# Statistical approach to study neutrino interactions.

Q. Gani1,a,∗, M. Hameeda2,3,b
1 Govt. College for Women Srinagar, 190003 India
2 Inter University Centre for Astronomy and Astrophysics Pune, 411007 India
3 Department of Physics, Govt. Degree College Tangmarg, 193402 India
a qudsiagani6<EMAIL_ADDRESS>
###### Abstract

Neutrino oscillation parameters can be understood in a better way by building
a more complete picture of neutrino interactions. This poses a series of
important theoretical and experimental challenges because of the elusive
nature of neutrino and an inherent difficulty in its detection. This work is
an attempt to study neutrino interactions through a purely theoretical
approach using fluctuation theory. Though experiments are trying hard to shed
more light on neutrino interactions, our knowledge of neutrino oscillation
parameters is not precise at different energy levels. In this context, one
cannot help but give a bold try to marvel at how far our theoretical
frameworks can extend.

PACS: 25.30.Pt, 13.15.+g, 14.60.Lm , 12.15.Mm, 14.60.Pq

KEYWORDS: cross-section, neutrino-nucleus interactions, neutrino oscillations,
neutrino mass and mixing, distribution function..

*corresponding author

###### Contents

  1. 1 Introduction
  2. 2 Gaussian Distribution and mean square fluctuations of neutrinos
  3. 3 Riemannian curvature of the $(m_{i},\Phi_{i})$ space and the neutrino interactions
  4. 4 Conclusions
  5. 5 Conflict of interest statement
  6. 6 Funding

##  1 Introduction

For decades there had been a problem that the number of measured neutrinos
coming from the sun due to nucleosynthesis is inconsistent with the prediction
from the standard solar model [1]. Similarly the observed deficit of muon-
neutrinos produced by cosmic ray high energy proton-proton collisions in the
atmosphere remained a puzzle for long [2]. In 1998, it was revealed at Super-
Kamiokande, Japan, that neutrinos can oscillate between their flavours [3].
First proposed by Pontecorvo in 1958, neutrino oscillations is a quantum
mechanical phenomenon [4] which arises since the mass eigen-states of
neutrinos $(\nu_{1},\nu_{2},\nu_{3})$ and their flavour eigen-states
$(\nu_{\mu},\nu_{e},\nu_{\tau})$ are not identical, due to which there is a
flip in their flavour or mass-mixing as these travel large distances. These
are related as:

$\nu_{l}=\Sigma U_{lm}\nu_{m}$ (1.1)

where m = 1, 2, 3 and $l$ = e, $\mu$, $\tau$. $U_{lm}$ is a unitary matrix,
called Pontecorvo-Maki-Nakagawa-Sakata or PMNS matrix [5]. It is a fundamental
result that neutrinos cannot oscillate unless these have mass. The discovery
of neutrino oscillation resolved the anomalies of both solar and atmospheric
neutrino deficits. The discovery of neutrino oscillations is one of the most
exciting recent developments in particle physics. Current and future neutrino
experiments are aiming to make precise measurements of the oscillation
parameters. Improving our understanding of neutrino interactions is crucial to
these precision studies of neutrino oscillations [6]. Currently around the
world, there are many ongoing experiments such as T2K [7], MINOS [8], SNO [9],
NO$\nu$A [10], Double Chooz [11], UNO [12], Super-K III [13], Hyper-K [14],
OPERA [15] etc. which are actively engaged in neutrino detection and
measurement of its oscillation parameters. Additionally, many experiments are
being planned to study the various aspects of neutrino interactions. These
include ICARUS [16], DUNE [17] etc. The Neutrino Factory, NuFact [18] has also
been conducting yearly workshops since 1999 to review the results of currently
operating experiments and design the next generation experiments. However, the
elusive nature of the neutrino and the inherent difficulty in its detection
has created many false signals. The detection of neutrino is inferred by the
particles it leaves behind, if at all it undergoes any interaction. Keeping in
view the importance of such studies, an attempt has been made to analyse the
neutrino interactions using fluctuation theory approach. The merit of using
this approach lies in the fact that neutrinos during the course of their
flight undergo fluctuations in their flux, flavor, mass and number.

##  2 Gaussian Distribution and mean square fluctuations of neutrinos

Suppose a $\mid\nu_{\alpha}>$ flavor neutrino beam is generated at the source
at space-time point (0,0) and is directed along X-axis to a detector at some
distance L from the generation point, then the probability for finding a
different flavor state $\mid\nu_{\beta}>$ at some space-time point (x,t) in
the beam at the detector will be given by square of the amplitude term
$\mid<\nu_{\beta}(x,t)\mid\nu_{\alpha}(0,0)>\mid$
Hence the oscillation probability is

$P(\nu_{\alpha}\rightarrow\nu_{\beta})=\mid<\nu_{\beta}(x,t)\mid\nu_{\alpha}(0,0)>\mid^{2}$
(2.2)

In terms of the mixing matrix relation, between the flavour states
$\mid\nu_{\alpha}>$ and $\mid\nu_{\beta}>$ and mass eigen-states
$\mid\nu_{1}>$ and $\mid\nu_{2}>$, the above equation may be written as

$P(\nu_{\alpha}\rightarrow\nu_{\beta})=2\cos^{2}\theta\sin^{2}\theta(1-\cos(\phi_{2}-\phi_{1}))$
(2.3)

where $\phi_{1}$ and $\phi_{2}$ are the phases of the wave-packets of each of
the mass eigen-states. The eigen-states of Hamiltonian are $\mid\nu_{1}>$ and
$\mid\nu_{2}>$ with eigen-values $m_{1}$ and $m_{2}$ for neutrinos at rest.
Neutrinos are produced in weak interactions in weak eigen-states of definite
lepton number $(\mid\nu_{e}>,\mid\nu_{\mu}>$ or $\mid\nu_{\tau}>)$ that are
not energy eigen-states. These two sets of states are related to each other by
a unitary matrix U given by [19]

$U=\left({\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\\\ \end{array}}\right)$

Where $\theta$ is an unspecified parameter known as the mixing angle and is to
be measured experimentally [20].
Now using the trigonometric identities
$\cos\theta\sin\theta=\frac{1}{2}\sin(2\theta)$ and
$2\sin^{2}\theta=1-\cos(2\theta)$, we find that

$P(\nu_{\alpha}\rightarrow\nu_{\beta})=\sin^{2}(2\theta)\sin^{2}\left(\frac{\phi_{2}-\phi_{1}}{2}\right)$
(2.4)

Using the expression for phase, where $i$ denotes the flavour state
$(e,\mu\hskip 5.69046ptor\hskip 5.69046pt\tau)$,

$\phi_{i}=E_{i}t-p_{i}x$ (2.5)

the phase difference is given by

$\phi_{2}-\phi_{1}=(E_{2}-E_{1})t-(p_{2}-p_{1})x$ (2.6)

Now we take into account relativistic neutrinos of momentum $p$, which can be
written as

$p_{i}=E_{i}\sqrt{1-\frac{m^{2}_{i}}{E^{2}_{i}}}$ (2.7)

Furthermore, for relativistic neutrinos, it is a very reasonable assumption
that t = x = L [21], so

$\phi_{2}-\phi_{1}=\left(\frac{m^{2}_{1}}{2E_{1}}-\frac{m^{2}_{2}}{2E_{2}}\right)L=\frac{\Delta
m^{2}L}{2E}$ (2.8)

Now, from equation (2.5), we get

$E_{i}=\frac{\phi_{i}}{t}+\frac{p_{i}x}{t}$ (2.9)

Again assuming t = x = L, we write

$E_{i}=\frac{\phi_{i}}{L}+p_{i}$ (2.10)

Equation (2.7) can be alternately written as

$P^{2}_{i}=E^{2}_{i}-m^{2}_{i}$ (2.11)

Using the same in equation (2.10), we get the energy eigen values of neutrinos
as

$E_{i}=\frac{\phi_{i}}{2L}+\frac{m^{2}L}{2\phi_{i}}$ (2.12)

Where we obtain $E_{i}$ as a function of two variables $\phi_{i}$ and $m_{i}$,
thus

${E_{i}}={E_{i}(\phi_{i},m_{i})}$ (2.13)

Expanding $\Delta{E_{i}}$ as a series upto the second-order terms, we get

$\Delta{E_{i}}=\frac{\partial{E_{i}}}{\partial\phi_{i}}\Delta{\phi_{i}}+\frac{\partial{E_{i}}}{\partial
m_{i}}\Delta{m_{i}}+\frac{1}{2}\left[\frac{\partial^{2}{E_{i}}}{\partial\phi_{i}^{2}}(\Delta\phi_{i})^{2}+\frac{\partial^{2}{E_{i}}}{\partial\phi_{i}\partial
m_{i}}\Delta\phi_{i}\Delta m_{i}+\frac{\partial^{2}{E_{i}}}{\partial
m_{i}^{2}}(\Delta m_{i})^{2}\right]$ (2.14)

Under extremum conditions, first derivatives of energy vanishes and we are
left with

$\Delta{E_{i}}=\frac{1}{2}\left[\frac{\partial^{2}{E_{i}}}{\partial\phi_{i}^{2}}(\Delta\phi_{i})^{2}+\frac{\partial^{2}{E_{i}}}{\partial\phi_{i}\partial
m_{i}}\Delta\phi_{i}\Delta m_{i}+\frac{\partial^{2}{E_{i}}}{\partial
m_{i}^{2}}(\Delta m_{i})^{2}\right]$ (2.15)

Figure 1: Neutrino interaction cross-sections

Now given the trend exhibited by neutrino interaction cross-sections obtained
from various experimental and simulation results [22] shown in figure 1, it is
very reasonable to express neutrino oscillation probability distribution as a
Guassian function where we know that the probability $W$ of a fluctuation is
given by [23]

$W\propto\exp{-\frac{R_{min}}{T}}$ (2.16)

Where $R_{min}$ is the work needed to carry out the change or fluctuation at a
temperature $T$.
In our case

$\frac{R_{min}}{T}=\frac{\Delta{E_{i}}}{T}$ (2.17)

Thus the probability of fluctuation is

$w\propto\exp{-\frac{\Delta{E_{i}}}{T}}$ (2.18)

Using equation (2.15) in equation (2.18) we obtain

$W\propto\exp{-\frac{1}{2T}\left[\frac{\partial^{2}{E_{i}}}{\partial\phi_{i}^{2}}(\Delta\phi_{i})^{2}+\frac{\partial^{2}{E_{i}}}{\partial\phi_{i}\partial
m_{i}}\Delta\phi_{i}\Delta m_{i}+\frac{\partial^{2}{E_{i}}}{\partial
m_{i}^{2}}(\Delta m_{i})^{2}\right]}$ (2.19)

Now the general Gaussian distribution is given as [23]

$W=\frac{\sqrt{\beta}}{(2\pi)^{n/2}}\exp{-\frac{1}{2}\beta_{ij}x_{i}x_{j}}$
(2.20)

where $\beta$ is the determinant of $\beta_{ij}$.
For two variables, the gaussian distribution can be written as

$W=\frac{\sqrt{\beta}}{2\pi}\exp{-\frac{1}{2}\left[\frac{(\Delta
x_{1})^{2}}{<(\Delta x_{1})^{2}>}+\frac{(\Delta x_{2})^{2}}{<(\Delta
x_{2})^{2}>}\right]}$ (2.21)

where $<(\Delta x_{1})^{2}>=\frac{1}{\beta_{11}}$ and $<(\Delta
x_{2})^{2}>=\frac{1}{\beta_{22}}$ which for the present case can be written as

$W=\frac{\sqrt{\beta}}{2\pi}\exp{-\frac{1}{2}\left[\frac{(\Delta\phi_{i})^{2}}{<(\Delta\phi_{i})^{2}>}+\frac{(\Delta
m_{i})^{2}}{<(\Delta m_{i})^{2}>}\right]}$ (2.22)

where

$\displaystyle<(\Delta\phi_{i})^{2}>=\frac{T}{\frac{\partial^{2}{E_{i}}}{\partial\phi_{i}^{2}}}=\frac{1}{\beta_{\phi\phi}},$
(2.23) $\displaystyle<(\Delta
m_{i})^{2}>=\frac{T}{\frac{\partial^{2}{E_{i}}}{\partial
m_{i}^{2}}}=\frac{1}{\beta_{mm}}$ (2.24)

and

$\beta=\beta_{\phi\phi}\beta_{mm}$ (2.25)

Further we assume that $\Delta{\phi}$ and $\Delta m$ are statistically
independent so that $<\Delta{\phi}\Delta{m}>=0$.
After calculations we find the mean square fluctuations as

$<(\Delta\phi_{i})^{2}>=\frac{\phi_{i}^{3}T}{m_{i}^{2}L}$ (2.26) $<(\Delta
m_{i})^{2}>=\frac{\phi_{i}T}{L}$ (2.27)

and the determinant $\beta$ as

$\beta=\frac{m_{i}^{2}L^{2}}{\phi_{i}^{4}T^{2}}$ (2.28)

Thus the Guassian distribution becomes

$W=\frac{m_{i}L}{2\pi\phi_{i}^{2}T}\exp\left(-\frac{1}{2T}\left[\frac{m_{i}^{2}L}{\phi_{i}^{3}}(\Delta\phi_{i})^{2}+\frac{L}{\phi_{i}}(\Delta
m_{i})^{2}\right]\right)$ (2.29)

Comparing the mean square fluctuations we find an interesting result i.e

$\frac{<(\Delta
m_{i})^{2}>}{m_{i}^{2}}=\frac{<(\Delta\phi_{i})^{2}>}{\phi_{i}^{2}}=\frac{\phi_{i}T}{m_{i}^{2}L}$
(2.30)

The phase of the wave-packet of each of the mass eigen-states is something
that differentiates neutrino mixing from real rotations. It is only present if
all the three flavours mix. Alternatively, one can state that if there is no
phase difference, there is no flavour oscillation or mass mixing. Hence the
mean square fluctuations in phase and mass should be equivalent as implied by
equation (2.30).
Now using this in equation (2.29) we can also write the probability as

$W=\frac{m_{i}L}{2\pi\phi_{i}^{2}T}\exp\left(-\frac{1}{T}\frac{m_{i}^{2}L}{\phi_{i}^{3}}(\Delta\phi_{i})^{2}\right)$
(2.31)

This is a simple and compact expression for neutrino oscillation probability
in vacuum. For a given length of flight $L$ of neutrinos at a temperature $T$,
on can find the oscillation probability from the following plot

Figure 2: Theoretical plot of neutrino oscillation probability

Now according to the most general expression for neutrino oscillation
probability given by the following expression, we find that neutrino
oscillations are sensitive only to the difference in the squares of their
masses [24].

$P(\nu_{\alpha}\rightarrow\nu_{\beta})=\sin^{2}(2\theta)\sin^{2}\left(1.27\Delta
m^{2}\frac{L}{E}\right)$ (2.32)

As of 2020, [25] the best-fit value of the difference of the squares of the
masses of mass eigenstates 1 and 2 is $\Delta m^{2}_{21}$ = 0.000074 $eV^{2}$,
while for eigenstates 2 and 3, it is $\Delta m^{2}_{32}$ = 0.00251 $eV^{2}$.
Since $\Delta m^{2}_{32}$ is the difference of two squared masses, at least
one of them must have a value which is the square root of this value. From
this we infer that there exists at least one neutrino mass eigenstate with a
mass of at least 0.05 $eV$ [26].
Now when this result is compared to the theoretical plot shown in figure 2, it
appears quite consistent. This can further throw a hint at neutrino masses.

##  3 Riemannian curvature of the $(m_{i},\Phi_{i})$ space and the neutrino
interactions

We see that the probability distribution given by equation (2.31) depends on
two parameters viz; $m_{i}$ and $\phi_{i}$, therefore we choose to work with
$(m_{i},\Phi_{i})$ space. The Gaussian distribution obtained in the previous
section can alternatively be written as

$W\propto\exp{-\frac{1}{2}\left[g_{\phi\phi}(\Delta\phi_{i})^{2}+g_{mm}(\Delta
m_{i})^{2}\right]}$ (3.33)

where we assume a space with coordinates $\phi_{i}$ and $m_{i}$ and define the
line element in this space as

$\Delta l^{2}=g_{\phi\phi}\Delta\phi_{i}^{2}+g_{mm}\Delta m_{i}^{2}$ (3.34)

and the metric elements of this space are expressed as

$g_{\phi\phi}=\frac{1}{T}\frac{\partial^{2}{E_{i}}}{\partial\phi_{i}^{2}}\\\
=\frac{m_{i}^{2}L}{\phi_{i}^{3}T}$ (3.35)
$g_{mm}=\frac{1}{T}\frac{\partial^{2}{E_{i}}}{\partial m_{i}^{2}}\\\
=\frac{L}{\phi_{i}T}$ (3.36)

and

$g=g_{\phi\phi}g_{mm}\\\ =\frac{m_{i}^{2}L^{2}}{\phi_{i}^{4}T}$ (3.37)

Knowing the metric elements of a space, we calculate the curvature using the
formula

$R=\frac{1}{\sqrt{g}}\left[\frac{\partial}{\partial
m_{i}}\left(\frac{1}{\sqrt{g}}\frac{\partial g_{\phi\phi}}{\partial
m_{i}}\right)+\frac{\partial}{\partial\phi_{i}}\left(\frac{1}{\sqrt{g}}\frac{\partial
g_{mm}}{\partial\phi}\right)\right]$ (3.38)

After some calculations we find

$R=0$ (3.39)

This is quite an interesting result. The curvature is zero when there are no
mutual interactions and this is in agreement with the fact that neutrinos are
indeed very feebly interacting particles. This can be gauged from the fact
that whereas the photons emitted from the solar core may require 40,000 years
to diffuse to the outer layers of the Sun, neutrinos generated in the fusion
reactions at the stellar core cross this distance practically unimpeded at
nearly the speed of light [27]. Neutrinos interact so weakly with the matter
that these can go through the whole of earth without deviation.
Furthermore, the order of a system is measured in terms of correlation
function. As for instance, the interactions can be determined by the density
correlation function which is related to number fluctuations as [23]

$\int\zeta dV=\frac{<(\Delta N_{i})^{2}>}{\bar{N_{i}}}-1$ (3.40)

For no mutual interactions i.e ideal approximation

$\frac{<(\Delta N_{i})^{2}>}{\bar{N_{i}}}-1=0$ (3.41)

This gives

${<(\Delta N_{i})^{2}>}=\bar{N_{i}}$ (3.42)

Now using mass in place of density, we may consider the assumption

$\frac{<(\Delta m_{i})^{2}>}{m_{i}^{2}}=\frac{<(\Delta
N_{i})^{2}>}{\bar{N_{i}}^{2}}=\frac{\phi_{i}T}{m^{2}L}$ (3.43)

from where we get

$\bar{N_{i}}=\frac{m_{i}^{2}L}{\phi_{i}T}$ (3.44)

Equation (3.43) further relates the mass mixing with the number fluctuation of
neutrinos so that we can write the extended result as

$\frac{<(\Delta m_{i})^{2}>}{m_{i}^{2}}=\frac{<(\Delta
N_{i})^{2}>}{\bar{N_{i}}^{2}}=\frac{<(\Delta\phi_{i})^{2}>}{\phi_{i}^{2}}$
(3.45)

This is obvious since a change in the phase or mass produces a neutrino of
different flavor and consequently alters the number of neutrinos corresponding
to a given flavor. All these are well established results and raise the scope
of fluctuation theory as an important tool in studying neutrino interactions.

##  4 Conclusions

We have tried to ground our theory in a domain where little or no data exists,
and have sought other means to assess its validity. The oscillation
probability given by equation (2.4) is derived on the basis of two important
assumptions;that the neutrino flavour and mass states are mixed and that a
coherent superposition of mass states is created at the weak vertex [28]. This
coherent superposition reflects the fact that we can’t experimentally resolve
which mass state was created at the vertex. If one could find that, then it
would also let us know the mass of the neutrino state that propagates to the
detector and then there would be no idea of superposition. The mass and phase
fluctuations are therefore identical which is also implied by our result in
equation (2.30).

An interesting class of theories called grand unified theories suggests that
all interactions (strong, electromagnetic, and weak) are unified at a large
energy scale and that the neutrino masses are inversely proportional to this
scale. A way to probe this very high energy scale is to search for very small
neutrino masses. One such attempt could be through the guassian distribution
approach which has been attempted in the present study.

Similarly a zero Rieman curvature for neutrino interactions as shown in
equation (3.39) leads to ideal gas like approximation for neutrinos to study
their associated thermodynamic properties.
Further, the neutrino number fluctuations expressed in equation (3.43) as a
consequence of change in mass or phase is also an obvious result from the
existing knowledge of neutrinos confirmed to us once again through the
fluctuation theory.

##  5 Conflict of interest statement

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

##  6 Funding

The work is supported and funded by Department of Science, Technology and
Innovation Council (JKST$\&$IC), government of J$\&$K, INDIA-190001.

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# Zero subsums in vector spaces over finite fields

Cosmin Pohoata Department of Mathematics, Yale University, USA. Email:
<EMAIL_ADDRESS>Dmitriy Zakharov Laboratory of Combinatorial and
Geometric Structures, MIPT, Russia. Email<EMAIL_ADDRESS>
###### Abstract

The Olson constant $\mathcal{O}L(\mathbb{F}_{p}^{d})$ represents the minimum
positive integer $t$ with the property that every subset
$A\subset\mathbb{F}_{p}^{d}$ of cardinality $t$ contains a nonempty subset
with vanishing sum. The problem of estimating
$\mathcal{O}L(\mathbb{F}_{p}^{d})$ is one of the oldest questions in additive
combinatorics, with a long and interesting history even for the case $d=1$.

In this paper, we prove that for any fixed $d\geq 2$ and $\epsilon>0$, the
Olson constant of $\mathbb{F}_{p}^{d}$ satisfies the inequality

$\mathcal{O}L(\mathbb{F}_{p}^{d})\leq(d-1+\epsilon)p$

for all sufficiently large primes $p$. This settles a conjecture of Hoi Nguyen
and Van Vu.

##  1 Introduction

For a subset $A$ of an additive group $G$, consider the set of all nonempty
subsums

$\Sigma^{*}(A):=\left\\{\sum_{x\in B}x\ |\ B\subset
A,B\neq\emptyset\right\\}.$

The _Olson constant_ $\mathcal{O}L(G)$ represents the minimum $t$ such that
every subset $A\subset G$ of cardinality $t$ satisfies $0\in\Sigma^{*}(A)$.
This is a well-known quantity in additive combinatorics, which is notoriously
difficult to estimate even for the most basic groups. Its nice history begins
in 1964 with Erdős and Heilbronn, who in [6] proved that there exists an
absolute constant $c$ such that $\mathcal{O}L(\mathbb{F}_{p})\leq c\sqrt{p}$,
where $p$ is an odd prime. In the same paper, they conjectured that their
result should generalize to arbitrary additive groups $G$ and that the optimal
constant $c$ in the inequality above is probably $c=\sqrt{2}$. A few years
later Szemerédi [19] settled the former conjecture in the affirmative. The
result of Erdős and Heilbronn for $\mathbb{F}_{p}$ and Szemerédi’s theorem for
general groups were both later refined by Olson in [14], [15] and [16], who
proved that $\mathcal{O}L(G)\leq 2\sqrt{|G|}$ and also introduced a remarkable
group ring approach (which has also recently resurfaced in the context of the
polynomial method developments around the cap set problem; see [17] and [18]).
Olson’s result was subsequently pushed further by Hamidoune and Zemor [9], who
proved that $\mathcal{O}L(G)\leq\sqrt{2|G|}+O(|G|^{1/3}\log|G|)$, and among
other things established the correct order of growth for
$\mathcal{O}L(\mathbb{F}_{p})$, up to lower order terms. In 2008, Nguyen,
Szemerédi and Vu [12] finally removed the lower terms in the primordial case
$G=\mathbb{F}_{p}$, therefore proving the optimal inequality
$\mathcal{O}L(\mathbb{F}_{p})\leq\sqrt{2p}$ for all sufficiently large primes
$p$. This work was also further refined in two separate rounds by Balandraud
in [3] and [4], who finally gave a short alternative argument which works for
all odd primes $p$ based on the quantitative Combinatorial Nullstellensatz
introduced by Karasev and Petrov in [11].

In this paper, we address the problem for $G=\mathbb{F}_{p}^{d}$, where $p$ is
an odd prime number, $d\geq 2$, and $\mathbb{F}_{p}^{d}$ denotes as usual the
vector space of $d$-dimensional vectors with coordinates from
$\mathbb{F}_{p}$. The situation in higher dimensions has been traditionally
known to be much more complicated. For $d=2$, the first important result only
appeared in 2004. In [8], Gao, Ruzsa and Thangadurai proved that
$\mathcal{O}L(\mathbb{F}_{p}^{2})=p+\mathcal{O}L(\mathbb{F}_{p})-1$ holds for
all primes $p>4.67\times 10^{34}$, thus establishing a beautiful connection
between $\mathcal{O}L(\mathbb{F}_{p}^{2})$ and $\mathcal{O}L(\mathbb{F}_{p})$.
In particular, given the successful story for $\mathbb{F}_{p}$, this result
also determines the Olson constant constant of $\mathbb{F}_{p}^{2}$ for large
primes. For higher dimensions, however, not much more is known. In the same
paper [8], Gao, Ruzsa and Thangadurai conjectured that

$\mathcal{O}L(\mathbb{F}_{p}^{d})=p+\mathcal{O}L(\mathbb{F}_{p}^{d-1})-1$ (1)

should hold in general for all $d\geq 2$ and for all sufficiently large primes
$p$, but this is still a (difficult) open problem. It is also perhaps worth
mentioning the curiosity that the assumption that $p$ is sufficiently large is
necessary this time around, see for instance the discussion from [7]. In 2011,
Nguyen and Vu [13] also studied this higher dimensional problem and proposed
the following asymptotic version of the conjecture: for any fixed $d\geq 2$
and $\epsilon>0$, the Olson constant of $\mathbb{F}_{p}^{d}$ satisfies the
inequality

$\mathcal{O}L(\mathbb{F}_{p}^{d})\leq(d-1+\epsilon)p$ (2)

for all sufficiently large primes $p$. Since
$\mathcal{O}L(\mathbb{F}_{p})=O(\sqrt{p})=o(p)$, it is clear that (1) implies
(2), but in some sense (2) should still capture all of the difficulties around
(1) when $d\geq 3$. Extending an elegant alternative approach they found for
the case $d=2$ of the Gao-Ruzsa-Thangadurai conjecture, Nguyen and Vu then
established this asymptotic conjecture when $d=3$; however, their argument has
various serious limitations already starting with $d\geq 4$, and so no further
progress has been made since.

Our main result is a resolution of this conjecture of Nguyen and Vu in all
dimensions $d\geq 2$ by using a new approach inspired by the second author’s
recent work on the Erdős-Ginzburg-Ziv problem [20].

###### Theorem 1

For any fixed $d\geq 2$ and $\epsilon>0$, the Olson constant of
$\mathbb{F}_{p}^{d}$ satisfies the inequality

$\mathcal{O}L(\mathbb{F}_{p}^{d})\leq(d-1+\epsilon)p$

for all sufficiently large primes $p$.

We include the proof of Theorem 1 in Section 3, after discussing terminology
and the required preliminary results in Section 2.

Before we move on however to the technical details, we end this section with a
high-level overview of the argument. Starting with a set
$X\subset\mathbb{F}_{p}^{d}$ of size $(d-1+\varepsilon)p$, where $p$ is a
sufficiently large prime number, the first important idea is to prove that one
can reduce the problem of finding a vanishing subsum in $\Sigma^{*}(X)$ to the
case when $X$ lies in a translate of the form
$v+[-K,K]^{l}\times\mathbb{F}_{p}^{d-l}$, for some
$l\in\left\\{1,\ldots,d-1\right\\}$, $v\in\mathbb{F}_{p}^{d}$, and where
$[-K,K]$ stands for the interval $\left\\{-K,-(K-1),\ldots,(K-1),K\right\\}$
–regarded as a subset of $\mathbb{F}_{p}$ (whose size does not depend on $p$).
The second idea is that if $0\not\in\Sigma^{*}(X)$, then one can also force
$X$ to satisfy some further refined structural properties. Roughly, we’ll be
able to assume among other things, for example, that $X$ must always have some
positive proportion of its elements outside the set
$\left\\{x\in\mathbb{F}_{p}^{d}~{}|~{}\xi(x)\in[-K,K]\right\\}$, for every
linear function $\xi\ :\ \mathbb{F}_{p}^{d}\to\mathbb{F}_{p}$ (except for some
trivial cases). The absence of “linear concentration” is crucial because the
third main idea is to consider the projection of this structured set $X$ onto
the first $l$ coordinates. The image $Y$ of this projection is a large
multiset in $\mathbb{F}_{p}^{l}$, so we can make use of tools such as the
Combinatorial Nullstellensatz to find a suitable subsequence whose sum of
elements vanishes and whose elements have various prescribed multiplicities.
Finally, in order to close the argument, we then need to use the rich
structure of $X$ to lift this auxiliary zero sum subsequence in $Y$ from the
previous step up to an actual proper subset of $X$ whose sum of elements
vanishes.

While sharing a rather similar philosophy with the method of Nguyen and Vu
from [13] (where projection is also important), finding the right framework to
project and lift to capture higher dimensional (additive) information and
establishing the precise structural results which allow our procedure to go
through for all dimensions $d\geq 2$ requires several new ideas, with both
algebraic and probabilistic ingredients.

##  2 Preliminaries

A function $\xi$ on the space $\mathbb{F}_{p}^{d}$ is called linear if it has
the form

$\xi(x_{1},\ldots,x_{d})=a_{0}+a_{1}x_{1}+\ldots+a_{d}x_{d},$ (3)

for some $a_{i}\in\mathbb{F}_{p}$. Linear functions $\xi_{1},\ldots,\xi_{l}$
are called linearly independent if their “linear parts”, i.e. the vectors
$(a_{1},\ldots,a_{d})$ from (3) are linearly independent in
$\mathbb{F}_{p}^{d}$.

If $\xi$ is a linear function and $K\in\mathbb{N}$ we denote by $H(\xi,K)$ the
set

$\left\\{x\in\mathbb{F}_{p}^{d}~{}|~{}\xi(x)\in[-K,K]\right\\},$

where $[-K,K]$ stands for the interval
$\left\\{-K,-(K-1),\ldots,(K-1),K\right\\}$, regarded as a subset of
$\mathbb{F}_{p}$. Given a linear function $\xi\ :\
\mathbb{F}_{p}^{d}\to\mathbb{F}_{p}$ and a multiset $X$ in
$\mathbb{F}_{p}^{d}$, for $K\in\mathbb{N}$ and $\delta\geq 0$ we say that $X$
is $(K,\delta)$-thick along $\xi$ if $|X\setminus H(\xi,K)|\geq\delta|X|$,
where the cardinality $|X|$ is calculated with multiplicities.

In what follows, $\delta\in(0,1),K^{\prime},K\in\mathbb{N}$ and
$g:\mathbb{N}\rightarrow\mathbb{N}$ is an increasing function. We use the
convention that $0$ is an element of $\mathbb{N}$. For $n\in\mathbb{N}$ we
denote by $[n]$ the set $\\{1,\ldots,n\\}$.

###  2.1 Tube decomposition

Definition: A set $X\subset\mathbb{F}_{p}^{d}$ is called
$(K,K^{\prime},\delta)$-tubular if there exists $l\in[0,d]$ and an affine
isomorphism $\psi:\mathbb{F}_{p}^{d}\rightarrow\mathbb{F}_{p}^{d}$ such that
$\psi X$ is contained in the set $[-K,K]^{l}\times\mathbb{F}_{p}^{d-l}$ and
satisfies the following property: $\psi X$ is $(K^{\prime},\delta)$-thick
along any linear function $\xi$ which is not constant on
$\\{0\\}\times\mathbb{F}_{p}^{d-l}$.

So, for instance, if $l=0$ then $X$ is $(K^{\prime},\delta)$-thick along any
non-constant linear function. If $l=d$ then, after an appropriate change of
coordinates, $X$ is contained in the box
$[-K,K]^{d}\subset\mathbb{F}_{p}^{d}$. In general, we allow some combination
of the above situations.

Note that a set $X$ is $(0,K,\delta)$-tubular if it is $(K,\delta)$-thick
along any linear map $\xi$ which is not constant on the affine hull of $X$.
Indeed, the space $\\{0\\}\times\mathbb{F}_{p}^{d-l}$ from the definition
above must coincide with the affine hull of $\psi X$. We shall say that in
this case $X$ is $(K,\delta)$-thick in its affine hull.

###### Lemma 1

Suppose that $\delta<2^{-d-1}$ and $K_{0}\geq 0$. Then for any set
$X\subset\mathbb{F}_{p}^{d}$ there exists $Y\subset X$ of size at least
$(1-2^{d+1}\delta)|X|$ which is $(K,g(K),\delta)$-tubular where
$K=g^{l}(K_{0})$ for some $l\in[0,d]$.

Proof of Lemma 1: Let $\xi_{1},\ldots,\xi_{l}$ be the maximal sequence of
linearly independent linear functions such that $X$ is
$(g^{i}(K_{0}),2^{i}\delta)$-thin along $\xi_{i}$ for any $i=1,\ldots,l$. Note
that we can have $l=0$, in which case we put $g^{0}(K_{0})=K_{0}$.

Consider $K=g^{l}(K_{0})$ and let

$Y=X\cap\bigcap_{i=1}^{l}H(\xi_{i},K).$

By the definition of $\xi_{i}$’s we have

$|Y|\geq|X|-|X|\sum_{i=1}^{l}2^{i}\delta\geq|X|(1-(2^{l+1}-1)\delta).$ (4)

Moreover, for any linear function $\eta$ which is linearly independent from
$\xi_{1},\ldots,\xi_{l}$, note that the set $X$ is
$(g(K),2^{l+1}\delta)$-thick along $\eta$. Consequently, by (4), the set
$Y\subset X$ is $(g(K),\delta)$-thick along $\eta$. Moreover, after an
appropriate change of coordinates, we have
$Y\subset[-K,K]^{l}\times\mathbb{F}_{p}^{d-l}$, where the copy of
$\mathbb{F}_{p}^{d-l}$ arises as the ($d-l$)-dimensional intersection of the
kernels of the maps $\xi_{1},\ldots,\xi_{l}$.

$\Box$

###### Lemma 2

For an increasing function $g:\mathbb{N}\rightarrow\mathbb{N}$, $K_{0}\geq 0$,
$\varepsilon>0$, and $d\geq 1$ there is some
$N=N(K_{0},d,\varepsilon,g)\in\mathbb{N}$, $\delta=\delta(K,\varepsilon,d)>0$,
$\mu=\mu(K,\varepsilon,d)>0$ such that the following holds. For any
(multi-)set $X\subset\mathbb{F}_{p}^{d}$ there is $l\in[0,N]$ and a
decomposition

$X=X_{0}\cup X_{1}\cup\ldots\cup X_{m},$ (5)

such that $|X_{0}|\leq\varepsilon|X|$ and for any $i\in[m]$ we have
$|X_{i}|\geq\mu|X|$ and $X_{i}$ is $(g(K),\delta)$-thick in its affine hull.
Here $K=g^{l}(K_{0})$, $\mu=\mu(K,\varepsilon,d)$ and
$\delta=\delta(K,\varepsilon,d)$.

Proof of Lemma 2: The proof is by induction on $d$. Take an arbitrary
(multi-)set $X\subset\mathbb{F}_{p}^{d}$. If $X$ is
$(g(K_{0}),\varepsilon/2)$-thick then there is nothing to prove since we can
take the decomposition $X_{0}=\emptyset$, $X_{1}=X$ and $l=0$. So we may
assume that $X$ is not $(g(K_{0}),\varepsilon/2)$-thick. Then after a change
of coordinates and removing at most $\varepsilon/2|X|$ elements from $X$ we
may assume that $X\subset[-g(K_{0}),g(K_{0})]\times\mathbb{F}_{p}^{d-1}$. For
each $y\in[-g(K_{0}),g(K_{0})]$ let
$X_{y}=X\cap\left(\\{y\\}\times\mathbb{F}_{p}^{d-1}\right)$. Remove from $X$
all sets $X_{y}$ such that $|X_{y}|<\varepsilon|X|/8g(K_{0})$, so that the
size of $X$ will decrease at most by $\varepsilon|X|/4$. Denote
$\varepsilon^{\prime}=\varepsilon/8g(K_{0})$.

Now we are going to apply the induction hypothesis to each of the remaining
sets $X_{y}$. Let $X_{y_{1}},\ldots,X_{y_{r}}$ be the list of all these sets,
where $r\leq 2g(K_{0})$. We apply induction to $X_{y_{1}}$ with
$\varepsilon=\varepsilon^{\prime}$, $K^{\prime}_{0}=g(K_{0})$ and
$g=g^{N_{1}}$ where $N_{1}$ will be determined later. In order for the number
$K=g^{l_{1}N_{1}}(K^{\prime}_{0})$ from the induction hypothesis to lie in the
interval $[K_{0},g^{N}(K_{0})]$ we need the following inequality:

$N>N(K^{\prime}_{0},d-1,\varepsilon^{\prime},g^{N_{1}})\cdot N_{1}.$ (6)

So we get a decomposition of the form $X_{y_{1}}=\bigcup_{i=0}^{m_{1}}X_{1,i}$
and there is some $l_{1}\leq
N(K^{\prime}_{0},d-1,\varepsilon^{\prime},g^{N_{1}})$ so that if we let
$K_{1}=g^{l_{1}N_{1}}(K^{\prime}_{0})$ then $X_{1,i}$ is
$(g^{N_{1}}(K_{1}),\delta_{1})$-thick in its affine hull for every
$i\in[m_{1}]$.

Now we apply the induction hypothesis to the set $X_{y_{2}}$ with
$\varepsilon=\varepsilon^{\prime}$, $K_{0}=K_{1}$ and $g=g^{N_{2}}$ where
$N_{2}$ will be determined later. To apply induction we need the following
inequality:

$N_{1}>N(K_{1},d-1,\varepsilon^{\prime},g^{N_{2}})\cdot N_{2}.$

We thus will obtain a decomposition $X_{y_{2}}=\bigcup_{i=0}^{m_{2}}X_{2,i}$
where $X_{2,i}$ is $(g^{N_{2}}(K_{2}),\delta_{2})$-thick in its affine hull
where $K_{2}=g^{N_{2}l_{2}}(K_{1})$ for some $l_{2}\leq
N(K_{1},d-1,\varepsilon^{\prime},g^{N_{2}})$. Moreover, we have
$\delta_{2}\gg_{K_{2},\varepsilon/8K_{0},d}1$.

Now observe that we have the following chain of inequalities:

$K_{1}\leq K_{2}\leq g^{N_{2}}(K_{2})\leq g^{N_{1}}(K_{1}).$

Thus, for every $i\in[m_{1}]$, the sets $X_{1,i}$ is
$(g^{N_{2}}(K_{2}),\delta_{1})$-thick in its affine hull. Also note that since
$K_{2}\geq K_{1}$ we have $\delta_{1}\gg_{K_{2},\varepsilon^{\prime},d}1$ as
well.

Applying induction in a similar manner $r-2$ more times to sets $X_{y_{j}}$
for $j=3,\ldots,r$ we will eventually get some $K_{r}=g^{l}(K_{0})$ where
$l\leq N$ such that all sets $X_{j,i}$ are $(g(K_{r}),\delta_{r})$-thick in
their affine hulls for some $\delta_{r}\gg_{K_{r},\varepsilon^{\prime},d}1$.
We will get a chain of inequalities of the form (6) which will give an upper
bound on the function $N(K_{0},d,\varepsilon,g)$. This concludes the proof.

$\Box$

We will need a stronger version of Lemma 2:

###### Lemma 3

For an increasing function $g:\mathbb{N}\rightarrow\mathbb{N}$, $K_{0}\geq 0$,
$\varepsilon>0$, and $d\geq 1$ there is some function
$N=N(K_{0},d,\varepsilon,g)\in\mathbb{N}$, $\delta=\delta(K,\varepsilon,d)>0$,
$\mu=\mu(K,\varepsilon,d)>0$ such that the following holds. For any set
$X\subset\mathbb{F}_{p}^{d}$ there is $l\in[0,N]$ and a decomposition

$X=X_{0}\cup X_{1}\cup\ldots\cup X_{m},$

such that $|X_{0}|\leq\varepsilon|X|$ and for any $i\in[m]$ we have
$|X_{i}|\geq\mu|X|$ and $X_{i}$ is $(g^{d+1}(K),\delta)$-thick in its affine
hull. Here $K=g^{l}(K_{0})$, $\mu=\mu(K,\varepsilon,d)$ and
$\delta=\delta(K,\varepsilon,d)$.

Moreover, for any $S\subset[m]$ the set $X_{S}=\bigcup_{i\in S}X_{i}$ is
$(K_{S},g(K_{S}),\mu)$-tubular where $K_{S}=g^{s}(K)$ for some $s\in[0,d]$.

Proof of Lemma 3: Let $g^{\prime}=g^{d+1}$ and apply Lemma 2 to $X$ with
$g^{\prime}$ instead of $g$, $\varepsilon/2$ instead of $\varepsilon$ and
$K_{0}=K_{0}$. We get a decomposition of the form (5) where sets $X_{i}$ are
$(g^{d+1}(K),\delta_{0})$-thick in their affine hulls for some
$\delta_{0}\gg_{K,\varepsilon,d}1$. Also we have $|X_{i}|\geq\mu_{0}|X|$ for
some $\mu_{0}\gg_{K,\varepsilon,d}1$.

Let $S_{1},S_{2},\ldots,S_{2^{m}-1}$ be the list of all non-empty subsets of
$[m]$ in any order. For $j=1,\ldots,2^{m}-1$, apply Lemma 1 consecutively to
sets $\bigcup_{i\in S}X_{i}$ with $K_{0}=K$, $g=g$ and

$\delta_{j}=\varepsilon\mu_{0}\delta_{0}2^{-d-2-m}2^{-(d+m+4)j}.$

Such choice of $\delta_{j}$ will guarantee us the following properties:

  1. 1.

The total number $R$ of removed elements will be at most

$R\leq 2^{d+1}|X|\sum_{j=1}^{2^{m}-1}\delta_{j}<\varepsilon|X|/2.$

  2. 2.

All sets $X_{i}$ will be $(g^{d+1}(K),\delta_{0}/2)$-thick since for every
$i\in[m]$:

$R\leq
2^{d+1}|X|\sum_{j=1}^{2^{m}-1}\delta_{j}\leq\varepsilon\mu_{0}\delta_{0}/2\leq|X_{i}|\delta_{0}/2.$

  3. 3.

For any $j$ the set $X_{S}=\bigcup_{i\in S_{j}}X_{i}$ will be
$(K_{S},g(K_{S}),\delta_{j}/2)$-tubular because the number of elements removed
from $X_{S}$ at steps $j^{\prime}>j$ is at most

$2^{d+1}|X|\sum_{j^{\prime}>j}\delta_{j^{\prime}}\leq\varepsilon\mu_{0}\delta_{0}2^{-(d+m+4)j}|X|\sum_{k=1}^{\infty}2^{-(d+m+4)k}\leq\varepsilon\mu_{0}\delta_{0}2^{-(d+m+3)j}|X|2^{-d-2-m}/2=\delta_{j}/2.$

We clearly have $\delta_{2^{m}-1}\gg_{\varepsilon,\mu_{0},\delta_{0},m}1$. But
$m\leq 1/\mu_{0}$ and so $\delta\gg_{K,\varepsilon,d}1$. Lemma is proved.

$\Box$

###  2.2 From tubes to subset sums

The main auxiliary result in this section is the following Proposition
inspired by the ideas from [20, Section 7.2].

###### Proposition 1

Let $d\geq 1$, $K\geq 1$, $\delta>0$ and $\mu>0$ and let $K_{2}>K$ be
sufficiently large with respect to parameters $K,d,\delta,\mu$. Let
$p>p_{0}(d,K,\delta,\mu)$ be a sufficiently large prime.

Fix some $l\in[0,d]$ and let $Y\subset[-K,K]^{l}$ be a non-empty set. For
$y\in Y$ let $X_{y}\subset\\{y\\}\times\mathbb{F}_{p}^{d-l}$ be an arbitrary
(multi-)set of size at least $\mu p$. Denote $X=\bigcup_{y\in Y}X_{y}$.
Suppose that $X$ is $(K_{2},\delta)$-thick along any linear function which is
not constant on $\\{0\\}\times\mathbb{F}_{p}^{d-l}$. Then there is some
$u_{0}\in\mathbb{F}_{p}^{l}$ and $k\in\mathbb{N}$ such that for every
$u\in\mathbb{F}_{p}^{d-l}$ there are subsets $S_{y}\subset X_{y}$ such that:

$\displaystyle\sum_{y\in Y}\sum_{x\in S_{y}}x=(u_{0},u),$

and $\sum_{y\in Y}|S_{y}|=k$.

We present the proof of Proposition 1 below. The argument is based on the
following lemma which was essentially proved by Alon and Dubiner [2, Corollary
2.3 and Proposition 2.4]. See also [20, Lemmas 3.1 and 3.2].

###### Lemma 4

Let $A\subset\mathbb{F}_{p}^{d}$ be a multiset which is $(K,\delta)$-thick
multiset along any linear function $\xi$ without constant term, for some
$K\geq 0$ and $\delta>0$. Then for any set $Y\subset\mathbb{F}_{p}^{d}$ of
size at most $p^{d}/2$ there is $a\in A$ such that

$|(Y+a)\setminus
Y|\geq\max\left\\{\frac{|Y|^{\frac{d-1}{d}}}{2},\,\frac{K\delta}{c_{0}p}|Y|\right\\},$

where $c_{0}\leq 10^{10}$ is an absolute constant.

Proof of Proposition 1: Let us first consider the case $l=0$. So, $X$ is
$(K_{2},\delta)$-thick along any non-constant linear function and $|X|\geq\mu
p$. Then it is not hard to see that the multiset $A=X-X$ is
$(K_{2},\delta)$-thick along any linear function without constant term.
Indeed, suppose that for some linear function $\xi$ more than
$(1-\delta)|X|^{2}$ differences $(x_{1}-x_{2})$ belong to $H(\xi,K_{2})$.
Then, by the pigeonhole principle, there is $x_{2}\in X$ such that more than
$(1-\delta)|X|$ vectors $x_{1}\in X$ belong to $x_{2}+H(\xi,K_{2})$. But this
contradicts the assumption that $X$ is $(K_{2},\delta)$-thick.

Now Lemma 4 can be applied to the multiset $A$. By choosing $K_{2}$
sufficiently large and applying Lemma 4 iteratively one can construct a
sequence of disjoint pairs $\\{a_{1},b_{1}\\},\ldots,\\{a_{l},b_{l}\\}\subset
X$ such that

$\\{a_{1},b_{1}\\}+\ldots+\\{a_{l},b_{l}\\}=\mathbb{F}_{p}^{d}.$

Indeed, at each step we apply Lemma 4 to
$Y_{i}=\\{a_{1},b_{1}\\}+\ldots+\\{a_{i},b_{i}\\}$ and obtain some
$(a_{i+1}-b_{i+1})\in A$ such that $Y_{i+1}$ is significantly larger than
$Y_{i}$. Details of this argument can be found in [20, Proposition 5.2]. So
the conclusion of Proposition 1 follows with $k=l$.

Now we consider the general case, that is, $l\in[d]$ is arbitrary. In this
case the multiset $X-X\subset\mathbb{F}_{p}^{d}$ is not $(K_{2},\delta)$-thick
and so we cannot apply Lemma 4. Instead, we are going to construct a certain
multiset $A\subset\\{0\\}\times\mathbb{F}_{p}^{d-l}$ which will be
$(K^{\prime},\delta^{\prime})$-thick along any linear function without
constant term. Then we will apply Lemma 4 to the set $A$ in a similar manner
as in the case $l=0$ to conclude the proof. To define $A$ let us consider the
set $\Lambda\subset\mathbb{Z}^{Y}$ consisting of all integer vectors
$(\lambda_{y})_{y\in Y}$ such that $\sum\lambda_{y}y=0$ and
$\sum\lambda_{y}=0$. For each $\lambda\in\Lambda$ let $\mathcal{J}^{\lambda}$
be the set of all pairs $(J_{1},J_{2})$, where $J_{i}\subset X$ and for every
$y\in Y$ we have

$\displaystyle(|J_{1}\cap X_{y}|,|J_{2}\cap
X_{y}|)=\begin{cases}(\lambda_{y},0),&\text{if }\lambda_{y}\geq 0\\\
(0,|\lambda_{y}|),&\text{if }\lambda_{y}<0.\end{cases}$

For a pair of (multi-)sets $(J_{1},J_{2})$ we denote by $\sigma(J_{1},J_{2})$
the sum of elements of $J_{1}$ minus the sum of elements of $J_{2}$. Note
that, by construction, for any $(J_{1},J_{2})\in\mathcal{J}^{\lambda}$ we have
$\sigma(J_{1},J_{2})\in\\{0\\}\times\mathbb{F}_{p}^{d-l}$.

The multiset $A\subset\\{0\\}\times\mathbb{F}_{p}^{d-l}$ is now defined as the
multiset of all sums $\sigma(J_{1},J_{2})$ over all
$(J_{1},J_{2})\in\mathcal{J}^{\lambda}$ and $\lambda\in\Lambda$ such that
$\|\lambda\|_{\infty}\leq T$. Here $\|\lambda\|_{\infty}$ denotes the maximum
of $|\lambda_{y}|$, $y\in Y$, and $T=T(d,K,\delta,\mu)>0$ is sufficiently
large.

It requires some work to show that $A$ is indeed
$(K^{\prime},\delta^{\prime})$-thick along any linear function on
$\\{0\\}\times\mathbb{F}_{p}^{d-l}$, so we isolate this fact as a separate
lemma.

###### Lemma 5

The multiset $A$ is $(K^{\prime},\delta^{\prime})$-thick along any linear
function without constant term on $\\{0\\}\times\mathbb{F}_{p}^{d-l}$. Here
$K^{\prime}$ and $\delta^{\prime}$ depend on parameters $K,K_{2},d,\delta,\mu$
in such a way that $K^{\prime}$ can be arbitrarily large compared to
$K,d,\delta,\mu$ and $\delta^{\prime}$ if one takes $K_{2}$ large enough.

The proof can be found in [20], where this fact is also used as a lemma, so
here we only present a sketch of that argument. We refer the reader to [20,
Lemma 7.4] for more details.

Proof of Lemma 5: Suppose that $A$ is not $(K^{\prime},\delta^{\prime})$-thick
along some linear function $\xi\ :\ \mathbb{F}_{p}^{d}\to\mathbb{F}_{p}$. This
means that $\xi(a)\in[-K^{\prime},K^{\prime}]$ for $(1-\delta^{\prime})|A|$
elements $a\in A$. Since $K^{\prime}$ is a constant with respect to $p$ and
$\delta^{\prime}$ is small, this is roughly the same as to say that $\xi(a)=0$
for any $a\in A$. Let us derive a contradiction from this assumption. If this
is the case, then for any $\lambda\in\Lambda$, $\|\lambda\|_{\infty}\leq T$,
and any $(J_{1},J_{2})\in\mathcal{J}^{\lambda}$ we have

$\xi(\sigma(J_{1},J_{2}))=0.$ (7)

Now let $y\in Y$ and choose some $x_{1}\in J_{1}\cap X_{y}$ and $x_{2}\in
X_{y}\setminus J_{1}\setminus J_{2}$. Then the pair
$(J_{1}\setminus\\{x_{1}\\}\cup\\{x_{2}\\},J_{2})$ also belongs to
$\mathcal{J}^{\lambda}$. Our assumption applied to these two pairs implies
that $\xi(x_{1}-x_{2})=0$ for all such $x_{1}\in J_{1}\cap X_{y}$ and
$x_{2}\in X_{y}\setminus J_{1}\setminus J_{2}$. By considering different pairs
$(J_{1},J_{2})$ it is not difficult to see that $\xi(x_{1}-x_{2})=0$ holds for
any $x_{1},x_{2}\in X_{y}$.

Extend $\xi$ to a linear function on $\mathbb{F}_{p}^{d}$ in the natural way,
namely, $\xi(u_{1},u_{2})=\xi(u_{2})$ for
$(u_{1},u_{2})\in\mathbb{F}_{p}^{l}\times\mathbb{F}_{p}^{d-l}$. Then we see
from the above that there are some elements $r_{y}\in\mathbb{F}_{p}$ such that
$\xi(X_{y})=r_{y}$ holds for any $y\in Y$. Thus, for any $\lambda\in\Lambda$
and $(J_{1},J_{2})\in\mathcal{J}^{\lambda}$ we have

$0=\xi(\sigma(J_{1},J_{2}))=\sum_{y\in Y}|J_{1}\cap X_{y}|r_{y}-|J_{2}\cap
X_{y}|r_{y}=\sum_{y\in Y}\lambda_{y}r_{y}.$

From the definition of $\Lambda$ and basic linear algebra it follows that the
vector $(r_{y})_{y\in Y}$ is a linear combination of vectors $(y_{i})_{y\in
Y}$ over all coordinates $i=1,\ldots,l$ and of the all ones vector $(1)_{y\in
Y}$. In other words, we have

$r_{y}=\alpha_{0}+\sum_{i=1}^{l}\alpha_{i}y_{i},$ (8)

for some $\alpha_{i}\in\mathbb{F}_{p}$ and all $y\in Y$. Define a linear
function $\eta$ on $\mathbb{F}_{p}^{d}$ as follows:

$\eta(x_{1},\ldots,x_{l},x_{l+1},\ldots,x_{d})=\alpha_{0}+\alpha_{1}x_{1}+\ldots+\alpha_{l}x_{l}+\xi(x_{l+1},\ldots,x_{d}).$

It is straightforward from (8) that $\eta(x)=0$ for any $x\in X_{y}$ and any
$y\in Y$. We conclude that $X$ lies on the hyperplane $\\{\eta=0\\}$, where
$\eta$ is a linear function which is not constant on
$\\{0\\}\times\mathbb{F}_{p}^{d-l}$. This contradicts the thickness assumption
on the multiset $X$. So the proof is complete under the assumption that
$\xi(A)=0$.

The general case, that is, the case when
$\xi(A)\subset[-K^{\prime},K^{\prime}]$, can be solved using similar ideas.
For instance, instead of (7) we have the condition that
$\xi(\sigma(J_{1},J_{2}))$ belongs to a short interval for almost all
$(J_{1},J_{2})\in\mathcal{J}^{\lambda}$ and for many choices of
$\lambda\in\Lambda$, $\|\lambda\|_{\infty}\leq T$. From this one can deduce
that there are some $r_{y}\in\mathbb{F}_{p}$ such that the difference
$\xi(x)-r_{y}$ is bounded by some constant for all $y\in Y$. This implies an
approximate version of (8) and so we can define $\eta$ is the same way as
before and verify that $X$ is concentrated on the set $H(\eta,K_{2})$.

$\hfill\square$

Returning to the proof of Proposition 1, we can now apply Lemma 4 to the
multiset $A$ and, as in the case $l=0$, construct a sequence of pairs
$(J_{1}^{1},J_{2}^{1}),\ldots,(J_{1}^{l},J_{2}^{l})$ such that all $J_{i}^{j}$
are disjoint and

$\\{\sigma(J_{1}^{1}),\sigma(J_{2}^{1})\\}+\ldots+\\{\sigma(J_{1}^{l}),\sigma(J_{2}^{l})\\}=\\{u_{0}\\}\times\mathbb{F}_{p}^{d-l},$

for some fixed vector $u_{0}$. It now follows that Proposition 1 holds with
$k$ equal to $|J_{1}^{1}|+\ldots+|J_{1}^{l}|$ (note that, by definition,
$|J_{1}^{i}|=|J_{2}^{i}|$), which therefore completes the proof.

$\Box$

###  2.3 High multiplicity case

The final lemma is a result about zero sums in sequences which can be regarded
as a generalization of Olson’s main result from [14].

###### Lemma 6

Let $Y\subset\mathbb{F}_{p}^{d}$ be an arbitrary set and let
$w:Y\rightarrow\mathbb{N}$ be a function such that $\sum_{y\in Y}w(y)\geq
d(p-1)+2r|Y|+1$ for some $r\geq 0$. Then, there exist coefficients
$a_{y}\in\mathbb{N}$, one for each $y\in Y$, such that
$a_{y}\in\\{0\\}\cup[r,w(y)-r]$ and $\sum_{y\in Y}a_{y}y=0$, while not all
$a_{y}$’s are simultaneously zero.

When $r=0$, notice that this indeed immediately implies that if $n>d(p-1)$,
then among any $n$ elements $v_{1},\ldots,v_{n}$ of $\mathbb{F}_{p}^{d}$ there
exists a nonempty subsequence with a zero subsum. To prove Lemma 6, we will
make use of Alon’s Combinatorial Nullstellensatz [1, Theorem 1.2], which we
recall for the reader’s convenience.

###### Lemma 7

Let $\mathbb{F}$ be an arbitrary field, and let $f=f(x_{1},\ldots,x_{n})$ be a
polynomial in $\mathbb{F}[x_{1},\ldots,x_{n}]$. Suppose the degree $\deg(f)$
of $f$ is $\sum_{i=1}^{n}{t_{i}}$, where each $t_{i}$ is a nonnegative
integer, and suppose that the coefficient of $\prod_{i=1}^{n}{x_{i\
}^{t_{i}}}$ in $f$ is nonzero. Then, if $S_{1},\ldots,S_{n}$ are the subsets
of $F$ with $|S_{i}|>t_{i}$, there exist $s_{1}\in S_{1}$,$\ldots$, $s_{n}\in
S_{n}$ so that

$f(s_{1},\ldots,s_{n})\neq 0.$

Proof of Lemma 6: For $y\in Y$ denote $A_{y}=\\{0\\}\cup[r,w(y)-r]$ and
consider the following $|Y|$-variate polynomial in $(\alpha_{y})_{y\in Y}$:

$P(\alpha_{y}~{}|~{}y\in Y)=\prod_{i=1}^{d}\left(1-\left(\sum_{y\in
Y}\alpha_{y}y_{i}\right)^{p-1}\right),$

where $y_{i}$ denotes the $i$-th coordinate of $y$ as an element in
$\mathbb{F}_{p}^{d}$. Note that $P(\alpha_{y}~{}|~{}y\in Y)$ is non-zero if
and only if $\sum_{y\in Y}\alpha_{y}y=0$, so the zero vector
$\overrightarrow{0}$ in $\mathbb{F}_{p}^{|Y|}$ is certainly not a zero of the
polynomial $P$. On the other hand, observe however that

$\sum_{y\in Y}(|A_{y}|-1)\geq\sum_{y\in Y}w(y)-2r|Y|>d(p-1),$

so by Lemma 7 applied in a slightly smaller cartesian product which is
strictly contained in $\prod_{y\in Y}A_{y}$ and which does not contain
$\overrightarrow{0}$, it follows that $P$ must take some other non-zero value
at a vector in $\prod_{y\in Y}A_{y}$ that does not have all coordinates equal
to $0$. This completes the proof of Lemma 6.

$\Box$

##  3 Proof of Theorem 1

Let $X\subset\mathbb{F}_{p}^{d}$ be an arbitrary set of size
$(d-1+\varepsilon)p$ where $p$ is a sufficiently large prime number. Let
$g:\mathbb{N}\rightarrow\mathbb{N}$ be a sufficiently fast growing function.

Apply Lemma 3 to $X$ with $\varepsilon^{\prime}=\varepsilon/2d$ and $g=g$,
$K=0$. After removing $X_{0}$ from $X$ we will obtain a set $X$ of size at
least $(d-1+\varepsilon/2)p$ and a decomposition $X=X_{1}\cup\ldots\cup X_{m}$
with several important properties. In particular, if $U_{i}$ denotes the
affine hull of the set $X_{i}$, recall that $X_{i}$ is
$(g^{d+1}(K),\delta)$-thick in $U_{i}$ for some
$\delta\gg_{K,\varepsilon,d}1$, for each $i\in\left\\{1,\ldots,m\right\\}$.
Moreover, for every $i$ we also have that $|X_{i}|\geq\mu|X|$, where
$\mu\gg_{K,\varepsilon,d}1$. Note that we may assume that $\mu$ is small
enough, namely, $\mu m<\varepsilon/100$. Furthermore, note that since $X$ is a
set, all spaces $U_{i}$ are non-zero dimensional.

Let $H\subset\mathbb{F}_{p}^{d}$ be a generic hyperplane passing through the
origin which intersects all affine spaces $U_{i}$. Such $H$ exists since
$m\ll_{K,d,\varepsilon}1$ and $p$ is large enough. Fix an arbitrary vector
$x_{i}\in H\cap U_{i}$ for each $i\in[m]$. Assign the weight $w_{i}=|X_{i}|$
and apply Lemma 6 to the set $Y=\\{x_{1},\ldots,x_{m}\\}$ with the weight
function $w$ and $r=\mu|X|/3$. We have $\mu<\varepsilon/10m$ and so

$\sum_{i=1}^{m}w_{i}=|X|\geq(d-1+\varepsilon/2)p>dp+2\mu|X|m/3.$

Thus, there are non-negative, not all zero, coefficients
$a_{i}\in\\{0\\}\cup[\mu|X|/3,w_{i}-\mu|X|/3]$ such that
$\sum_{i=1}^{m}a_{i}x_{i}=0$. Let $S\subset[m]$ be the set of $i\in[m]$ for
which $a_{i}>0$. By Lemma 3, the set $X_{S}=\bigcup_{i\in S}X_{i}$ is
$(K_{S},g(K_{S}),\mu)$-tubular for $K_{S}=g^{l}(K)$ and $l\in[0,d]$. So after
a linear change of coordinates, there is a vector
$v\in\mathbb{F}_{p}^{l}\times\\{0\\}$ such that

$X_{S}\subset v+[-K_{S},K_{S}]^{l}\times\mathbb{F}_{p}^{d-l}.$

Denote by $\pi$ the projection onto first $l$ coordinates. The condition that
$X_{i}$ is $(g^{d+1}(K),\delta)$-thick in $U_{i}$ and the fact that
$K_{S}<g^{d+1}(K)$ implies that $\pi(U_{i})$ is a single point for any
$i\in[m]$. Denote this point by $y_{i}\in v+[-K_{S},K_{S}]^{l}$ and observe
that

$\sum_{i=1}^{m}a_{i}y_{i}=0,$ (9)

since $\pi$ is a linear operator.

Denote by $Y\subset[-K_{S},K_{S}]^{l}$ the set obtained from the projection
$\pi(X)$ and by shifting by $v$. Note that for any $y\in Y$ the set
$X_{y}=X_{S}\cap\left(\\{y\\}\times\mathbb{F}_{p}^{d-l}\right)$ has size at
least $\mu|X|\geq\mu|X_{S}|$ by Lemma 3. Moreover, the set $X_{S}$ is
$(g(K_{S}),\delta)$-thick along any linear function which is non-constant on
$\\{0\\}\times\mathbb{F}_{p}^{d-l}$. For $y\in Y$ denote by $a_{y}$ the sum of
all numbers $a_{i}$ over all $i$ is such that $y=y_{i}-v$.

###### Proposition 2

If $p$ is large enough then there are sets $Z_{y}\subset X_{y}$, such that for
any $y\in Y$ we have $|Z_{y}|\in[\mu|X_{y}|/20,\mu|X_{y}|/10]$ and the set
$Z=\bigcup_{y\in Y}Z_{y}$ is $(g(K_{S}),\delta/4)$-thick along any linear
function which is not constant on $\\{0\\}\times\mathbb{F}_{p}^{d-l}$.

We prove Proposition 2 by using a probabilistic argument, where we make use of
the following Chernoff bound (see, for example, [10, Corollary 21.7]).

###### Lemma 8

Let $X$ be a random variable with the binomial distribution
$\operatorname{Bin}(N,p)$ and $\eta\in(0,1)$. Then

$\displaystyle\operatorname{Pr}\left(X\leq(1-\eta)\mathbb{E}[X]\right)$
$\displaystyle\leq$
$\displaystyle\exp\left(-\frac{\eta^{2}}{2}\mathbb{E}[X]\right)$
$\displaystyle\operatorname{Pr}\left(X\geq(1+\eta)\mathbb{E}[X]\right)$
$\displaystyle\leq$
$\displaystyle\exp\left(-\frac{\eta^{2}}{3}\mathbb{E}[X]\right)$

Proof of Proposition 2: Choose sets $Z_{y}\subset X_{y}$ at random according
to the binomial distribution $\operatorname{Bin}(|X_{y}|,\mu/15)$. It follows
from Lemma 8 and the fact that $|X_{y}|\gg p$ that with high probability we
have $|Z_{y}|\in[\mu|X_{y}|/20,\mu|X_{y}|/10]$ for all $y\in Y$. We will show
that for any fixed linear function the event that $Z=\bigcup_{y\in Y}Z_{y}$ is
not $(g(K_{S}),\delta/4)$-thick has probability exponentially small in $p$.
Since there are only $O(p^{d})$ linear functions on $\mathbb{F}_{p}^{d}$ this
will be enough to prove Proposition 2. Fix a linear function $\xi$ which is
not constant on $\\{0\\}\times\mathbb{F}_{p}^{d-l}$ and denote
$X^{\prime}_{y}=X_{y}\setminus H(\xi,g(K_{S}))$. Since the set $X_{S}$ is
$(g(K_{S}),\delta)$-thick along $\xi$, the set $X^{\prime}=\bigcup_{y\in
Y}X^{\prime}_{y}$ has size at least $\delta|X_{S}|$. Note that the expected
size of the intersection $Z_{y}\cap X^{\prime}_{y}$ is asymptotically equal to
$\frac{|Z_{y}||X^{\prime}_{y}|}{|X_{y}|}$ and so, provided that
$|X^{\prime}_{y}|\gg p$, by Lemma 8 the probability of the event that
$|Z_{y}\cap X^{\prime}_{y}|<\frac{|Z_{y}||X^{\prime}_{y}|}{1.5|X_{y}|}$ is at
most $e^{-cp}$ for some $c\gg 1$. Therefore, the probability that

$\sum_{y\in Y}|Z_{y}\cap X^{\prime}_{y}|<\sum_{y\in
Y}\frac{|Z_{y}||X^{\prime}_{y}|}{1.5|X_{y}|}$

is at most $|Y|e^{-cp}$. But since the set $Z_{y}$ has size in the interval
$[\mu|X_{y}|/20,\mu|X_{y}|/10]$ the right hand side is at least

$(\mu/20)\sum_{y\in
Y}|X^{\prime}_{y}|/1.5\geq(\mu/20)\delta|X|/1.5>\delta|Z|/4,$

which means that $Z$ is $(g(K_{S}),\delta/4)$-thick along $\xi$ with
probability at least $1-|Y|e^{-cp}$. This completes the proof. $\Box$

Fix sets $Z_{y}$ as in Proposition 2. Now let the function $g$ grow so fast
that we have $g(K)>K_{2}$ where $K_{2}=K_{2}(K,d,\delta/4,\mu^{2}/10)$ is the
function from Proposition 1. We can then apply Proposition 1 to sets $Z_{y}-v$
with $u=0$ to get some $k_{y}\in\mathbb{N}$ such that

$\sum_{y\in Y}k_{y}(y-v)=u_{0},~{}~{}\sum_{y\in Y}k_{y}=k,$

where $u_{0}$ and $k$ are from the statement of Proposition 1. Note that
$k_{y}\leq|Z_{y}|\leq\mu|X_{y}|/10$.

For each $y\in Y$ fix a subset $A_{y}\subset X_{y}\setminus Z_{y}$ of size
$a_{y}-k_{y}$ (which is possible thanks to the estimates on $a_{i}$). Let
$u=(u_{1},u_{2})\in\mathbb{F}_{p}^{d}$ denote the following vector:

$u=\sum_{y\in Y}\sum_{x\in A_{y}}x.$

From (9) and from the definition of the $k_{y}$’s we see that, in fact,
$u_{1}=-u_{0}-kv$.

By the conclusion of Proposition 1, applied to the vector
$-u_{2}\in\mathbb{F}_{p}^{d-l}$, we obtain some sets $S_{y}\subset Z_{y}$ such
that $\sum_{y\in Y}|S_{y}|=k$ and

$\sum_{y\in Y}\sum_{x\in S_{y}}(x-v)=(u_{0},-u_{2}),$

After rearranging, this rewrites as

$\sum_{y\in Y}\sum_{x\in S_{y}}x=(u_{0}+kv,-u_{2})=-\sum_{y\in Y}\sum_{x\in
A_{y}}x.$

So we see that the set $B=\bigcup_{y\in Y}A_{y}\cup S_{y}$ has zero sum.
Theorem 1 is proved.

## References

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# Master-slave synchronization of Bose-Einstein condensate in 1D tilted
bichromatical optical lattice

E. Tosyali${}^{\textsf{\footnotesize{\mbox{?}}}}$, F.
Aydogmus${}^{\textsf{\footnotesize{\mbox{?}}}}$

(Received September 7, 2019, in final form September 26, 2019)

###### Abstract

This paper investigates the synchronization of chaotic behavior in a model of
Bose-Einstein condensate (BEC) held in a 1D tilted bichromatical optical
lattice potential by using the active control technique. The synchronization
is presented in the master-slave configuration which implies that the master
system evolves freely and drives the dynamics of the slave system. Also the
numerical simulations are given to indicate the practicability and the
effectiveness of the used controllers.

Key words: synchronization, Bose-Einstein condensate, Gross-Pitaevskii
equation, optical lattice potential

###### Abstract

Ó ñòàòò äîñëäæóòüñÿ ñèíõðîíçàöÿ õàîòèíî¿ ïîâåäíêè ó ìîäåë êîíäåíñàòó Áîçå-
Åéíøòåéíà, ïîìùåíîãî ó ïîòåíöàë 1D ïîõèëî¿ áõðîìàòèíî¿ îïòèíî¿ ãðàòêè ç
âèêîðèñòàííÿì ìåòîäó àêòèâíîãî êîíòðîëþ. Ñèíõðîíçàöÿ ïðåäñòàâëåíà êîíôãóðàöþ
‘‘êåðâíèé-êåðîâàíèé’’, ÿêà ïåðåäáàà, ùî êåðâíà ñèñòåìà ðîçâèâàòüñÿ âëüíî òà
çàïóñêà äèíàìêó êåðîâàíî¿ ñèñòåìè. Ïðåäñòàâëåíî èñëîâ ñèìóëÿö¿, ùî
ïäòâåðäæóþòü ïðàêòèíñòü òà åôåêòèâíñòü çàñòîñîâàíèõ êîíòðîëåðâ.

Ключов слова: ñèíõðîíçàöÿ, êîíäåíñàò Áîçå-Åéíøòåéíà, ðâíÿííÿ Ãðîññà-
Ïòàâñüêîãî, îïòèíèé ãðàòêîâèé ïîòåíöàë

##  1 Introduction

Chaos is a well known nonlinear phenomenon in many scientific disciplines,
ranging from physics to engineering. A chaotic system has unpredictable and
complex behaviors. Especially, there has been an increasing interest in the
study of chaos synchronization. Chaos synchronization has great potential in
physics, chemical and biological systems, secure communications, power
electrical systems and so on [1, 2, 3, 4, 5, 6, 7, 8]. Carroll and Pecora
first introduced the idea of synchronizing two chaotic systems in 1990 [1].
Due to the connection between control and synchronization, numerous control
methods are used to synchronize the chaotic systems. Among these methods,
active control has been widely accepted as one of the efficient methods for
chaos synchronization [9, 10]. Especially, it is very attractive in studying
the control problems of the Bose-Einstein condensate (BEC) system due to its
rich dynamics. The BEC system in an optical lattice shows many good properties
as a typical nonlinear system [11, 12]. It is well known that the mean-field
theory is a successful theory commonly used to describe the BEC. This approach
can quite accurately describe the static and dynamical properties of BEC. The
relevant model is a variant of the famous nonlinear Schrödinger equation, the
so-called Gross Pitaevskii (GP) equation [13, 14].

In this paper, first we introduce the GP equation with 1D tilted bichromatical
optical lattice potential. The differential equations system derived from GP
equation is obtained by the use of a particular solution similar to solve the
nonlinear Schrödinger equation. Then, we focus on the synchronization of BEC
system via an active control technique in the master-slave scheme. In the
master-slave scheme, the behavior of the slave system is controlled with the
operation of a master system, i.e., the master-slave configuration means that
the master system evolves freely and drives the dynamics of the slave system
[15]. The controller provides that the states of the controlled chaotic slave
system exponentially synchronize with the state of the master system.
Furthermore, numerical simulations for synchronization of the BEC system are
performed to verify the results. In figure 1, the schematic representation of
the master-slave configuration is shown.

##  2 Active control and synchronization

Active control method is one of the most frequently used approaches for
synchronization between two chaotic systems in the master-slave scheme [16].
If we consider a master system

$\frac{\mathrm{d}a}{\mathrm{d}x}={\bf{A}}a+g(a)\,,$ (2.1)

where $a$ state vector and ${\bf{A}}$ is a constant system matrix, and $g(a)$
is nonlinear function. And the slave system

$\frac{\mathrm{d}b}{\mathrm{d}x}={\bf{B}}b+f(b)+u(x)\,,$ (2.2)

where $b$ is state vector, ${\bf{B}}$ is a constant system matrix, and $f(b)$
is a nonlinear function, and $u(x)$ is an active control function.

$MasterSystem$start$e(x)$$SlaveSystem$$Controller$+-u(x) Figure 1: The master-
slave synchronization configuration.

In the master-slave synchronization configuration, the error state is obtained
from $e=b-a$. Thus, the error dynamics is written as follows:

$\frac{\mathrm{d}e}{\mathrm{d}x}=\frac{\mathrm{d}b}{\mathrm{d}x}-\frac{\mathrm{d}a}{\mathrm{d}x}={\bf{C}}e+G(a,b)+u(x)\,,$
(2.3)

where ${\bf{C}}={\bf{\bar{B}}}-{\bf{\bar{A}}}$ are the common parts of the
system matrices, the non-common parts and nonlinear functions are gathered in
$G(a,b)$ as follows:

$G(a,b)=f(a)-g(b)+(B-\bar{B})b-(A-\bar{A})a\,,$ (2.4)

and $u(x)$ is the controller matrix. The feedback from the controller $u(t)$
is designed to get the error $e$ to decay to zero. Thus, active controller
should eliminate nonlinear terms and non-common parts. We define the active
control function

$u(x)=-G(a,b)+\tau(x)\,,$ (2.5)

where $\tau(x)=-{\bf{K}}e$ is a linear controller and $\bf{K}$ is a linear
gain matrix. After substitution of equation (2.5) into (2.3) we get

$\dot{e}={\bf{C}}e+\tau(x)\,,$ (2.6)

we replace $\tau(x)$ in to equation (2.6)

$\dot{e}=({\bf{C}}-{\bf{K}})e$ (2.7)

If the eigenvalues of matrix ${\bf{C}}-{\bf{K}}$ are negative real or complex
with negative parts, the system is synchronized and asymptotically stable at
the origin. The controller manipulates the slave system and the controlled
chaotic slave system exponentially synchronizes with the state of the master
system as shown in figure 1.

##  3 The Gross-Pitaevskii equation

In the ultra-cold temperature regime, the nonlinear Schrödinger equation,
known as the GP equation [14, 17], models quite accurately the static and
dynamical properties of BEC with the macroscopic wave function
$\Psi=\Psi(x,t)$. There are many different methods of reducing the original
three-dimensional generalized GP equation to quasi-one dimensional GP
equation. We consider a BEC described with quasi-1D cubic nonlinear GP
equation which is used in the regime of shock waves [31, 18, 19, 20, 21],
which is in the form of

$\mathrm{i}\hbar\frac{\partial}{\partial
t}\Psi\left(x,t\right)=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}\Psi\left(x,t\right)+\left[V_{\text{ext}}\left(x\right)+g_{\text{1D}}^{\prime}\left|\Psi\left(x,t\right)\right|^{2}\right]\Psi\left(x,t\right),$
(3.1)

where $V_{\text{ext}}$ ia an external potential confining the BEC, $m$ is the
mass of the atoms of the condensate, $g_{\text{1D}}^{\prime}$ describes the
quasi-1D interaction between the atoms in the condensate and is given by

$g_{\text{1D}}^{\prime}=\frac{m\omega_{r}g_{\text{3D}}}{2\piup
a_{r}^{2}}=2a_{s}\hbar\omega_{r}\,,$

where $g_{\text{3D}}=4\piup a_{s}\hbar^{2}/m$ is the atom-atom ineraction and
proportional to the $a_{s}$. $a_{s}$ is the $s$-wave scattering length between
the atoms. The $s$-wave scattering length is positive or negative for
repulsive and attractive interactions, respectively. In our case, it is
negative. $w_{r}$ is ground state of a harmonic frequency of the oscillator.
Taking an experimentally suitable frequency $\omega_{0}$ as the units of
frequencies $\omega_{1},\omega_{2}$ and normalizing space, and the wave
function with $\omega_{0}^{-1}$,
$l_{0}=\sqrt{\hbar/\left(m\omega_{0}\right)}$, and $1/\sqrt{l_{0}}$. GP
equation becomes as follows:

$\mathrm{i}\hbar\frac{\partial}{\partial
t}\Psi\left(x,t\right)=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}\Psi\left(x,t\right)+\left[V\left(x\right)+g_{\text{1D}}\left|\Psi\left(x,t\right)\right|^{2}\right]\Psi\left(x,t\right),$
(3.2)

Here, interaction is reduced
$g_{\text{1D}}=2\omega_{r}a_{s}/\left(\omega_{0}l_{0}\right)$ and the
potential $V_{\text{ext}}$ is normalized by $\hbar\omega_{0}$. We use
parameters $\omega_{r}=19$ Hz and $\omega_{0}=250$ Hz that are used in the
experiments [22]. From these parameters
$\sqrt{\hbar/\left(m\omega_{0}\right)}\approx 3.3$ $\muup$m for 23Na. The
$s$-wave length is chosen $5-10$ nm [23], therefore, $g_{\text{1D}}$ could be
from $0.011$ to $0.022$.

##  4 Tilted bichromatic optical lattice potential

We choose the external trapping potential as follows:

$V_{\text{ext}}\left(x\right)=V(x)+Fx\,,$ (4.1)

where the constant force $F$ describes $F=ma$. Here, $m$ is the mass of atoms
and $a$ is the acceleration wich generates a tilted optical potential as shown
in figure 2, accelerates the atoms in the $x$ direction with linearly
increasing flow density and leads to the atoms tunnelling out of the traps
[24, 25, 26, 27]. Moreover, $V(x)$ is the optical potential. In this study,
the optical potential is given in the form of

$V(x)=V_{1}\cos^{2}(\omega_{1}x)+V_{2}\cos^{2}(\omega_{2}x)\,,$ (4.2)

here, $V_{1}$ and $V_{2}$ are the respective amplitudes. In order to obtain a
simple description and for a better understanding of the BEC dynamics, we
consider only the solution of equation (3.2) as below [27, 28]

$\Psi\left(x,t\right)=\Phi\left(x\right)\mathrm{e}^{\frac{-\mathrm{i}\mu
t}{\hbar}},$ (4.3)

here, $\mu$ is the chemical potential of the condensate and $\Phi(x)$ is a
real function independent of time. $\Phi(x)$ is normalized to the total number
of particles in the system, i.e.,

$\int\left|\Phi\left(x\right)\right|^{2}\mathrm{d}x=N,$ (4.4)

where $N$ is the particle number. Substitution of equations (4.2) and (4.3)
into equation (3.2) , yields

$\displaystyle\mu\Phi\left(x\right)$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}\Phi\left(x\right)$ (4.5) $\displaystyle+$
$\displaystyle\left[V_{1}\cos^{2}\left(w_{1}x\right)+V_{2}\cos^{2}\left(w_{2}x\right)+Fx+g_{\text{1D}}\left|\Phi\left(x\right)\right|^{2}\right]\Phi\left(x\right),$

For simplicity, we rescale the wave function and present dimensionless
parameters as $\upsilon_{1}={2mV_{1}}/{\hbar^{2}}$,
$\upsilon_{2}={2mV_{2}}/{\hbar^{2}}$, $\gamma={2m\mu}/{\hbar^{2}}$,
$\zeta={2mF}/{\hbar^{2}}$, $\eta={2mg_{0}}/{\hbar^{2}}$. The system equation
can be written in the following form

$\frac{\mathrm{d}^{2}\Phi}{\mathrm{d}x^{2}}=\left[\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta\left|\Phi\right|^{2}\right]\Phi\,,$ (4.6)

Inserting $\Phi(x)=\phi(x)\mathrm{e}^{\text{i}\theta(x)}$ into (4.6) leads to
two coupled equations which comes from separetly analyzing the real and the
imaginary parts,

$\frac{\mathrm{d}^{2}\phi}{\mathrm{d}x^{2}}=\phi\left(\frac{\mathrm{d}\theta}{\mathrm{d}x}\right)^{2}+\left[\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta\left|\phi\right|^{2}\right]\phi\,,$ (4.7a)
$\frac{\mathrm{d}^{2}\theta}{\mathrm{d}x^{2}}+2\frac{1}{\phi}\frac{\mathrm{d}\theta}{\mathrm{d}x}\frac{\mathrm{d}\phi}{\mathrm{d}x}=0.$
(4.7b)

where $\phi(x)$ and $\theta(x)$ are real functions. The first derivative
$2{\mathrm{d}\theta}/{\mathrm{d}x}$ in equation (4.7b) shows the velocity and
$\phi^{2}=n$ is the number of density of atoms. Therefore, equation (4.7b)
denotes that there exist a steady current $J$ which is obtained from the first
integration constant of equation (4.7b) as below

$J=2\phi^{2}\frac{\mathrm{d}\theta}{\mathrm{d}x}\,.$ (4.8)

Equation (4.8) represents a steady superfluid in the system. If we put $J$
into equation (4.7a), we have a nonlinear equation

$\frac{\mathrm{d}^{2}\phi}{\mathrm{d}x^{2}}=\frac{J^{2}}{4\phi^{3}}+\left[\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta\left|\phi\right|^{2}\right]\phi.$ (4.9)

It is difficult to obtain the exact solution of eqaution (4.9) because of its
complexity. Thus, numerical solutions were performed. For numerical
calculation, we reduce the system to the first-order by the transformation

$\frac{\mathrm{d}x_{1}}{\mathrm{d}x}=y_{1}\,,$ (4.10a)
$\frac{\mathrm{d}y_{1}}{\mathrm{d}x}=\frac{J^{2}}{4x_{1}^{3}}+\left[\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta\left|x_{1}\right|^{2}\right]x_{1}.$ (4.10b)

where $\phi=x_{1}$ and ${\mathrm{d}\phi}/{\mathrm{d}x}=y_{1}$. The
superposition of two laser beams of different frequencies can generate a
bichromatic potential. Therefore, bichromatic potential can be referred to as
double well potential. In this study, we add an extra tilted force to the
bichromatic potential. This causes an acceleration of bosons in the lattice
potential. Bosons can be allowed to jump from one trap to another.

(a) (b)

Figure 2: Plot of the bichromatic optical lattice potential with the
parameters $\nu_{1}=1,\nu_{2}=0.8,\omega_{1}=2\piup,\omega_{2}=5\piup$, (a)
$\zeta=0$, (b) $\zeta=0.1$.

Recently, the regular and chaotic solutions of the BEC system in 1D tilted
bichromatical optical lattice potential have been investigated by constructing
its Poincare sections in phase space [29]. Moreover, the dynamics of a Bose-
Einstein condensate system under the Gaussian white noise has been studied
[30].

##  5 Chaos synchronization in BEC

Chaos synchronization means making two (or more) systems oscillate in the same
manner by using the active control parameters [1]. Master-slave control
technique is the most frequently used synchronization approach used for the
chaotic systems. In the master-slave scheme, the behaviors of the slave system
are controlled with the operation of a master system [15]. The master and
slave systems are defined as follows:

$\frac{\mathrm{d}x_{1}}{\mathrm{d}x}=y_{1}\,,$ (5.1a)
$\frac{\mathrm{d}y_{1}}{\mathrm{d}x}=\frac{J^{2}}{4x_{1}^{3}}+\left[\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta\left|x_{1}\right|^{2}\right]x_{1}\,,$ (5.1b)

and

$\frac{\mathrm{d}x_{2}}{\mathrm{d}x}=y_{2}+u_{1}(x)\,,$ (5.2a)
$\frac{\mathrm{d}y_{2}}{\mathrm{d}x}=\frac{J^{2}}{4x_{2}^{3}}+\left[\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta\left|x_{2}\right|^{2}\right]x_{2}+u_{2}(x)\,,$ (5.2b)

where $\phi=x_{2}$, ${\mathrm{d}\phi}/{\mathrm{d}x}=y_{2}$, $u_{1}$ and
$u_{2}$ are the nonlinear controllers, so that two chaotic systems can be
synchronized. In this case, the error dynamics are $e_{1}=x_{2}-x_{1}$ and
$e_{2}=y_{2}-y_{1}$. The error dynamics is determined by subtracting equations
(5.1a), (5.1b) from equation (5.2a), (5.2b)

$\displaystyle\frac{\mathrm{d}e_{1}}{\mathrm{d}x}=e_{2}+u_{1}(x)\,,$ (5.3a)
$\displaystyle\frac{\mathrm{d}e_{2}}{\mathrm{d}x}$ $\displaystyle=$
$\displaystyle\frac{J^{2}}{4x_{2}^{3}}-\frac{J^{2}}{4x_{1}^{3}}+\left[\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta\left|x_{2}\right|^{2}\right]e_{1}+u_{2}(x).$ (5.3b)

We write the error functions as below,

$u_{1}(x)=\tau_{1}(x)\,,$ (5.4a)
$u_{2}(x)=-\frac{J^{2}}{4x_{2}^{3}}+\frac{J^{2}}{4x_{1}^{3}}-\eta(x_{2}^{2}-x_{1}^{2})+\tau_{2}(x).$
(5.4b)

Substituting control functions into equation (5.5a) and (5.5b), the error
system becomes,

$\displaystyle\frac{\mathrm{d}e_{1}}{\mathrm{d}x}$
$\displaystyle=e_{2}+\tau_{1}(x)\,,$ (5.5a)
$\displaystyle\frac{\mathrm{d}e_{2}}{\mathrm{d}x}$
$\displaystyle=\left[\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta\right]e_{1}+\tau_{2}(x).$ (5.5b)

The system can be controlled with control inputs $\tau_{1}$ and $\tau_{2}$ as
function of $e_{1}$ and $e_{2}$. These feedbacks stabilize the system where
$\lim_{x\to\infty}||e(x)||=0$. We choose control inputs $\tau_{1}$ and
$\tau_{2}$ as follows:

$\begin{bmatrix}\tau_{1}\\\ \tau_{2}\end{bmatrix}=D\begin{bmatrix}e_{1}\\\
e_{2}\end{bmatrix},$ (5.6)

where $D=\begin{bmatrix}a&b\\\ c&d\end{bmatrix}$ is a $2\times 2$ constant
feedback matrix. The system can be written as follows:

$\begin{bmatrix}\frac{\mathrm{d}e_{1}}{\mathrm{d}x}\\\
\frac{\mathrm{d}e_{2}}{\mathrm{d}x}\end{bmatrix}=C\begin{bmatrix}e_{1}\\\
e_{2}\end{bmatrix},$ (5.7)

where $C$ is is the coefficient matrix as below

$C=\begin{bmatrix}0+a&1+b\\\
\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta+c&0+d\end{bmatrix}.$ (5.8)

Based on the Routh-Hurwitz criterion [31], Lyapunov stability theory and
active control strategy, two identical BEC are synchronized. If we choose
$a=-1$, $b=-1$, $d=-1$,
$c=-\left[\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta\right]$,

$\tau_{1}(x)=-e_{1}-e_{2}\,,$ (5.9a)
$\tau_{2}(x)=-\left[\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta\right]e_{1}-e_{2}\,.$ (5.9b)

Finally we get a slave system as below,

$\frac{\mathrm{d}x_{2}}{\mathrm{d}x}=y_{2}+(-e_{1}-e_{2})\,,$ (5.10a)
$\begin{split}\frac{\mathrm{d}y_{2}}{\mathrm{d}x}&=\frac{J^{2}}{4x_{1}^{3}}+\left[\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta\left|x_{2}\right|^{2}\right]x_{2}\\\
&-\eta(x_{2}^{3}-x_{1}^{3})-\left[\upsilon_{1}\cos^{2}\left(w_{1}x\right)+\upsilon_{2}\cos^{2}\left(w_{2}x\right)+\zeta
x-\gamma+\eta\right]e_{1}-e_{2}\,.\end{split}$ (5.10b)

##  6 Numerical simulations

In this section, we investigate the simulation results for chaotic
synchronization of the systems (5.1a), (5.1b) and (5.2a), (5.2b). In the
numerical simulations, we have written a Maxima code that applies the fourth
order Runge Kutta algorithm which is used to solve the sets of differential
equations related to the master and slave systems with step size 0.1 and step
length 3500. We selected the parameters of the BEC system as $\zeta=0.1$,
${J}=0.4$,
$v_{1}=1,v_{2}=0.8,w_{1}=2\piup,w_{2}=5\piup,\gamma=0.5,\eta=-0.015$,
respectively. Shock wave dynamics can be found in our recently published paper
for these parameters [29].

|
---|---
(a) | (b)

Figure 3: (Colour online) Spatial evolution of $(e_{1},e_{2})$ with the
controller activated at the $x=0$ for (a) $(e_{1}(0),e_{2}(0))=(-0.8,1.3)$ and
(b) $(e_{1}(0),e_{2}(0))=(-0.839,-0.97)$.

|
---|---

(a) (b)

|
---|---

(c) (d)

Figure 4: (Colour online) Synchronized master-slave systems using the active
control method for $0\leq x\leq 30$, $\zeta=0.1$, ${J}=0.4$,
$v_{1}=1,v_{2}=0.8,w_{1}=2\piup,w_{2}=5\piup,\gamma=0.5,\eta=-0.015$ (a)
$\left(x_{1}(0),x_{2}(0)\right)=\left(1,0.2\right)$, (b)
$\left(y_{1}(0),y_{2}(0)\right)=\left(-1,0.3\right)$, (c)
$\left(x_{1}(0),x_{2}(0)\right)=\left(0.56,0.279\right)$, (d)
$\left(y_{1}(0),y_{2}(0)\right)=\left(0.3,0.67\right)$.

For the numerical calculations, we choose the arbitrary possible intial
condtions between $-1$ and $1$. In the chaotic regime, the velocity field and
atomic density may not be determined exactly in experiment due to fluctuations
of the atomic thermal cloud [32, 33]. Thus, we choose possible intial
conditions randomly for numerical calculations. In figure 3 we plot the error
functions from equation (5.5a) and (5.5b) for two different initial conditions
(a) $(e_{1}(0),e_{2}(0))=(-0.8,1.3)$ and (b)
$(e_{1}(0),e_{2}(0))=(-0.839,-0.97)$, respectively. Control functions start at
$x=0$. As a result, the system rapidly tends to synchronization. Figures 4, 5
and 6 represent the solution of master and slave systems. In figure 4 spatial
evolutions with the controller are given for the initial conditions (a)
$\left(x_{1}(0),x_{2}(0)\right)=\left(1,0.2\right)$, (b)
$\left(y_{1}(0),y_{2}(0)\right)=\left(-1,0.3\right)$, (c)
$\left(x_{1}(0),x_{2}(0)\right)=\left(0.56,0.279\right)$, (d)
$\left(y_{1}(0),y_{2}(0)\right)=\left(0.3,0.67\right)$ and
$x_{\text{max}}=30$. We solve the master and slave system from $x=0$ to
$x_{\text{max}}=3500$ with the step size $0.1$ in figures 5 and 6. According
to spatial evolution, the system shows chaotic dispersive shock-wave-like
dynamics in the phase space [29]. In addition, the system is synchronized
along the flow.

|
---|---

(a) (b)

|
---|---

(c) (d)

Figure 5: (Colour online) Chaotic synchronization the master-slave systems
using active control method for $\zeta=0.1$, ${J}=0.4$,
$v_{1}=1,v_{2}=0.8,w_{1}=2\piup,w_{2}=5\piup,\gamma=0.5,\eta=-0.015$ and the
initial conditions for master systems (a) $x_{1}(0)=1$ and (c) $y_{1}(0)=-1$,
slave systems (b) $x_{2}(0)=0.2$ and (d) $y_{2}(0)=0.3$.

|
---|---

(a) (b)
(c) (d)

Figure 6: (Colour online) Chaotic synchronization the master-slave systems
using active control method for $\zeta=0.1$, ${J}=0.4$,
$v_{1}=1,v_{2}=0.8,w_{1}=2\piup,w_{2}=5\piup,\gamma=0.5,\eta=-0.015$ and the
initial conditions for master systems (a) $x_{1}(0)=0.56$ and (c)
$y_{1}(0)=0.3$, slave systems (b) $x_{2}(0)=0.279$ and (d) $y_{2}(0)=0.67$.

##  7 Conclusion

It is known that the chaos in BEC systems can undermine the stability of the
condensates playing a destructive role [26]. Thus, the studies on predicting
and controlling the chaos on BEC systems are quite important. An active
control method can be used in time and space dependent systems. Recently,
synchronization of BEC via an active control method was investigated for the
time dependence [34, 35].In this study, we have studied the synchronization of
the BEC system held in a 1D tilted bichromatical optical lattice potential via
the master-slave active control technique for real space dependence. We first
present the system under 1D tilted bichromatical optical lattice potential,
and then we investigate chaotic synchronization in BEC. The chaotic
synchronization system consists of the master and slave system. The control
functions provide the situation when the states of the chaotic slave system
exponentially synchronize with the state of the master system with different
initial conditions. The initial values of the master-slave system are taken as
$(x_{1}[0],y_{1}[0])=(1,-1)$ and $(x_{2}[0],y_{2}[0])=(0.2,0.3)$ and
$(x_{1}[0],y_{1}[0])=(0.56,0.3)$ and $(x_{2}[0],y_{2}[0])=(-0.279,-0.67)$,
respectively, for numerical simulation. The synchronization errors rapidly
converge to zero. From the physical point of view, the technique used is based
on the Routh-Hurwitz criterion and is quite useful in the ultra-cold quantum
superfluid atoms. Numerical results verify the accuracy of the applied method,
and the obtained results show that two idential BEC could be synchronized by
an active control method for real space dependence.

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