for $k>N$, has no
eigenvalues in the intervals $(-\infty,\mu_{1}(t_{0})-r]$ and
$\left[\mu_{j}(t_{0})+r,\mu_{j+1}(t_{0})-r\right]$, for $j=1,2,...,p$, since
$A_{t_{0}}$ has no eigenvalues in those intervals. Therefore, the eigenvalues
of $A_{t_{k}}$, for $k>N$, lying in
$\left(\mu_{j}(t_{0})-r,\mu_{j}(t_{0})+r\right)$ are
$\lambda_{s_{j-1}+1}(t),\lambda_{s_{j-1}+2}(t),...,\lambda_{s_{j}}(t)$, for
$j=1,2,...,p$.

Now, we are ready to prove the main results for this continuous family.

###### Theorem 2

Let $\left\\{A_{t}:t\in E\right\\}$ be the continuous family of the operators
$A_{t}\in SBC(H)$. Then, the eigenvalues (4) of $A_{t}$ continuously depend on
$t\in E$.

Proof. Consider the sequence $r_{s}\rightarrow 0$. By Theorem 1, there exists
$N_{s}$ such that, if $k>N_{s}$ then the eigenvalues
$\lambda_{s_{j-1}+1}(t_{k}),\lambda_{s_{j-1}+2}(t_{k}),...,\lambda_{s_{j}}(t_{k})$
lie in $\left(\mu_{j}(t_{0})-r_{s},\mu_{j}(t_{0})+r_{s}\right)$. Therefore, we
have $\left|\lambda_{n}(t_{k})-\lambda_{n}(t_{0})\right|<r_{s}$, for
$k>N_{s}$, since $n\in[s_{j-1}+1,s_{j}]$ and
$\lambda_{n}(t_{0})=\mu_{j}(t_{0})$ due to Notation 1. Now letting $s$ tend to
infinity, we obtain that $\lambda_{n}(t_{k})\rightarrow\lambda_{n}(t_{0})$ as
$k\rightarrow\infty$, for any sequence $\left\\{t_{k}\in
E:k\in\mathbb{N}\right\\}$ converging to $t_{0}$. Thus $\lambda_{n}$ is
continuous at $t_{0}$.

Now, suppose that $\lambda_{n}(t_{0})$ is a simple eigenvalue. Then, by
Theorem 1, for every $r>0$ satisfying (7), there exists $N>0$ such that the
operator $A_{t_{k}}$, for $k>N$, has a unique eigenvalue (counting the
multiplicity) in the interval
$\left(\lambda_{n}(t_{0})-r,\lambda_{n}(t_{0})+r\right)$ and this eigenvalue
is $\lambda_{n}(t_{k})$. Since $\lambda_{n}(t_{k})$ is a simple eigenvalue, we
have

$\int_{\gamma}(A_{t_{k}}-\lambda
I)^{-1}fd\lambda=(f,\Psi_{n,t_{k}})\Psi_{n,t_{k}},$ (8)

for $f\in H$, $k=0$, and $k>N$, where $\gamma$ is a closed curve that encloses
only the eigenvalue $\lambda_{n}(t_{k})$ and $\Psi_{n,t_{k}}$ is a normalized
eigenfunction corresponding to $\lambda_{n}(t_{k})$. Using (6) and (8), we
obtain the following relations:

$(f,\Psi_{n,t_{k}})\Psi_{n,t_{k}}\rightarrow(f,\Psi_{n,t_{0}})\Psi_{n,t_{0}}$

as $t_{k}\rightarrow t_{0}$, for each $f\in H$. Here replacing $f$ by
$\Psi_{n,t_{0}}$, we obtain

$(\Psi_{n,t_{0}},\Psi_{n,t_{k}})\Psi_{n,t_{k}}\rightarrow\Psi_{n,t_{0}}$ (9)

and $\left|(\Psi_{n,t_{0}},\Psi_{n,t_{k}})\right|\rightarrow 1$ as
$t_{k}\rightarrow t_{0}$. Since a normalized eigenfunction is still normalized
if it is multiplied by a factor of absolute value $1$, $\Psi_{k,t_{k}}$ and
$\Psi_{k,t_{0}}$ can be chosen so that

$\arg(\Psi_{n,t_{0}},\Psi_{n,t_{k}})=0,$ (10)

that is,
$\left|(\Psi_{n,t_{0}},\Psi_{n,t_{k}})\right|=(\Psi_{n,t_{0}},\Psi_{n,t_{k}})$.
Thus

$(\Psi_{n,t_{0}},\Psi_{n,t_{k}})\rightarrow 1,$

as $t_{k}\rightarrow t_{0}$. This, together with (9) implies that

$\left\|\Psi_{n,t_{k}}-\Psi_{n,t_{0}}\right\|\leq\left\|(1-(\Psi_{n,t_{0}},\Psi_{n,t_{k}}))\Psi_{n,t_{k}}\right\|+\left\|(\Psi_{n,t_{0}},\Psi_{n,t_{k}})\Psi_{n,t_{k}}-\Psi_{n,t_{0}}\right\|\rightarrow
0,$

as $t\rightarrow t_{0}$. It means that, $\Psi_{n,t}$ is continuous at $t_{0}$.
Thus, we have

###### Theorem 3

Let $\left\\{A_{t}:t\in E\right\\}$ be the continuous family of the operators
$A_{t}\in SBC(H)$ and $t_{0}\in E$. If $\lambda_{n}(t_{0})$ is a simple
eigenvalue, then the eigenvalues $\lambda_{n}(t)$ are simple in some
neighborhood of $t_{0}$ and the corresponding normalized eigenfunctions
$\Psi_{n,t}$ can be chosen so that
$\left\|\Psi_{n,t}-\Psi_{n,t_{0}}\right\|\rightarrow 0$ as $t\rightarrow
t_{0}$.

Now, we consider the application of Theorems 1-3 to the differential operators
$T_{t}$ for $t\in F^{\star}$ defined in $L_{2}^{m}(F)$ by (1) and (2), where
$F$ and $F^{\star}$ is the fundamental domains of the lattices $\Omega$ and
$\Gamma.$ Without loss of generality, we assume that the measure of the
fundamental domain $F$ of the lattice $\Omega$ is $1$.

###### Theorem 4

Suppose that (3) holds and there exist $\varepsilon>0$ and $c>0$ such that

$\left\|Tu+cu-
Bu\right\|+\left\|u\right\|\geq\varepsilon{\textstyle\sum\limits_{\left|\alpha\right|\leq
2s-1}}\left\|D_{\alpha}u\right\|,$ (11)

for all $u\in D(T_{t})$ and $t\in F^{\ast}$, where $D(T_{t})$ is the domain of
definition of $T_{t}$, $Tu$ and $B$ are defined in (1). Then
$\left\\{T_{t}:t\in F^{\ast}\right\\}$ is a continuous family of the operators
in the sense of Definition 2 and hence Theorems 1-3 continue to hold for this
family.

Proof. For the simplicity of the notation, denote $T_{t}+cI-B$ by $A_{t}$. By
Theorem 2.23 (c) of [2, Chap. 4, page 206], it is enough to prove that, the
sequence $\left\\{A_{t_{n}}\right\\}$ converges to $A_{t_{0}}$ in the
generalized sense, for any sequence $\left\\{t_{n}\right\\}\subset F^{\ast}$
converging to $t_{0}$. If $u\in D(A_{t_{0}})$, then $e^{i\left\langle
t_{n}-t_{0},x\right\rangle}u\in D(A_{t_{n}})$. By the product rule of
differentiation, we have

$A_{t_{n}}e^{i\left\langle t_{n}-t_{0},x\right\rangle}u=e^{i\left\langle
t_{n}-t_{0},x\right\rangle}A_{t_{0}}u+\left|t_{n}-t_{0}\right|Su,$ (12)

where $e^{i\left\langle t_{n}-t_{0},x\right\rangle}A_{t_{0}}u$ is the sum of
the terms of $A_{t_{n}}e^{i\left\langle t_{n}-t_{0},x\right\rangle}u$ for
which no differentiation is applied to $e^{i\left\langle
t_{n}-t_{0},x\right\rangle}$ and $\left|t_{n}-t_{0}\right|Su$ is the sum of
the other terms of $A_{t_{n}}e^{i\left\langle t_{n}-t_{0},x\right\rangle}u,$
that is, is the sum of the terms of $A_{t_{n}}e^{i\left\langle
t_{n}-t_{0},x\right\rangle}u$ for which some differentiations are applied to
$e^{i\left\langle t_{n}-t_{0},x\right\rangle}$. It is clear that, $Su$ is the
differential expression of order $\leq 2s-1$, with bounded coefficients.
Therefore, there exists $M$ such that

$\left\|Su\right\|<M{\textstyle\sum\limits_{\left|\alpha\right|\leq
2s-1}}\left\|D_{\alpha}u\right\|.$ (13)

Now, using (11)-(13), (5) and Definition 1, we prove that
$\left\\{A_{t_{n}}\right\\}$ converges to $A_{t_{0}}$ in the generalized
sense. Let $S_{t}$ be the unit sphere of $G\left(A_{t}\right)$. If
$v=(u,A_{t_{0}}u)\in S_{t_{0}}$, then $w=(e^{i\left\langle
t_{n}-t_{0},x\right\rangle}u,A_{t_{n}}e^{i\left\langle
t_{n}-t_{0},x\right\rangle}u)\in G(A_{t_{n}})$ and

$\left\|w-v\right\|\leq\left\|e^{i\left\langle
t_{n}-t_{0},x\right\rangle}u-u\right\|+\left\|A_{t_{n}}e^{i\left\langle
t_{n}-t_{0},x\right\rangle}u-A_{t_{0}}u\right\|.$ (14)

For the second term of (14), we have

$\left\|A_{t_{n}}e^{i\left\langle
t_{n}-t_{0},x\right\rangle}u-A_{t_{0}}u\right\|\leq\left\|A_{t_{n}}e^{i\left\langle
t_{n}-t_{0},x\right\rangle}u-e^{i\left\langle
t_{n}-t_{0},x\right\rangle}A_{t_{0}}u\right\|+\left\|\left(e^{i\left\langle
t_{n}-t_{0},x\right\rangle}-1\right)A_{t_{0}}u\right\|.$ (15)

On the other hand, using (12), (13) and (11), we obtain

$\left\|A_{t_{n}}e^{i\left\langle
t_{n}-t_{0},x\right\rangle}u-e^{i\left\langle
t_{n}-t_{0},x\right\rangle}A_{t_{0}}u\right\|=\left|t_{n}-t_{0}\right|\left\|Su\right\|<\left|t_{n}-t_{0}\right|M\left({\textstyle\sum\limits_{\left|\alpha\right|\leq
2s-1}}\left\|D_{\alpha}u\right\|\right)\leq$ (16)
$\frac{\left|t_{n}-t_{0}\right|M}{\varepsilon}\left(\left\|A_{t_{0}}u\right\|+\left\|u\right\|\right).$

Moreover, the inclusion $(u,A_{t_{0}}u)\in S_{t_{0}}$ implies that
$\left\|A_{t_{0}}u\right\|+\left\|u\right\|<2$. This inequality, together with
(14)-(16) implies that there exists $C>0$ such that

$\sup_{v\in S_{t_{0}}}\left(\inf_{w\in
G(A_{t_{n}})}\left\|w-v\right\|\right)\leq C\left|t_{n}-t_{0}\right|.$

In the same way, we prove that

$\sup_{v\in S_{t_{n}}}\left(\inf_{w\in
G(A_{t_{0}})}\left\|w-v\right\|\right)\leq C\left|t_{n}-t_{0}\right|.$

Therefore by (5), $g(A_{t_{n}},A_{t_{o}})\rightarrow 0$ as
$n\rightarrow\infty$. Hence, by Definition 2, $\left\\{T_{t}:t\in
F^{\ast}\right\\}$ is a continuous family in the generalized sense.

By the standard estimation, one can find a large class of differential
operators for which (11) and (3) hold and hence Theorems 1-3 continue to hold.
For example, we find a class of the partial differential operators satisfying
(11) and (3) in the following manner. Let us write (1) in the form

$Tu=Lu+Pu+Bu,$ (17)

where

$Lu={\textstyle\sum\limits_{\left|\alpha\right|=2s}}Q_{\alpha}D_{\alpha}u,\text{
}Pu={\textstyle\sum\limits_{\left|\alpha\right|\leq 2s-1}}Q_{\alpha}(x)u.$

First we note that, there exists $c_{1}>0$ for which

$\left\|Pu\right\|\leq c_{1}{\textstyle\sum\limits_{\left|\alpha\right|\leq
2s-1}}\left\|D_{\alpha}u\right\|,$ (18)

since the coefficients of the expression $Pu$ are matrices with bounded
entries. Here and in subsequent relations we denote by $c_{1},c_{2},...$ the
positive constants. Then, we find a condition on the main term $Lu$ of (16)
for which, there exists $c>0$ such that

$\left\|Lu+cu\right\|\geq\left(c_{1}+\varepsilon\right){\textstyle\sum\limits_{\left|\alpha\right|\leq
2s-1}}\left\|D_{\alpha}u\right\|.$ (19)

It is clear that (11) follows from (19) and (18). The condition on $L$, namely
on the principal symbol of (1) is the following. Suppose that

$L={\textstyle\sum\limits_{\left|\alpha\right|=2s}}q_{\alpha}I_{m}D_{\alpha},$
(20)

where $I_{m}$ is $m\times m$ unit matrix, and there exists $c_{2}>0$ such that

${\textstyle\sum\limits_{\left|\alpha\right|=2s}}q_{\alpha}\left(\xi\right)^{\alpha}\geq
c_{2}\left|\xi\right|^{2s},$ (21)

for each $\xi\in\mathbb{R}^{d}$, where $q_{\alpha}\in\mathbb{R}$ and
$\left(\xi\right)^{\alpha}=\xi_{1}^{\alpha_{1}}\xi_{2}^{\alpha_{2}}...\xi_{d}^{\alpha_{d}}$.
Now we prove that if (20) and (21) hold, then there exists $c>0$ such that
(19) and hence (11) hold. Moreover, we prove that (3) also holds and obtain
the following consequence of Theorem 4.

###### Corollary 1

Suppose that $L$ has the form (20), inequality (21) holds, and that $Pu$ is a
formally self-adjoint differential expression. Then, the Bloch eigenvalues (4)
of the differential operator $T_{t}$ generated in $L_{2}^{m}(F)$ by (1) and
(2) continuously depend on $t\in F^{\ast}$. Moreover, if $\lambda_{n}(t_{0})$
is a simple eigenvalue, then the eigenvalues $\lambda_{n}(t)$ are simple in
some neighborhood of $t_{0}$ and the corresponding normalized eigenfunctions
$\Psi_{n,t}$ can be chosen so that
$\left\|\Psi_{n,t}-\Psi_{n,t_{0}}\right\|\rightarrow 0$ as $t\rightarrow
t_{0}$.

Proof. First we prove the validity of (19) if (20) and (21) hold. Instead of
(19), we prove

$\left\|Lu+cu\right\|^{2}\geq\left(c_{1}+\varepsilon\right)^{2}\left({\textstyle\sum\limits_{\left|\alpha\right|\leq
2s-1}}\left\|D_{\alpha}u\right\|\right)^{2},$ (22)

for all $u\in L_{2}^{m}(F)$ such that $\partial^{\alpha}u\in L_{2}^{m}(F)$,
for $\left|\alpha\right|\leq 2s$. It follows from (20) and (21) that

$\left\|L\left(e^{i\left\langle\gamma,x\right\rangle}e_{k}\right)+ce_{k}e^{i\left\langle\gamma,x\right\rangle}\right\|^{2}\geq\left(c_{2}\left|\gamma\right|^{2s}+c\right)^{2},$

where $\\{e_{1},e_{2},...,e_{m}\\}$ is the standard basis of $\mathbb{R}^{m}$
and $\gamma\in\Gamma$. Using this and the decomposition

$u={\textstyle\sum\limits_{\gamma\in\Gamma}}\left({\textstyle\sum\limits_{k=1,2,...,m}}u_{\gamma,k}e^{i\left\langle\gamma,x\right\rangle}e_{k}\right)$
(23)

of$\ u$ by the orthonormal basis
$\left\\{e^{i\left\langle\gamma,x\right\rangle}e_{k}:\gamma\in\Gamma,\text{
}k=1,2,...,m\right\\}$, we obtain that

$\left\|\left(Lu\right)+cu\right\|^{2}\geq{\textstyle\sum\limits_{\gamma\in\Gamma}}\left({\textstyle\sum\limits_{k=1,2,...,m}}\left|u_{\gamma,k}\left(c_{2}\left|\gamma\right|^{2s}+c\right)\right|^{2}\right).$
(24)

On the other hand, using (23) we have

$\left({\textstyle\sum\limits_{\left|\alpha\right|\leq
2s-1}}\left\|D_{\alpha}u\right\|\right)^{2}\leq
c_{3}{\textstyle\sum\limits_{\left|\alpha\right|\leq
2s-1}}\left\|D_{\alpha}u\right\|^{2}=c_{3}{\textstyle\sum\limits_{\gamma\in\Gamma}}\left({\textstyle\sum\limits_{k=1,2,...,m}}\left|u_{\gamma,k}{\textstyle\sum\limits_{\left|\alpha\right|\leq
2s-1}}\left|\gamma\right|^{\alpha}\right|^{2}\right).$ (25)

Therefore, (22) follows from (24) and (25), if we prove that there exists
$c>0$ such that

$c_{2}\left|\gamma\right|^{2s}+c>\sqrt{c_{3}}\left(c_{1}+\varepsilon\right){\textstyle\sum\limits_{\left|\alpha\right|\leq
2s-1}}\left|\gamma\right|^{\alpha},$ (26)

for all $\gamma\in\Gamma$. Hence, it remains to prove (26). Clearly, there
exists $c_{4}$ such that $c_{2}\left|\gamma\right|^{2s}$ is greater than the
right-hand side of (26), for $\left|\gamma\right|>c_{4}$. Besides, the number
of $\gamma\in\Gamma$ satisfying $\left|\gamma\right|\leq c_{4}$ is finite.
Therefore, there exists $c>0$ such that (26) holds, for
$\left|\gamma\right|\leq c_{4}$. Thus, (26) and hence (22), (19) and (11)
hold.

Now, we prove that (3) holds, too. Let $L_{t}$ and $P_{t}$ be respectively the
operators generated by the differential expression $Lu$ and $Pu$ and boundary
conditions (2). Since the orthonormal basis
$\left\\{e^{i\left\langle\gamma+t,x\right\rangle}e_{k}:\gamma\in\Gamma,k=1,2,...,m\right\\}$
of $L_{2}^{m}(F)$ is the set of the eigenfunctions of $L_{t}$ and all
eigenvalues of $L_{t}$ are nonnegative numbers, we have $L_{t}\in
SBC(L_{2}^{m}(F))$.

Now, we prove that $T_{t}\in SBC(L_{2}^{m}(F))$, that is, (3) holds. It
readily follows from the proof of (26) that, there exists $c>0$ such that

$c_{2}\left|\gamma\right|^{2s}+c>2\sqrt{c_{3}}\left(c_{1}+\varepsilon\right){\textstyle\sum\limits_{\left|\alpha\right|\leq
2s-1}}\left|\gamma\right|^{\alpha}.$

Therefore, arguing as in the proof of (19), we see that, there exist $c_{5}$
such that

$\left\|Pu+Bu\right\|\leq
c_{5}\left\|u\right\|+\tfrac{1}{2}\left\|Lu+cu\right\|.$ (27)

It shows that, $P_{t}+B$ is relatively bounded with respect to $L_{t}+cI$ with
relative bound smaller than $1$. Therefore, by Theorem 4.11 of [2] (Chap.5,
page 291), $T_{t}+cI=L_{t}+cI+P_{t}$ $+B$ is self-adjoint and bounded from
below. It remains to prove that $T_{t}+cI$ has a compact resolvent. For this,
we use Theorem 3.17 of [2, Chap. 4, page 214] which implies that, if (27)
holds and there exists $\mu\in\rho(L+cI)$ such that

$c_{5}\left\|\left(L_{t}+cI-\mu
I\right)^{-1}\right\|+\frac{1}{2}\left\|\left(L_{t}+cI\right)\left(L_{t}+cI-\mu
I\right)^{-1}\right\|<1,$ (28)

then $T_{t}+cI$ has a compact resolvent. Since $L_{t}+cI$ is a self-adjoint
operator, it is clear and well-known that (see, for example (4.9) of [2, Chap.
5, page 290]), if $\mu=ic_{6},$ then

$\left\|\left(L_{t}+cI-\mu I\right)^{-1}\right\|\leq\frac{1}{c_{6}},\text{
}\left\|\left(L_{t}+cI\right)\left(L_{t}+cI-\mu I\right)^{-1}\right\|\leq 1.$

Thus, if $c_{6}>2c_{5}$, then (28) holds. Hence, $T_{t}+cI$ is a below-bounded
self-adjoint operator with a compact resolvent and (3) holds. Now, the proof
follows from Theorem 4.

Now we show that in some cases, instead of (10), one can constructively define
normalized eigenfunctions $\Psi_{n,t}$ that continuously depend on $t.$ First
of all, let us note the following obvious statement. If $\lambda_{n}(t)$ is a
simple eigenvalue, then the set of all normalized eigenfunctions corresponding
to $\lambda_{n}(t)$ is
$\left\\{e^{i\alpha}\Psi_{n,t}:\alpha\in[0,2\pi)\right\\},$ where $\Psi_{n,t}$
is a fixed normalized eigenfunction. If the eigenvalue $\lambda_{n}(a)$ is
simple, then there exists a neighborhood $U(a)$ of $a$ such that for $t\in
U(a)$ the eigenvalue $\lambda_{n}(t)$ is also simple and the equality

$\int_{\gamma}(T_{t}-\lambda
I)^{-1}e^{i(a,x)}d\lambda=(e^{i(a,x)},e^{i\alpha}\Psi_{n,t})e^{i\alpha}\Psi_{n,t}=(e^{i(a,x)},\Psi_{n,t})\Psi_{n,t}$
(29)

is true for any choice of the normalized eigenfunction $\Psi_{n,t},$ where
$\gamma$ is a closed curve that encloses only the eigenvalue $\lambda_{n}(t).$
Since the projection operator onto the subspace corresponding to the
eigenvalue $\lambda_{n}(t)$ continuously depends on $t\in U(a)$ and the norm
is a continuous function, it follows from (29) that
$\left|(\Psi_{n,t},e^{i(a,x)})\right|$ is also continuous function with
respect to $t$ in $U(a)$ for any normalized eigenfunction $\Psi_{n,t}.$ This
with the inequality

$\left|\left|(\Psi_{n,t},e^{i(t,x)})\right|-\left|(\Psi_{n,t},e^{i(a,x)})\right|\right|\leq\left|(\Psi_{n,t},e^{i(t,x)})-(\Psi_{n,t},e^{i(a,x)})\right|\leq\left\|e^{i(t,x)}-e^{i(a,x)})\right\|$
(30)

implies the following obvious statement.

###### Proposition 1

If $\lambda_{n}(a)$ is a simple eigenvalue, then the function
$\left|(\Psi_{n,t},e^{i(t,x)})\right|$ does not depend on choice of the
normalized eigenfunction $\Psi_{n,t}$ and is continuous in some neighborhood
$U(a)$ of $a$, where $U(a)\subset F^{\ast}$ and any set $E$ satisfying the
conditions:

$(a)$ $\left\\{\gamma+t:t\in E,\text{
}\gamma\in\Gamma\right\\}=\mathbb{R}^{d}$ and $(b)$ if $t\in E,$ then
$\gamma+t\notin E$ for any $\gamma\in\Gamma\backslash\left\\{0\right\\}$ can
be used as the fundamental domain $F^{\star}$ of $\Gamma.$

If, in addition, there exist $\varepsilon>0$ such that

$\left|(\Psi_{n,t},e^{i(t,x)})\right|>\varepsilon$ (31)

for all $t\in U(a)$ and the normalized eigenfunction $\Psi_{n,t}$ is chosen so
that

$\arg(\Psi_{n,t}(x),e^{i(t,x)})=0,$ (32)

then $(\Psi_{n,t},e^{i(t,x)})=\left|(\Psi_{n,t},e^{i(t,x)})\right|,$ and hence
by Proposition 1, $(\Psi_{n,t},e^{i(t,x)})$ continuously depends on $t$ in
some neighborhood of $a.$ This with (30) and (31) implies that

$\frac{(\Psi_{n,t}(x),e^{i(a,x)})}{(\Psi_{n,a}(x),e^{i(a,x)})}=:\alpha(t)\rightarrow
1$

as $t\rightarrow a.$ Therefore, using (31) and the continuity of the right
side of (29) we obtain

$\left\|(e^{i(a,x)},\Psi_{n,t})\Psi_{n,t}-(e^{i(a,x)},\Psi_{n,a})\Psi_{n,a}\right\|\rightarrow
0$

and $\left\|\alpha(t))\Psi_{n,t}-\Psi_{n,a}\right\|\rightarrow 0$ as
$t\rightarrow a.$ Thus we have

$\left\|\Psi_{n,t}-\Psi_{n,a}\right\|\leq\left\|(1-\alpha(t))\Psi_{n,t}\right\|+\left\|\alpha(t))\Psi_{n,t}-\Psi_{n,a}\right\|\rightarrow
0$

as $t\rightarrow a.$ In other words, the following statement is proved.

###### Proposition 2

If $\lambda_{n}(a)$ is a simple eigenvalue and (31) holds, then the normalized
eigenfunction $\Psi_{n,t}$ satisfying (32) continuously depends on $t$ in some
neighborhood of $a.$

###### Remark 1

Note that the constructive choice (32) is also used in [6]. Namely in [6] for
the case when the right side of (1) is equal to $-\Delta+q$ and $m=1$ (for the
Schrödinger operator $L$) in the neighborhood of the sphere
$\left\\{t\in\mathbb{R}^{d}:\left|t\right|=\rho\right\\},$ where $\rho$ is a
large number, I constructed a set $B$ such that if $t\in B,$ then there exists
unique eigenvalue $\lambda_{n(t)}(t)$ that is simple and close to
$\left|t\right|^{2}$ and the corresponding normalized eigenfunction
$\Psi_{n,t}$ satisfies the asymptotic formula

$\left|(\Psi_{n,t},e^{i(t,x)})\right|^{2}=1+O(\rho^{-\delta})>\tfrac{1}{2}$
(33)

for some $\delta>0.$ Moreover, the normalized eigenfunction was chosen so that
(32) holds (see [6] page 55). In [6] the choice (32) was made in order to
write (33) in an elegant form $\Psi_{n,t}=e^{i(t,x)}+O(\rho^{-\delta})$ .
However, in Proposition 2 we show that the choice (32) ensures the continuity
of $\Psi_{n,t}.$ Note that Proposition 2 is also new for the Schrödinger
operator. However, Proposition 1 for the Schrödinger operator is clear, since
it follows directly from the continuities of the function $e^{i(t,x)},$ the
projection operator, and the norm. Proposition 1 and (33) were used to prove
that $n(t)=n(a)$ for all $t\in U(a)$ (see (5.11) from [6]), where $U(a)\subset
B$ and satisfies condition $(b)$ (see Lemma 5.1$(b)$ from [6] and Proposition
1), i.e., $U(a)\subset\left(B\cap F^{\star}\right)$ for some fundamental
domain $F^{\ast}.$

###### Acknowledgement 1

The work was supported by the Scientific and Technological Research Council of
Turkey (Tubitak, project No. 119F054).

## References

  * [1] M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations. Edinburg. Scottish Acedemic Press 1973.
  * [2] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1980.
  * [3] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Volume 4, Academic Press, New York, 1987.
  * [4] M. M. Skriganov, Geometrical and Arithmetical Methods in the Spectral Theory of the Multidimensional Periodic Operators, Proc. Steklov Math. Inst., 171, 1984.
  * [5] O. A. Veliev, Multidimensional Periodic Schrödinger Operator, Springer, Switzerland, 2019.
  * [6] O. A. Veliev, Perturbation theory for the periodic multidimensional Schrödinger operator and the Bethe-Sommerfeld Conjecture, International Journal of Contemporary Math. Sciences, Vol.2, no.2, 19 (2007).


# Stability through non-shadows

Jun Gao Extremal Combinatorics and Probability Group (ECOPRO), Institute for
Basic Science (IBS), Daejeon, South Korea. Emails: {jungao, hongliu,
<EMAIL_ADDRESS>Supported by IBS-R029-C4. Hong Liu11footnotemark: 1
Zixiang Xu11footnotemark: 1

###### Abstract

We study families $\mathcal{F}\subseteq 2^{[n]}$ with restricted intersections
and prove a conjecture of Snevily in a stronger form for large $n$. We also
obtain stability results for Kleitman’s isodiametric inequality and families
with bounded set-wise differences. Our proofs introduce a new twist to the
classical linear algebra method, harnessing the non-shadows of $\mathcal{F}$,
which may be of independent interest.

##  1 Introduction

###  1.1 Restricted intersections

The celebrated theorem of Erdős, Ko and Rado [8] states that when $n\geq 2k$,
every $k$-uniform intersecting family $\mathcal{F}\subseteq 2^{[n]}$ has size
at most $\binom{n-1}{k-1}$. We can view the Erdős-Ko-Rado theorem as a result
on families with restricted intersections where the empty intersection is
forbidden. A family $\mathcal{F}\subseteq 2^{[n]}$ is _Sperner_ if for any
distinct $A,B\in{\mathcal{F}}$, $A\not\subseteq B$. Another cornerstone result
in extremal set theory is Sperner’s theorem [30] from 1928, stating that any
Sperner family in $2^{[n]}$ has size at most $\binom{n}{\lfloor n/2\rfloor}$.
In Sperner’s theorem, $|A\setminus B|=0$ is forbidden.

What can we say if we instead prescribe all possible pairwise intersection
sizes? Formally, for a subset $L$ of non-negative integers, a family
$\mathcal{F}$ is $L$-intersecting if for any distinct
$F,F^{\prime}\in\mathcal{F}$, $|F\cap F^{\prime}|\in L$. Results of this type
date back to the work of Fisher [9], who considered the $L$-intersecting
problem when $L$ consists of a single element. More precisely, Fisher [9]
proved that a uniform family with this property cannot contain more sets than
the size of its underlying set. Later, this problem was extensively studied
via linear algebra methods by e.g., Frankl-Wilson [14], Snevily [29], Ray-
Chaudhuri-Wilson [26] and Alon-Babai-Suzuki [1]. We also refer the interested
readers to the recent survey [15].

The following fundamental result was proved by Ray-Chaudhuri and Wilson [26]
in 1975.

######  Theorem 1.1 (Ray-Chaudhuri-Wilson [26]).

Let $L=\\{\ell_{1},\ell_{2},\ldots,\ell_{s}\\}$ be a set of $s$ non-negative
integers and $k$ be a positive integer. Let $\mathcal{F}$ be a $k$-uniform
$L$-intersecting family, then $|\mathcal{F}|\leq\binom{n}{s}$.

Snevily [28] proposed the following conjecture in 1995.

######  Conjecture 1.2 (Snevily [28]).

Let $L=\\{\ell_{1},\ell_{2},\ldots,\ell_{s}\\}$ be a set of $s$ non-negative
integers and $K=\\{k_{1},k_{2},\ldots,k_{r}\\}$ be a set of $r$ positive
integers with $\max{\ell_{i}}<\min{k_{j}}$. Let
$\mathcal{F}\subseteq\bigcup_{i=1}^{r}\binom{[n]}{k_{i}}$ be an
$L$-intersecting family, then $|\mathcal{F}|\leq\binom{n}{s}$.

This conjecture, if true, would be a generalization of the Ray-Chaudhuri-
Wilson theorem. Indeed, Theorem 1.1 shows that Conjecture 1.2 is true when
$|K|=1$. Using the multilinear polynomial methods, Alon, Babai and Suzuki [1]
proved that families as above satisfy
$|\mathcal{F}|\leq\binom{n}{s}+\cdots+\binom{n}{s-r+1}$ and Snevily [27]
showed that $|\mathcal{F}|\leq\binom{n-1}{s}+\cdots+\binom{n-1}{0}$. Both of
these bounds verify Conjecture 1.2 asymptotically, that is,
$|\mathcal{F}|\leq(1+o(1))\binom{n}{s}$ when $n\rightarrow\infty$.

In support of Conjecture 1.2, Snevily [28] proved the special case when
$L=\\{0,1,\ldots,s-1\\}$, and for the general case that
$|\mathcal{F}|\leq\sum_{i=s-2r+1}^{s}\binom{n-1}{i}$, implying that
$|\mathcal{F}|\leq\binom{n}{s}+\binom{n}{s-1}$ for large $n$. For more results
related to Conjecture 1.2, we refer the readers to [6, 15, 18, 19, 31] and the
reference therein.

Observe that, if a set system $\mathcal{F}\subseteq 2^{[n]}$ satisfies the
conditions in Conjecture 1.2, then $\mathcal{F}$ is Sperner. By LYM inequality
(Theorem 2.2), It is clear that Conjecture 1.2 is true when $n<2s-1$. We
propose a stronger conjecture for Sperner systems as follows.

######  Conjecture 1.3.

Let $n,s$ be two integers with $n\geq 2s-1$,
$L=\\{\ell_{1},\ell_{2},\ldots,\ell_{s}\\}$ be a set of $s$ non-negative
integers and $\mathcal{F}\subseteq 2^{[n]}$ be an $L$-intersecting Sperner
system, then $|\mathcal{F}|\leq\binom{n}{s}$.

Clearly, Conjecture 1.3 implies Conjecture 1.2. Our first result verifies
Conjecture 1.3 when $0\in L$ for reasonably large $n$.

######  Theorem 1.4.

Let $L=\\{\ell_{1},\ell_{2},\ldots,\ell_{s}\\}$ be a set of $s$ non-negative
integers and $\mathcal{F}\subseteq 2^{[n]}$ be an $L$-intersecting Sperner
family, we have $|\mathcal{F}|\leq\sum_{i=0}^{s}\binom{n-1}{i}$. In
particular, if $0\in L$ and $n\geq 3s^{2}$, then
$|\mathcal{F}|\leq\binom{n}{s}$, with equality holds if and only if
$L=\\{0,1,2,\ldots,s-1\\}$ and $\mathcal{F}=\binom{[n]}{s}$.

Note that when $L=\\{0\\}$, $\mathcal{F}=\\{\\{1\\},\\{2\\},\ldots,\\{n\\}\\}$
is a $\\{0\\}$-intersecting Sperner family, showing that the above upper bound
$|\mathcal{F}|\leq\sum_{i=0}^{s}\binom{n-1}{i}$ cannot be improved in general.

What if $0\notin L$? We say a family
$\mathcal{F}=\\{F_{1},F_{2},\ldots,F_{m}\\}\subseteq 2^{n}$ has _rank_ $t$ if
$\max_{i\in[m]}{|F_{i}|}=t$, and say $\mathcal{F}$ is _trivially_
$\ell$-intersecting if there is a set $A$ with size $\ell$ such that
$A\subseteq\bigcap_{i=1}^{m}F_{i}$. We obtain better bounds when $0\notin L$
and $\mathcal{F}$ has relatively small rank.

######  Theorem 1.5.

Let $L=\\{\ell_{1},\ell_{2},\ldots,\ell_{s}\\}$ be a set of $s$ positive
integers with $\ell_{1}<\ell_{2}<\cdots<\ell_{s}$ and $\mathcal{F}\subseteq
2^{[n]}$ be an $L$-intersecting Sperner family with rank $M$. Then either
$\mathcal{F}$ is trivially $\ell_{1}$-intersecting and then
$|\mathcal{F}|\leq\sum_{i=0}^{s}{\binom{n-2}{i}}={\binom{n}{s}}-\Omega_{s}(n^{s-1})$,
or $|\mathcal{F}|=O_{M,s}(n^{s-1})$.

Combining Theorems 1.4 and 1.5, we see that Conjecture 1.2 is true when $n$ is
sufficiently large.

######  Corollary 1.6.

Let $L=\\{\ell_{1},\ell_{2},\ldots,\ell_{s}\\}$ be a set of $s$ non-negative
integers and let $K=\\{k_{1},k_{2},\ldots,k_{r}\\}$ be a set of $r$ positive
integers with $\max{\ell_{i}}<\min{k_{j}}$. There exists $n_{0}$ such that for
any $n\geq n_{0}$ and $L$-intersecting family
$\mathcal{F}\subseteq\bigcup_{i=1}^{r}\binom{[n]}{k_{i}}$,
$|\mathcal{F}|\leq\binom{n}{s}$.

###  1.2 Restricted symmetric/set-wise differences

For set systems with bounded symmetric differences, Kleitman [22] proved the
following classical result: for integers $n>d$ and a family
$\mathcal{F}\subseteq 2^{[n]}$ with $|A\triangle B|\leq d$ for any
$A,B\in\mathcal{F}$,

$|\mathcal{F}|\leq\begin{cases}\sum\limits_{i=0}^{k}\binom{n}{i}&d=2k;\\\
2\sum\limits_{i=0}^{k}\binom{n-1}{i}&d=2k+1.\end{cases}$

The bounds in both cases above are optimal. For the even case, the upper bound
can be attained by the radius-$k$ Hamming ball
$\mathcal{K}(n,k):=\\{F:F\subseteq[n],|F|\leq k\\}$, and for the odd case,
consider the family
$\mathcal{K}_{y}(n,k):=\\{F:F\subseteq[n],|F\setminus\\{y\\}|\leq k\\}$. The
original proof of Kleitman is combinatorial. Recently, Huang, Klurman and
Pohoata [17] provided an algebraic proof via the Cvetković bound [7] and
extended the results when the allowed symmetric differences lie in a set of
consecutive integers.

For a family $\mathcal{F}\subseteq 2^{[n]}$ and a subset $S\subseteq[n]$, we
define the _translate_ of $\mathcal{F}$ by $S$ as $\mathcal{F}\triangle
S:=\\{F\triangle S:F\in\mathcal{F}\\}$. Recently, Frankl [12] proved the
following stability result for Kleitman’s theorem.

######  Theorem 1.7 (Frankl [12]).

Let $n\geq d+2$, $d\geqslant 0$ and let $\mathcal{F}\subseteq 2^{[n]}$ with
$|A\triangle B|\leq d$ for any $A,B\in\mathcal{F}$.

  1. 1.

If $d=2k$ and $\mathcal{F}$ is not contained in any translate of
$\mathcal{K}(n,k)$, then
$|\mathcal{F}|\leqslant\sum_{i=0}^{k}\binom{n}{i}-\binom{n-k-1}{k}+1$.

  2. 2.

If $d=2k+1$ and $\mathcal{F}$ is not contained in any translate of
$\mathcal{K}_{y}(n,k)$, then $|\mathcal{F}|\leq
2\sum_{i=0}^{k}\binom{n-1}{i}-\binom{n-k-2}{k}+1$.

Frankl’s proof is combinatorial and relatively involved, making use of his
earlier stability result for Katona theorem [13]. We prove a finer stability
result for Kleitman’s theorem for the odd case.

######  Theorem 1.8.

Let $\mathcal{F}\subseteq 2^{[n]}$ be a family of subsets of $[n]$ with
$|A\triangle B|\leq 2k+1$ for any $A,B\in\mathcal{F}$, then either
$|\mathcal{F}|\leq 2\sum_{i=0}^{k}\binom{n}{i}-2\binom{n-5k-1}{k}$, or
$\mathcal{F}$ is contained in some translate of $\mathcal{K}(n,k+1)$.
Furthermore, if $\mathcal{F}$ is contained in some translate of
$\mathcal{K}(n,k+1)$, then either $\mathcal{F}$ is contained in some translate
of $\mathcal{K}_{y}(n,k)$, or $|\mathcal{F}|\leq
2\sum_{i=0}^{k}\binom{n-1}{i}-\binom{n-k-2}{k}+1$.

Note that if $\mathcal{F}$ is not contained in any translate of
$\mathcal{K}(n,k+1)$, our upper bound $|\mathcal{F}|\leq
2\sum_{i=0}^{k}\binom{n}{i}-2\binom{n-5k-1}{k}$ is better than the bound
$|\mathcal{F}|\leq 2\sum_{i=0}^{k}\binom{n-1}{i}-\binom{n-k-2}{k}+1$ in
Theorem 1.7 when $n\geq 10k^{2}$. We remark that our method does also apply to
the even case. However, it yields a slighly weaker bound
$|\mathcal{F}|\leq\sum_{i=0}^{k}\binom{n+1}{i}-\binom{n-5k}{k}$ when
$\mathcal{F}$ is not a translate of $\mathcal{K}(n,k)$.

Another type of well-studied restricted intersection problem is to bound the
set-wise differences. In 1983, Katona [21] asked if we require $|A\setminus
B|\leq k$ for any distinct $A,B\in\mathcal{F}$, then what is the largest such
$\mathcal{F}$? Frankl [10] proved that for a set $L$ of non-negative integers,
given a family $\mathcal{F}\subseteq 2^{[n]}$ so that for any distinct
$A,B\in\mathcal{F}$, $|A\setminus B|\in L$, then
$|\mathcal{F}|\leq\sum_{i=0}^{|L|}\binom{n}{i}$.

Our method also yields stability result for families with bounded set-wise
differences as follows.

######  Theorem 1.9.

Let $n,t,k$ be positive integers with $k<t\leq\frac{k+n}{2}$ and
$\mathcal{F}=\\{F_{1},F_{2},\ldots,F_{m}\\}\subseteq 2^{[n]}$ be a family with
rank $t$ and $|F_{i}\setminus F_{j}|\leq k$ for any
$F_{i},F_{j}\in\mathcal{F}$, then either $\mathcal{F}$ is trivially
$(t-k)$-intersecting and $|\mathcal{F}|\leq\sum_{i=0}^{k}\binom{n-(t-k)}{i}$,
or $|\mathcal{F}|\leq\sum_{i=0}^{k}\binom{n}{i}-\binom{n-t-2k}{k}$. In the
latter case, if $n$ is sufficiently large in terms of $t$ and $k$, then
$|\mathcal{F}|\leq(1-c)\binom{n-(t-k)}{k},$ for some constant $c>0$.

We remark that the upper bound on the rank $t\leq\frac{k+n}{2}$ is not
restrictive. Indeed, for any sets $A,B$, $|A\setminus B|=|B^{c}\setminus
A^{c}|$. Thus, when $t>\frac{k+n}{2}$, we can consider the family consisting
of _complement_ $F^{c}$ of all sets $F\in\mathcal{F}$ instead.

For families with set-wise differences lying in $L$, where $|L|=k$, Frankl
[10] conjectured that if the family is Sperner, then the upper bound
$|\mathcal{F}|\leq\sum_{i=0}^{k}\binom{n}{i}$ can be improved to
$\binom{n}{k}$. As a corollary, we show that this conjecture holds for
$L=\\{1,2,\ldots,k\\}$ and large $n$. The following corollary can also be
deduced from the main results in [11].

######  Corollary 1.10.

Given integer $k$ and Sperner family $\mathcal{F}\subseteq 2^{[n]}$ with
$|A\setminus B|\leq k$ for any $A,B\in\mathcal{F}$, if $n$ is sufficiently
large, then $|\mathcal{F}|\leq\binom{n}{k}$, with equality holds if and only
if $\mathcal{F}=\binom{[n]}{k}$.

###  1.3 Our method

Linear algebra method has proven to be a powerful tool in extremal set theory,
see e.g., [1, 2, 3, 4, 6, 19, 25, 26, 29]. Given a family ${\mathcal{F}}$, in
this approach we associate members of ${\mathcal{F}}$ with multilinear
polynomials that are linearly independent. We can then bound the cardinality
of the family by the dimension of the underlying space. Improvements can often
be made by cleverly adding an extra set of polynomials, which together with
the original ones are still linearly independent.

We introduce a new twist to this classical method, utilizing the non-shadows
of the family to find a large collection of polynomials that can be added.
There are several advantages of our variation. First of all, the linear
independence can usually be verified easily. A more important feature is that,
as the new polynomials are associated with non-shadows of ${\mathcal{F}}$, we
can gather additional structural information of ${\mathcal{F}}$, which not
only offers optimal bounds in many scenarios but also provides a stability
result as shown in Theorems 1.8 and 1.9.

Organization. The rest of this paper is organized as follows. In Section 2 we
will collect some useful tools and give a short proof of a special case of the
Katona intersection theorem (Theorem 2.4) to illustrate our method. We will
prove the stability results, Theorems 1.8 and 1.9, in Section 3. The proofs of
Theorems 1.4 and 1.5 will be presented in Section 4.

Notations. In this paper, we usually regard $[n]=\\{1,2,\ldots,n\\}$ as the
ground set. For a set $A\subseteq[n]$, we write $A^{c}$ for its complement,
that is, $A^{c}=[n]\setminus A$. For a subset $A\subseteq[n]$, we will use
$\binom{A}{k}$ to denote the family of all subsets of $A$ with size $k$ and
$\binom{A}{\leq k}$ to denote the family of all subsets of $A$ with size at
most $k$. For a pair of vectors $\bm{x}=\\{x_{1},x_{2},\ldots,x_{n}\\}$ and
$\bm{y}=\\{y_{1},y_{2},\ldots,y_{n}\\}$, we define their inner product as
$\bm{x}\cdot\bm{y}=\sum_{i=1}^{n}x_{i}y_{i}$. We use $\bm{1}_{n}$ to denote
the length-$n$ all-ones vector $(1,1,\ldots,1)$ and omit the subscript when
the dimension is clear. Given a polynomial $f$ in $n$ variables
$x_{1},x_{2},\ldots,x_{n}$, define its _multilinear reduction_ $\tilde{f}$ to
be the polynomial obtained from $f$ by replacing each $x_{i}^{t}$ term by
$x_{i}$ for any positive integer $t$ and $1\leq i\leq n$. For a family
$\mathcal{F}\subseteq 2^{[n]}$, we denote the _$k$ -shadow_ of $\mathcal{F}$
as $\partial_{k}{\mathcal{F}}:=\\{T\in\binom{[n]}{k}:T\subseteq F\ \text{for\
some\ }F\in\mathcal{F}\\}$.

##  2 Preliminaries

###  2.1 Some useful tools

Hilton and Milner [16] showed the following result, which states that the size
of non-trivial intersecting families is noticeably smaller than that of the
trivial ones.

######  Theorem 2.1 (Hilton-Milner [16]).

Let $n,k$ be positive integers with $n>2k$. If
$\mathcal{F}\subseteq\binom{[n]}{k}$ is an intersecting family and
$\bigcap_{F\in\mathcal{F}}F=\emptyset$, then we have

$|\mathcal{F}|\leq\binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1.$

For Sperner families, the Lubell–Yamamoto–Meshalkin inequality, discovered by
Bollobás [5], Lubell [23], Mešalkin [24] and Yamamoto [32] independently, is
useful.

######  Theorem 2.2 (LYM inequality).

Let $\mathcal{F}\subseteq 2^{[n]}$ be a Sperner family of subsets of $[n]$,
then

$\sum\limits_{A\in\mathcal{F}}\frac{1}{\binom{n}{|A|}}\leq 1.$

We also need the triangular criterion when we want to prove a sequence of
polynomials are linearly independent.

######  Proposition 2.3.

Let $f_{1},f_{2},\ldots,f_{m}$ be functions in a linear space. If
$v^{(1)},v^{(2)},\ldots,v^{(m)}$ are vectors such that $f_{i}(v^{(i)})\neq 0$
for $1\leq i\leq m$ and $f_{i}(v^{(j)})=0$ for $i>j$, then
$f_{1},f_{2},\ldots,f_{m}$ are linearly independent.

###  2.2 Warm up: Katona intersection theorem

As a warm-up and illustration of our method, we will give a short proof of a
special case of the Katona intersection theorem.

######  Theorem 2.4 (Katona [20]).

Suppose $\mathcal{F}\subseteq\binom{[n]}{k+1}$ is an intersecting family. Then
$|\partial_{k}{\mathcal{F}}|\geq|\mathcal{F}|$.

###### Proof of Theorem 2.4.

Suppose $\mathcal{F}=\\{F_{1},F_{2},\ldots,F_{m}\\}\subseteq\binom{[n]}{k+1}$
is an intersecting family. Let $\bm{a}_{i},\bm{b}_{i}$ be the characteristic
vectors of $F_{i},F_{i}^{c}$, respectively. For $\bm{x}=(x_{1},\ldots,x_{n})$
and $i\in[m]$, we define
$g_{i}(\bm{x})=\prod_{j=1}^{k}(\bm{x}\cdot\bm{b}_{i}-j)$ and let
$f_{i}=\tilde{g}_{i}$ be the multilinear reduction of $g_{i}$. Note that
$\ker(f_{i})$ corresponds to characteristic vectors of subsets $F\subseteq[n]$
with $|F\setminus F_{i}|\in[k]$,

Let $\\{S_{1},\ldots,S_{r}\\}=\binom{[n]}{\leq k-1}$ with $|S_{i}|\leq|S_{j}|$
for $i<j$, so $r=\sum_{i=0}^{k-1}\binom{n}{i}$. For each $S_{i}$, let
$f_{S_{i}}$ be the multilinear reduction of
$g_{S_{i}}(\bm{x}):=(\bm{x}\cdot\bm{1}-k-1)\cdot\prod_{\ell\in
S_{i}}x_{\ell}$, and $\bm{a}_{S_{i}}$ be the characteristic vector of $S_{i}$.
Let
$\\{T_{1},T_{2},\ldots,T_{h}\\}=\binom{[n]}{k}\setminus\partial_{k}{\mathcal{F}}$
be the collection of all $k$-sets that are not $k$-shadow of $\mathcal{F}$.
For each $T_{i}$, we define $f_{T_{i}}(\bm{x}):=\prod_{j\in T_{i}}x_{j}$, and
$\bm{a}_{T_{i}}$ to be the characteristic vector of $T_{i}$. Here we can see
that $\ker(f_{T_{i}})$ corresponds to characteristic vectors of subsets of
$[n]$ which do not contain $T_{i}$, and $\ker(g_{S_{i}})$ corresponds to
characteristic vectors of subsets of $[n]$ which contain $k+1$ elements or do
not contain $S_{i}$.

######  Claim 2.5.

The polynomials
$\\{f_{1},\ldots,f_{m},f_{S_{1}},\ldots,f_{S_{r}},f_{T_{1}},\ldots,f_{T_{h}}\\}$
are linearly independent.

###### Proof of claim.

We shall show that these polynomials satisfy the triangular criterion in
Proposition 2.3. First note that $f_{i}(\bm{a}_{i})=(-1)^{k}k!\neq 0$ as
$\bm{a}_{i}\cdot\bm{b}_{i}=0$ for each $i$. For $i\neq j$, we have
$f_{i}(\bm{a}_{j})=0$ as $\bm{a}_{j}\cdot\bm{b}_{i}=|F_{j}\cap
F_{i}^{c}|=|F_{j}\setminus F_{i}|\in[k]$. Next,
$f_{S_{i}}(\bm{a}_{S_{i}})=(|S_{i}|-k-1)\neq 0$. Since $|F_{i}|=k+1$, we know
that $f_{S_{i}}(\bm{a}_{j})=0$. By definition, if $i<j$, we have
$|S_{i}|\leq|S_{j}|$ and $S_{j}\setminus S_{i}\neq\emptyset$. Thus for any
$i<j$, we have $f_{S_{j}}(\bm{a}_{S_{i}})=0$. For $f_{T_{i}}$’s, we have
$f_{T_{i}}(\bm{a}_{T_{i}})=1\neq 0$, and for any
$T\in\\{T_{1},\ldots,T_{i-1},T_{i+1}\ldots,T_{h},S_{1},S_{2},\ldots,S_{r}\\}$,
we can see $|T|\leq|T_{i}|,T\neq T_{i}$ and $T_{i}\setminus T\neq\emptyset$,
so we have $f_{T_{i}}(\bm{a}_{T})=0$. Lastly, since $T_{i}\not\subseteq F_{j}$
for any $i,j$, we have $f_{T_{i}}(\bm{a}_{j})=0$. Hence, Proposition 2.3
implies that
$\\{f_{1},\ldots,f_{m},f_{S_{1}},\ldots,f_{S_{r}},f_{T_{1}},\ldots,f_{T_{h}}\\}$
are linearly independent. ∎

Since all of these polynomials have degree at most $k$, we have
$|\mathcal{F}|+r+h\leq\sum_{i=0}^{k}\binom{n}{i}$. As
$r=\sum_{i=0}^{k-1}\binom{n}{i}$ and
$h=\binom{n}{k}-|\partial_{k}{\mathcal{F}}|$, we have
$|\mathcal{F}|\leq|\partial_{k}{\mathcal{F}}|$ as desired.

∎

##  3 Stability

###  3.1 Families with bounded symmetric differences

In this section, we will prove Theorem 1.8. Let $\mathcal{F}_{o}$ consist of
all sets of odd size in $\mathcal{F}$ and $\mathcal{F}_{e}$ consist of all
sets of even size in $\mathcal{F}$. Observe that for any subsets
$X,Y,Z\subseteq[n]$, $(X\triangle Y)\triangle(X\triangle Z)=Y\triangle Z$.
Hence, by translating all sets in ${\mathcal{F}}$ by some appropriate
$U\in{\mathcal{F}}$, we may assume $\emptyset\in\mathcal{F}$ and $|F|\leq
2k+1$ for any $F\in\mathcal{F}$. Then without loss of generality, we can
further assume that $|\mathcal{F}_{o}|\leq|\mathcal{F}_{e}|$.

Let $\mathcal{F}_{e}=\\{F_{1},\ldots,F_{m}\\}$ and $\bm{a}_{i},\bm{b}_{i}$ be
the characteristic vectors of $F_{i},F_{i}^{c}$, respectively. For
$\bm{x}=(x_{1},\ldots,x_{n})$ and $i\in[m]$, we define
$g_{i}(\bm{x})=\prod_{\ell=1}^{k}(\bm{x}\cdot\bm{b}_{i}+(\bm{1}-\bm{x})\cdot\bm{a}_{i}-2\ell)$
and $f_{i}=\tilde{g}_{i}$. It is easy to see that $\ker(g_{i})$ corresponds to
characteristic vectors of subsets $F\subseteq[n]$ with $|F_{i}\triangle
F|\in\\{2,4,\ldots,2k\\}$.

By definition, $f_{i}(\bm{a}_{i})=(-2)^{k}k!\neq 0$ as
$\bm{a}_{i}\cdot\bm{b}_{i}=0$ for each $i$. As each set in $\mathcal{F}_{e}$
has even size,
$\bm{a}_{i}\cdot\bm{b}_{j}+\bm{b}_{i}\cdot\bm{a}_{j}=|F_{i}\triangle
F_{j}|\in\\{2,4,\ldots,2k\\}$ for any $i\neq j$, so we have
$f_{i}(\bm{a}_{j})=0$. By Proposition 2.3, $\\{f_{i}\\}_{i=1}^{m}$ are
linearly independent, which implies $m\leq\sum_{i=0}^{k}\binom{n}{i}$ since
the polynomials $\\{f_{i}\\}_{i=1}^{m}$ lie in the space of $n$-variate
polynomials of degree at most $k$.

Let
$\mathcal{S}=\binom{[n]}{k}\setminus\partial_{k}{\mathcal{F}}_{e}:=\\{T\in\binom{[n]}{k}:T\not\subseteq
F_{i},\text{ for all }i\in[m]\\}$. For each subset $T\in\mathcal{S}$, we
define a polynomial $h_{T}(\bm{x})=\prod_{i\in T}x_{i}$ and let $\bm{a}_{T}$
be the characteristic vector of $T$. Similarly, $\ker(h_{T})$ corresponds to
characteristic vectors of subsets of $[n]$ which do not contain $T$ as a
subset. The following claim will establish an upper bound
$|\mathcal{F}_{e}|\leq\sum_{i=0}^{k}\binom{n}{i}-|\mathcal{S}|$.

######  Claim 3.1.

The polynomials $\\{h_{T}\\}_{T\in\mathcal{S}}$ and $\\{f_{i}\\}_{i=1}^{m}$
are linearly independent.

###### Proof of claim.

For any $T\in\mathcal{S}$ and $i\in[m]$, since $T\not\subseteq F_{i}$, we have
$h_{T}(\bm{a}_{i})=0$. For any $T,S\in\mathcal{S}$, it is easy to check that
$h_{T}(a_{S})\neq 0$ if and only if $T=S$. By Proposition 2.3,
$\\{h_{T}\\}_{T\in\mathcal{S}}$ and $\\{f_{i}\\}_{i=1}^{m}$ are linearly
independent. ∎

If $|\mathcal{S}|\geq\binom{n-5k-1}{k}$, then we have

$|\mathcal{F}|\leq 2|\mathcal{F}_{e}|\leq
2\big{(}\sum_{i=0}^{k}\binom{n}{i}-|\mathcal{S}|\big{)}\leq
2\sum_{i=0}^{k}\binom{n}{i}-2\binom{n-5k-1}{k},$

yielding the first alternative.

We may then assume $|\mathcal{S}|<\binom{n-5k-1}{k}$. Relabelling
$\mathcal{F}_{e}$ if necessary, we may assume
$2k+1\geq|F_{1}|\geq|F_{2}|\geq\cdots\geq|F_{m}|$. As $|F_{1}|$ is maximum, we
have $|F_{i}\setminus F_{1}|\leq|F_{1}\setminus F_{i}|$ for any $1\leq i\leq
m$. Since $|F_{1}\triangle F_{i}|\leq 2k+1$ and both of $|F_{1}|$ and
$|F_{i}|$ are even, we have $|F_{i}\setminus F_{1}|\leq k$ and $|F_{1}|\leq
2k$. We can then draw the following claim.

######  Claim 3.2.

If $|F_{i}\setminus F_{1}|=k$, then we have $|F_{1}\setminus F_{i}|=k$, and
$|F_{1}|=|F_{i}|$.

Since $|\mathcal{S}|<\binom{n-5k-1}{k}$ and $n-|F_{1}|>n-5k-1$, there exists
some $k$-set $R\subseteq F_{1}^{c}$ such that
$R\in\partial_{k}{\mathcal{F}_{e}}$. Consequently, there exists some
$E\in\mathcal{F}_{e}$ such that $R\subseteq E$ and $|E\setminus
F_{1}|\geq|R|=k$. In fact, $|E\setminus F_{1}|=k$ as $|E\setminus F_{1}|\leq
k$. Claim 3.2 then implies that $|F_{1}|=|E|$. Fix such $E$ and let
$A:=F_{1}\cap E$.

######  Claim 3.3.

For any $F\in\mathcal{F}$, we have $|F\triangle A|\leq k+1$.

###### Proof of claim.

Suppose that there exists $F\in\mathcal{F}$ such that $|F\triangle A|\geq
k+2$. Since $|F_{1}|\leq 2k$, $|E\setminus F_{1}|=k$ and $|F|\leq 2k+1$, we
have $|F_{1}\cup E\cup F|\leq 5k+1$. Since $|\mathcal{S}|<\binom{n-5k-1}{k}$,
there exists some $k$-set $Q\subseteq[n]\setminus(F_{1}\cup E\cup F)$ such
that $Q\in\partial_{k}{\mathcal{F}}_{e}$. Thus there is some
$T\in\mathcal{F}_{e}$ such that $Q\subseteq T$ and
$|T\cap([n]\setminus(F_{1}\cup E\cup F))|\geq k$. As $|T\setminus F_{1}|\leq
k$, we have $T\cap([n]\setminus(F_{1}\cup E\cup F))=T\setminus F_{1}$.
Therefore $T\setminus F_{1}$, $E\setminus A$ and $F_{1}\setminus A$ are
pairwise disjoint sets of size $k$. By Claim 3.2, $|T\triangle F_{1}|=2k$,
$|T\triangle E|=2k$ and $T=A\cup(T\setminus
F_{1})=A\cup(T\cap([n]\setminus(F_{1}\cup E\cup F)))$. Then we have

$|T\triangle F|=|T\cap([n]\setminus(F_{1}\cup E\cup F))|+|A\triangle F|\geq
k+k+2=2k+2,$

a contradiction. ∎

By Claim 3.3, if $|\mathcal{S}|<\binom{n-5k-1}{k}$, then $\mathcal{F}$ should
be contained in a translate of $\mathcal{K}(n,k+1)$. Without loss of
generality, assume $\mathcal{F}$ is contained in $\mathcal{K}(n,k+1)$. Let
$\mathcal{H}=\\{H\in\mathcal{F},|H|=k+1\\}$. Note that $\mathcal{H}$ is an
intersecting family, because the symmetric difference of any pair of subsets
in $\mathcal{F}$ is bounded by $2k+1$. If $\mathcal{H}$ is a non-trivial
intersecting family, by Hilton-Milner Theorem, we have

$|\mathcal{F}|\leq\sum_{i=0}^{k}\binom{n}{i}+|\mathcal{H}|\leq\sum_{i=0}^{k}\binom{n}{i}+\binom{n-1}{k}-\binom{n-k-2}{k}+1=2\sum\limits_{i=0}^{k}\binom{n-1}{i}-\binom{n-k-2}{k}+1.$

Otherwise, there exists some $y\in[n]$ such that $y\in H$ for any
$H\in\mathcal{H}$, which implies $\mathcal{F}$ is contained in
$\mathcal{K}_{y}(n,k)$. This completes the proof.

###  3.2 Families with bounded set-wise differences

In this section, we prove Theorem 1.9 and then apply it to show Corollary
1.10.

###### Proof of Theorem 1.9.

Let $\mathcal{F}=\\{F_{1},F_{2},\ldots,F_{m}\\}\subseteq 2^{[n]}$ be a family
with rank $t$ and $|F_{i}\setminus F_{j}|\leq k$ for any
$F_{i},F_{j}\in\mathcal{F}$. Without loss of generality, we assume that
$|F_{j}|\geq|F_{i}|$ for any $i>j$, then we have $|F_{1}|=t$ and
$|F_{j}\setminus F_{i}|\geq 1$ for any $i>j$.

Let $\bm{a}_{i},\bm{b}_{i}$ be the characteristic vectors of
$F_{i},F_{i}^{c}$, respectively. For $\bm{x}=(x_{1},\ldots,x_{n})$ and
$i\in[m]$, we define $g_{i}(\bm{x})=\prod_{j=1}^{k}(\bm{x}\cdot\bm{b}_{i}-j)$
and $f_{i}=\tilde{g}_{i}$. Here $\ker(f_{i})$ corresponds to characteristic
vectors of subsets $F\subseteq[n]$ with $|F\setminus F_{i}|\in[k]$.

By definition, $f_{i}(\bm{a}_{i})=(-1)^{k}k!\neq 0$ as
$\bm{a}_{i}\cdot\bm{b}_{i}=0$ for each $i$. For $i>j$, we have
$f_{i}(\bm{a}_{j})=0$ as $\bm{a}_{j}\cdot\bm{b}_{i}=|F_{j}\setminus
F_{i}|\in[k]$. By Proposition 2.3, $\\{f_{i}\\}_{i=1}^{m}$ are linearly
independent, which implies $m\leq\sum_{j=0}^{k}\binom{n}{j}$.

Let $\mathcal{S}=\binom{[n]}{k}\setminus\partial_{k}{\mathcal{F}}$. For each
subset $T\in\mathcal{S}$, we define a polynomial $h_{T}(\bm{x})=\prod_{i\in
T}x_{i}$. The following claim can be derived the same way as Claim 3.1.

######  Claim 3.4.

The polynomials $\\{h_{T}\\}_{T\in\mathcal{S}}$ and $\\{f_{i}\\}_{i=1}^{m}$
are linearly independent.

Claim 3.4 implies that $m\leq\sum_{i=0}^{k}\binom{n}{i}-|\mathcal{S}|$. Thus,
we shall assume $|\mathcal{S}|<\binom{n-t-2k}{k}$, otherwise the second
alternative holds.

######  Claim 3.5.

For any pair of distinct $E,F\in\mathcal{F}$ with $|F|=t,|E\setminus F|=k$, we
must have $|E|=t$ and $|F\setminus E|=k$.

###### Proof of claim.

Note that $\max_{i\in[m]}\\{|F_{i}|\\}=t$, thus we have $k\geq|F\setminus
E|\geq|E\setminus F|\geq k$ and so $|F\setminus E|=k$. Also, $|E|=|E\cup
F|-|F\setminus E|=|E\cup F|-|E\setminus F|=|F|=t$. ∎

By our assumption, we have
$\binom{|F_{1}^{c}|}{k}=\binom{n-t}{k}>|\mathcal{S}|$, hence there exists some
$F\in\mathcal{F}$ such that $|F\cap F_{1}^{c}|=|F\setminus F_{1}|=k$. By Claim
3.5, we have $|F|=t$ and $|F\cap F_{1}|=t-|F\setminus F_{1}|=t-k$. Let $F\cap
F_{1}=A$. We can further extract more structural properties of $\mathcal{F}$
as follows.

######  Claim 3.6.

For any $F^{\prime}\in\mathcal{F}$, if $|F^{\prime}\cap F_{1}^{c}|=k$, then we
have $F^{\prime}\cap F_{1}=A$.

###### Proof of claim.

Since $|F_{1}^{c}\setminus(F^{\prime}\cup F)|\geq n-t-2k$ and
$|\mathcal{S}|<\binom{n-t-2k}{k}$, there exist
$T\in\binom{F_{1}^{c}\setminus(F^{\prime}\cup F)}{k}$ and $E\in\mathcal{F}$
such that $T\subseteq E$. Since $|T|=k$, we have $E\setminus
F^{\prime}=E\setminus F=T\subseteq F_{1}^{c},$ by Claim 3.5 we have $|E|=t$.
Also note that $F\cap E\subseteq F_{1}$, so $|F\cap E|=|F\cap E\cap
F_{1}|=t-k$, which implies $E\cap F_{1}=F\cap F_{1}=A$. Applying the same
analysis on the (ordered) triple $(F^{\prime},E,F_{1})$ instead of
$(E,F,F_{1})$, we have $F^{\prime}\cap F_{1}=E\cap F_{1}=A$ as desired. ∎

Next, we will show that for any $F^{\prime}\in\mathcal{F}$, we have
$A\subseteq F^{\prime}\cap F_{1}$, and so $\mathcal{F}$ is trivially
$(t-k)$-intersecting with
$A\subseteq\bigcap_{F^{\prime}\in\mathcal{F}}F^{\prime}$. Suppose there exists
some $F^{\prime}$, such that $A\setminus(F^{\prime}\cap F_{1})\neq\emptyset$,
then since $|F_{1}^{c}\setminus F^{\prime}|\geq n-t-k$, and
$|\mathcal{S}|<\binom{n-t-2k}{k}$, there exist subsets
$T\in\binom{F_{1}^{c}\setminus F^{\prime}}{k}$ and $E\in\mathcal{F}$ such that
$T\subseteq E$. By Claim 3.6 we know that $E\cap F_{1}=A$, so $|E\setminus
F^{\prime}|$ = $|T\cup(A\setminus F^{\prime})|\geq k+1$, which is a
contradiction.

Then we have that $A\subseteq F_{i}$ for all $i\in[m]$. Since $|F_{i}|\leq t$,
we have $|F_{i}\setminus A|\leq k$, which implies that
$m\leq\sum_{i=0}^{k}\binom{n-(t-k)}{i}$. The proof of Theorem 1.9 is finished.
∎

###### Proof of Corollary 1.10.

Suppose $\mathcal{F}=\\{F_{1},F_{2},\ldots,F_{m}\\}\subseteq 2^{[n]}$ is a
rank $t$ Sperner family such that $|F_{i}\setminus F_{j}|\leq k$. Suppose
first that $t>\frac{k+n}{2}$, as $|F_{i}\setminus F_{j}|\leq k$ for any
$F_{i},F_{j}\in\mathcal{F}$, we know $\min_{i\in[m]}\\{|F_{i}|\\}\geq t-k$.
Define $\mathcal{F}^{\prime}:=\\{F_{1}^{c},F_{2}^{c},\ldots,F_{m}^{c}\\}$.
Then $\max_{i\in[m]}\\{|F_{i}^{c}|\\}=n-\min_{i\in[m]}\\{|F_{i}|\\}\leq
n-(t-k)\leq\frac{k+n}{2}$ and $|F_{i}^{c}\setminus F_{j}^{c}|=|F_{j}\setminus
F_{i}|\leq k$ for any $F_{i}^{c},F_{j}^{c}\in\mathcal{F}^{\prime}$, so we can
assume $t\leq\frac{k+n}{2}$.

If $t>k$, by Theorem 1.9, we have either

$|\mathcal{F}|\leq\sum_{i=0}^{k}\binom{n}{i}-\binom{\frac{n-5k}{2}}{k}<\binom{n}{k},$

when $n\gg k$, or there exists $A$ with $|A|=t-k$ and $A\subseteq F_{i}$ for
any $i\in[m]$, which implies $\\{F_{i}\setminus A\\}_{i=1}^{m}$ is a Sperner
family with size at most $k$. By Theorem 2.2, we have

$1\geq\sum_{F\in\mathcal{F}}\frac{1}{\binom{n-(t-k)}{|F\setminus
A|}}\geq\sum_{F\in\mathcal{F}}\frac{1}{\binom{n-(t-k)}{k}},$

which implies $|\mathcal{F}|\leq\binom{n-(t-k)}{k}<\binom{n}{k}$.

If $t\leq k$. By Theorem 2.2, we have

$1\geq\sum_{F\in\mathcal{F}}\frac{1}{\binom{n}{|F|}}\geq\sum_{F\in\mathcal{F}}\frac{1}{\binom{n}{k}},$

which implies $|\mathcal{F}|\leq\binom{n}{k}$, the equality holds if and only
if $\mathcal{F}=\binom{[n]}{k}$. ∎

##  4 Sperner families with restricted intersections

In this section, we will prove Theorem 1.4 and Theorem 1.5, respectively.

###  4.1 Proof of Theorem 1.4

Let $L=\\{\ell_{1},\ell_{2},\ldots,\ell_{s}\\}$ be a set of $s$ non-negative
integers with $\ell_{1}<\ell_{2}<\cdots<\ell_{s}$. Let
$\mathcal{F}=\\{F_{1},\ldots,F_{m}\\}$ be an $L$-intersecting Sperner family.
By relabelling, we can assume there exists some integer $r$ such that $1\in
F_{i}$ if and only if $1\leq i\leq r$. For each $1\leq i\leq m$, let
$\bm{a}_{i}$ be the characteristic vector of $F_{i}$. For
$\bm{x}=(x_{1},\ldots,x_{n})$ and $i\in[m]$, we define
$g_{i}(\bm{x})=\prod_{\ell\in L,\ell\neq|F_{i}|}(\bm{x}\cdot\bm{a}_{i}-\ell)$
and $f_{i}(\bm{x})$ be the multilinear reduction of $g_{i}(\bm{x})$ with
$x_{1}=1$, that is, $f_{i}(\bm{x})$ is obtained from $g_{i}(\bm{x})$ by
replacing each $x_{i}^{t}$ term by $x_{i}$ for any $2\leq i\leq n$ and $t\geq
2$, and $x_{1}^{t}$ term by $1$ for any $t\geq 1$.

Note that $\bm{a}_{i}\cdot\bm{a}_{j}=|F_{i}\cap F_{j}|$, and we replace
$x_{1}^{t}$ with $1$ in our multilinear reduction to get $f_{i}$, this acts as
if $1\in F_{j}$ when we evaluate $f_{i}(\bm{a}_{j})$ and hence

$f_{i}(\bm{a}_{j})=\prod_{\ell\in
L,\ell\neq|F_{i}|}(|F_{i}\cap(F_{j}\cup\\{1\\})|-\ell).$

Moreover, observe that if $i>j$ and $r\geq i$, then
$|F_{i}\cap(F_{j}\cup\\{1\\})|=|F_{i}\cap F_{j}|$ as $1\in F_{j}$, and if
$i>j$ and $r<i$, then $|F_{i}\cap(F_{j}\cup\\{1\\})|=|F_{i}\cap F_{j}|$ also
holds as $1\notin F_{i}$. Therefore, if $i>j$, we have
$f_{i}(\bm{a}_{j})=\prod_{\ell\in L,\ell\neq|F_{i}|}(|F_{i}\cap F_{j}|-\ell)$.
Since $\mathcal{F}$ is a Sperner family, we know $|F_{i}\cap F_{j}|\in
L\setminus\\{|F_{i}|\\}$, which implies that $f_{i}(\bm{a}_{j})=0$ for $i>j$.
If $i=j$, then $f_{i}(\bm{a}_{j})=\prod_{\ell\in
L,\ell\neq|F_{i}|}(|F_{i}|-\ell)\neq 0$. By Proposition 2.3, the polynomials
$\\{f_{i}\\}_{i=1}^{m}$ are linearly independent, which implies
$m\leq\sum_{j=0}^{s}\binom{n-1}{j}$ since for $i\in[m]$, $f_{i}(\bm{x})$ has
$n-1$ variables $x_{2},x_{3},\ldots,x_{n}$ and degree at most $s$. The first
result then follows.

For the second part, let $\ell_{1}=0\in L$. We will prove
$|\mathcal{F}|\leq\binom{n}{s}$ by induction on $s$, where $s=|L|$ and $n\geq
3s^{2}$. When $s=1$, by the first result, we have
$|\mathcal{F}|\leq\sum_{j=0}^{s}\binom{n-1}{j}=\binom{n}{1}$, and it is easy
to see that equality holds if and only if $\mathcal{F}={[n]\choose 1}$.

Now, suppose $s\geq 2$ and the result holds for any positive integer less than
$s$. If $L=\\{0,1,2,\ldots,s-1\\}$, then for any $S\in\binom{[n]}{s}$, there
exists at most one set in $\mathcal{F}$ containing $S$. Construct a new family
$\mathcal{F^{\prime}}:=\\{F^{\prime}_{1},\ldots,F^{\prime}_{m}\\}$, where
$F^{\prime}_{i}$ is an arbitrary subset of $F_{i}$ with size $s$ if
$|F_{i}|>s$, and $F^{\prime}_{i}=F_{i}$ otherwise. Since $\mathcal{F}$ is a
Sperner family, $\mathcal{F^{\prime}}$ is also a Sperner family with
$|F^{\prime}_{i}|\leq s$. Then by Theorem 2.2, we have $m\leq\binom{n}{s}$. It
is easy to see that the equality holds if and only if
$\mathcal{F}=\mathcal{F}^{\prime}=\binom{[n]}{s}$. Next, we assume that
$L\neq\\{0,1,2,\ldots,s-1\\}$.

Let $\mathcal{H}=\\{F:F\in\mathcal{F},|F|\leq s\\}$ and
$L^{\prime}=L\cap\\{0,1,2,\ldots,s-1\\}$. Then $\mathcal{H}$ is an
$L^{\prime}$-intersecting Sperner family, with $0\in L^{\prime}$ and
$|L^{\prime}|\leq s-1$. By the induction hypothesis, we have
$|\mathcal{H}|\leq\binom{n}{s-1}$.

For each $x\in[n]$, let $\mathcal{F}_{x}=\\{F\in\mathcal{F}:x\in F\\}$, and
$L\setminus\\{0\\}=\\{\ell_{2},\ell_{3},\ldots,\ell_{s}\\}$. Then
$\mathcal{F}_{x}$ is an $L\setminus\\{0\\}$-intersecting Sperner family with
$|L\setminus\\{0\\}|\leq s-1$, and by the first part
$|\mathcal{F}_{x}|\leq\sum_{i=0}^{s-1}\binom{n-1}{i}$. Then via double
counting, we have

$(m-|\mathcal{H}|)(s+1)\leq\sum_{x\in[n]}|\mathcal{F}_{x}|\leq
n\sum_{i=0}^{s-1}\binom{n-1}{i}.$

Since $n\geq 3s^{2},s\geq 2$, we have
$\sum_{i=0}^{s-2}\binom{n-1}{i}\leq\binom{n-1}{s-2}\cdot\sum_{i=0}^{s-2}2^{-i}\leq
2\binom{n-1}{s-2}$. Therefore, we have

$\displaystyle m$
$\displaystyle\leq|\mathcal{H}|+\frac{n}{s+1}\sum_{i=0}^{s-1}\binom{n-1}{i}\leq\binom{n}{s-1}+\frac{n}{s+1}\binom{n-1}{s-1}+\frac{n}{s+1}\sum_{i=0}^{s-2}\binom{n-1}{i}$
$\displaystyle\leq\binom{n}{s-1}+\frac{s}{s+1}\binom{n}{s}+\frac{2n}{s+1}\binom{n-1}{s-2}=\binom{n}{s-1}+\frac{s}{s+1}\binom{n}{s}+\frac{2(s-1)}{s+1}\binom{n}{s-1}$
$\displaystyle=\bigg{(}\frac{3s-1}{s+1}\cdot\frac{s}{n-s+1}+\frac{s}{s+1}\bigg{)}\binom{n}{s}\leq\bigg{(}\frac{1}{s+1}\cdot\frac{3s^{2}-s}{3s^{2}-s+1}+\frac{s}{s+1}\bigg{)}\binom{n}{s}<\binom{n}{s}.$

###  4.2 Proof of Theorem 1.5

Let $T\in\binom{[n]}{\ell_{1}}$ be the set satisfying that
$|\\{F\in\mathcal{F}:T\subseteq F\\}|$ is the largest among all sets in
$\binom{[n]}{\ell_{1}}$. Let $\mathcal{H}=\\{F\in\mathcal{F}:T\subseteq F\\}$.
If $\mathcal{H}=\mathcal{F}$, then ${\mathcal{F}}$ is trivially
$\ell_{1}$-intersecting and by Theorem 1.4, as $|T|=\ell_{1}\geq 1$, we have
$\\{F\setminus T:F\in\mathcal{F}\\}\subseteq 2^{[n]\setminus T}$ and so

$|\mathcal{F}|=\big{|}\\{F\setminus
T:F\in\mathcal{F}\\}\big{|}\leq\sum_{i=0}^{s}\binom{n-2}{i}=\binom{n}{s}-\binom{n-1}{s-1}+\sum_{i=0}^{s-2}\binom{n-2}{i}=\binom{n}{s}-\Omega_{s}(n^{s-1}).$

We may then assume ${\mathcal{H}}\neq{\mathcal{F}}$ and so there exists some
$E\in\mathcal{F}$ with $T\not\subseteq E$. For any $F\in\mathcal{H}$ we have
$|F\cap E|\geq\ell_{1}$, so $(E\setminus T)\cap F\neq\emptyset$, for otherwise
$T\subseteq E$. Consequently, for each $x\in E\setminus T$, letting
$\mathcal{F}_{x}=\\{F:F\in\mathcal{H},x\in F\\}$, we have
$\mathcal{H}\subseteq\bigcup_{x\in E\setminus T}\mathcal{F}_{x}$. Since for
any sets $A,B\in\mathcal{F}_{x}$, $\\{x\\}\cup T\subseteq A\cap B$, $F_{x}$ is
an $\\{\ell_{2},\ell_{3},\ldots,\ell_{s}\\}$-intersecting Sperner family.
Then, by Theorem 1.4, we have
$|\mathcal{F}_{x}|\leq\sum_{i=0}^{s-1}\binom{n-1}{i}$ and $|\mathcal{H}|\leq
M\sum_{i=0}^{s-1}\binom{n-1}{i}$.

Take an arbitrary set $X\in\mathcal{F}$. For any set $F\in\mathcal{F}$, we
have $|X\cap F|\geq\ell_{1}$, thus we can partition the whole family
$\mathcal{F}$ into subfamilies $\mathcal{H}_{Y}$ according to which
$\ell_{1}$-subset $Y\in\binom{X}{\ell_{1}}$ its members contain. Note that for
each such ${\mathcal{H}}_{Y}$, $|\mathcal{H}_{Y}|\leq|\mathcal{H}|$. As
$|X|\leq M$, we have
$|\mathcal{F}|\leq\binom{M}{\ell_{1}}|\mathcal{H}|=O_{M,s}(n^{s-1})$.

## Acknowledgement

We would like to express our gratitude to the anonymous reviewer for the
detailed and constructive comments which are helpful to the improvement of the
technical presentation of this paper. We also thank Yongtao Li and Tuan Tran
for helpful discussions.

## References

  * [1] N. Alon, L. Babai, and H. Suzuki.  Multilinear polynomials and Frankl–Ray-Chaudhuri–Wilson type intersection theorems.  J. Combin. Theory Ser. A, 58(2):165–180, 1991.
  * [2] L. Babai.  A short proof of the nonuniform Ray-Chaudhuri-Wilson inequality.  Combinatorica, 8(1):133–135, 1988.
  * [3] E. Bannai, E. Bannai, and D. Stanton.  An upper bound for the cardinality of an $s$-distance subset in real Euclidean space. II.  Combinatorica, 3(2):147–152, 1983.
  * [4] A. Blokhuis.  A new upper bound for the cardinality of $2$-distance sets in Euclidean space.  In Convexity and graph theory (Jerusalem, 1981), volume 87 of North-Holland Math. Stud., pages 65–66. North-Holland, Amsterdam, 1984\.
  * [5] B. Bollobás.  On generalized graphs.  Acta Math. Acad. Sci. Hungar., 16:447–452, 1965.
  * [6] W. Y. C. Chen and J. Liu.  Set systems with $L$-intersections modulo a prime number.  J. Combin. Theory Ser. A, 116(1):120–131, 2009.
  * [7] D. M. Cvetković.  Chromatic number and the spectrum of a graph.  Publ. Inst. Math. (Beograd) (N.S.), 14(28):25–38, 1972.
  * [8] P. Erdős, C. Ko, and R. Rado.  Intersection theorems for systems of finite sets.  Quart. J. Math. Oxford Ser. (2), 12:313–320, 1961.
  * [9] R. A. Fisher.  An examination of the different possible solutions of a problem in incomplete blocks.  Ann. Eugenics, 10:52–75, 1940.
  * [10] P. Frankl.  Bounding the size of a family knowing the cardinality of differences.  Studia Sci. Math. Hungar., 20(1-4):33–36, 1985.
  * [11] P. Frankl.  Antichains of fixed diameter.  Mosc. J. Comb. Number Theory, 7(3):3–33, 2017.
  * [12] P. Frankl.  A stability result for families with fixed diameter.  Combin. Probab. Comput., 26(4):506–516, 2017.
  * [13] P. Frankl.  A stability result for the Katona theorem.  J. Combin. Theory Ser. B, 122:869–876, 2017.
  * [14] P. Frankl and R. M. Wilson.  Intersection theorems with geometric consequences.  Combinatorica, 1(4):357–368, 1981.
  * [15] D. Gerbner and B. Patkós.  Extremal finite set theory.  Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL, 2019.
  * [16] A. J. W. Hilton and E. C. Milner.  Some intersection theorems for systems of finite sets.  Quart. J. Math. Oxford Ser. (2), 18:369–384, 1967.
  * [17] H. Huang, O. Klurman, and C. Pohoata.  On subsets of the hypercube with prescribed Hamming distances.  J. Combin. Theory Ser. A, 171:105156, 21, 2020.
  * [18] K.-W. Hwang and Y. Kim.  A proof of Alon-Babai-Suzuki’s conjecture and multilinear polynomials.  European J. Combin., 43:289–294, 2015.
  * [19] K.-W. Hwang and N. N. Sheikh.  Intersection families and Snevily’s conjecture.  European J. Combin., 28(3):843–847, 2007.
  * [20] G. Katona.  Intersection theorems for systems of finite sets.  Acta Math. Acad. Sci. Hungar., 15:329–337, 1964.
  * [21] G. O. H. Katona.  Open problems.  Matematikus Kurir, 1983.
  * [22] D. J. Kleitman.  On a combinatorial conjecture of Erdős.  J. Combinatorial Theory, 1:209–214, 1966.
  * [23] D. Lubell.  A short proof of Sperner’s lemma.  J. Combinatorial Theory, 1:299, 1966.
  * [24] L. D. Mešalkin.  A generalization of Sperner’s theorem on the number of subsets of a finite set.  Teor. Verojatnost. i Primenen, 8:219–220, 1963.
  * [25] D. Mubayi and Y. Zhao.  On the VC-dimension of uniform hypergraphs.  J. Algebraic Combin., 25(1):101–110, 2007.
  * [26] D. K. Ray-Chaudhuri and R. M. Wilson.  On $t$-designs.  Osaka Math. J., 12(3):737–744, 1975.
  * [27] H. S. Snevily.  On generalizations of the de Bruijn-Erdős theorem.  J. Combin. Theory Ser. A, 68(1):232–238, 1994.
  * [28] H. S. Snevily.  A generalization of the Ray-Chaudhuri-Wilson theorem.  J. Combin. Des., 3(5):349–352, 1995.
  * [29] H. S. Snevily.  A sharp bound for the number of sets that pairwise intersect at $k$ positive values.  Combinatorica, 23(3):527–533, 2003.
  * [30] E. Sperner.  Ein Satz über Untermengen einer endlichen Menge.  Math. Z., 27(1):544–548, 1928.
  * [31] X. Wang, H. Wei, and G. Ge.  A strengthened inequality of Alon-Babai-Suzuki’s conjecture on set systems with restricted intersections modulo $p$.  Discrete Math., 341(1):109–118, 2018.
  * [32] K. Yamamoto.  Logarithmic order of free distributive lattice.  J. Math. Soc. Japan, 6:343–353, 1954.