HRI-P-08-12-004

A comparison of the sensitivities of the parameters

with atmospheric neutrinos for different analysis methods

Abhijit Samanta 111E-mail address<EMAIL_ADDRESS>
Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211 019,
India

###### Abstract

In the atmospheric neutrino experiments the primary problems are the huge
uncertainties of flux, very rapid fall of flux with increase of energy, the
energy dependent wide resolutions of energy and zenith angle between true
neutrinos and reconstructed neutrinos. These all in together make the choice
of binning of the data for chi-square analysis complicated. The large iron
calorimeter has the ability to measure the energy and the direction of the
muon with high resolution. From the bending of the track in the magnetic field
it can also distinguish its charge. We have analyzed the atmospheric neutrino
oscillation generating events by Nuance and then considering the muons
produced in the charge current interactions as the reconstructed neutrinos.
This practically takes into account the major problem of wide resolutions. We
have binned the data in three ways: i) in the grids of $\log E-\log L$ plane,
ii) in the grids of $\log E-\cos\theta_{\rm zenith}$ plane, and iii) in the
bins of $\log(L/E)$. We have performed a marginalized $\chi^{2}$ study over
$\Delta m_{32}^{2},~{}\theta_{13}$ and $\theta_{23}$ for neutrinos and anti-
neutrinos separately for each method and finally compared the results.

PACS 14.60.Pq

Keywords: neutrino oscillation, atmospheric neutrino, INO

##  1 Introduction

The atmospheric neutrino anomaly was first observed by IMB in 1986 and then
confirmed by Kamiokande in 1988 [1, 2]. Finally, the neutrino oscillation was
discovered in 1998 with atmospheric neutrino experiment [3]. The atmospheric
neutrinos are produced by the interactions of the cosmic rays with the
atmosphere. At the neutrino energies above a few GeV, the effect of geo-
magnetic field on the cosmic rays is negligible and then the atmospheric
neutrino flux can be predicted to be up-down symmetric. The flight lengths for
up and down going neutrinos are very different. The atmospheric neutrino
experiments exploit these features to study the neutrino oscillation.

The atmospheric neutrinos are also equally important in the precision era of
neutrino physics. The main thrust is now on the precise measurements of
oscillation parameters. This helps to identify the right track to understand
the underlying principle that gives the neutrino masses and their mixing. In
the recent years, the studies of neutrinos has become a popular tool to probe
the physics beyond the standard model. In the standard oscillation picture,
there are six parameters. The present 1$\sigma$, 2$\sigma$ and 3$\sigma$
confidence level ranges from global $3\nu$ oscillation analysis (2008) 222The
CP-violating phase $\delta_{CP}$ is still unconstrained. [4] are tabulated in
table 1.

Parameter | $\Delta m_{21}^{2}/10^{-5}\mathrm{\ eV}^{2}$ | $\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{13}$ | $\sin^{2}\theta_{23}$ | $|\Delta m_{31}^{2}|/10^{-3}\mathrm{\ eV}^{2}$
---|---|---|---|---|---
Best fit | 7.67 | 0.312 | 0.016 | 0.466 | 2.39
$1\sigma$ range | 7.48 – 7.83 | 0.294 – 0.331 | 0.006 – 0.026 | 0.408 – 0.539 | 2.31 – 2.50
$2\sigma$ range | 7.31 – 8.01 | 0.278 – 0.352 | $<0.036$ | 0.366 – 0.602 | 2.19 – 2.66
$3\sigma$ range | 7.14 – 8.19 | 0.263 – 0.375 | $<0.046$ | 0.331 – 0.644 | 2.06 – 2.81
Table 1: Global 3$\nu$ oscillation analysis (2008)

This spectacular achievement is very stimulating to uncover the facts which
are still missing. To determine the mass ordering (sign of $\Delta
m_{32}^{2}$) 333$\Delta m^{2}_{32}=m_{3}^{2}-m_{2}^{2}$., the values of
$\theta_{13}$ and $\delta_{CP}$ with good precision, the octant of
$\theta_{23}$ with atmospheric neutrinos as well as neutrinos from artificial
beams, there are many ongoing and planned experiments: INO [5], UNO [6], T2K
[7], NOvA [8], Hyper-Kamiokande [9] and many others. In the current few years,
a large fraction of effort in particle physics research has gone to study the
physics potential of these detectors [10]. The current research activity [11,
12, 13, 14, 15, 16, 17, 18, 19, 20] shows the uniqueness in physics potential
of the large magnetized Iron CALorimeter (ICAL) detector at the India-based
Neutrino Observatory (INO). It should be noted here that its position at
PUSHEP has a special feature. It gives the magic baseline from CERN for beam
experiments, which provides the oscillation probabilities relatively
insensitive to the yet unconstrained CP phase compared to all other baselines
and permits to make the precise measurements of the masses and their mixing
avoiding the degeneracy issues [16]. On the other hand, ICAL can detect
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$ separately using the magnetic field for
charge current events. The oscillation study with atmospheric neutrinos is the
primary goal of ICAL at INO. Before going into the detailed techniques of the
analysis methods, we will first discuss the basic nature of atmospheric
neutrino oscillation and the detection characteristics of ICAL detector.

###  1.1  The atmospheric neutrino oscillation and the ICAL detector

The present atmospheric neutrino data from the pioneering Super Kamiokande
(SK) experiment are well explained by two flavor oscillation [21, 22].
However, one expects the reflection of $\nu_{\mu}\rightarrow\nu_{e}$
oscillation in data for standard 3-flavor framework in the data if
$\theta_{13}$ is nonzero. Neglecting the $\Delta m_{21}^{2}$ term the
oscillation probability can be expressed as:

$\displaystyle\mbox{P}(\nu_{\mu}\rightarrow\nu_{e})$ $\displaystyle=$
$\displaystyle\mbox{P}(\nu_{e}\rightarrow\nu_{\mu})$ $\displaystyle=$
$\displaystyle\sin^{2}\theta_{23}\sin^{2}2\theta_{13}\sin^{2}\left(\frac{1.27\Delta
m^{2}L}{E}\right)$ $\displaystyle\mbox{P}(\nu_{\mu}\rightarrow\nu_{\mu})$
$\displaystyle=$ $\displaystyle 1$ (1)
$\displaystyle-4\cos^{2}\theta_{13}\sin^{2}\theta_{23}(1-\cos^{2}\theta_{13}\sin^{2}\theta_{23})$
$\displaystyle\times\sin^{2}\left(\frac{1.27\Delta m^{2}L}{E}\right)$

These oscillation probabilities are derived for vacuum. Since it involves
electron neutrino, the oscillation will be modulated by the matter effect [23,
24]. Then,

$\displaystyle\mbox{P}(\nu_{\mu}\rightarrow\nu_{e})$ $\displaystyle=$
$\displaystyle\mbox{P}(\nu_{e}\rightarrow\nu_{\mu})$ (2) $\displaystyle=$
$\displaystyle\sin^{2}\theta_{23}\sin^{2}2\theta_{13}^{M}\sin^{2}\left(\frac{1.27\Delta{m^{2}}_{M}L}{E}\right).$

The symbol ‘M’ denotes effective parameters in matter. The effective mixing
angle is

$\displaystyle\sin^{2}2\theta_{13}^{M}$ $\displaystyle=$
$\displaystyle\frac{\sin^{2}2\theta_{13}}{(\cos 2\theta_{13}-A_{CC}/\Delta
m^{2})^{2}+\sin^{2}2\theta_{13}}$ (3)

and

$\displaystyle\Delta{m^{2}}_{M}$ $\displaystyle=$
$\displaystyle\sqrt{\left(\Delta m^{2}\cos
2\theta_{13}-A_{CC}\right)^{2}+\left(\Delta m^{2}\sin
2\theta_{13}\right)^{2}}$ (4)

with

$\displaystyle A_{CC}$ $\displaystyle=$ $\displaystyle 2\sqrt{2}G_{F}N_{e}E,$
(5)

where $G_{F}$ is the Fermi constant, $N_{e}$ is the electron density of the
medium and $E$ is neutrino energy [25]. The matter potential term $A_{CC}$ has
the same absolute value, but opposite sign for neutrinos and anti-neutrinos.
The Mikheyev-Smirnov-Wolfenstein (MSW) resonance occurs when neutrino passes
through the matter (see eq. 3). It happens for Normal Hierarchy (NH) with
neutrinos and for Inverted Hierarchy (IH) with anti-neutrinos. The resonance
energy corresponding to a baseline can be seen in [26].

The muon neutrino (anti-neutrino) produces $\mu^{-}$ ($\mu^{+}$) in Charge
Current (CC) weak interactions. The magnetized ICAL can distinguish $\mu^{+}$
and $\mu^{-}$ with the magnetic field. The energy ($E$) and zenith angle
($\theta_{\rm zenith}$) or baseline ($L$) resolutions of the muons are very
high at ICAL [5]. The hadron energy can also be measured at ICAL. However, its
resolution is very poor and strongly depends on thickness of the iron layers.

The atmospheric neutrinos are expected to be very useful in precision studies
for its very wide energy range (MeV $-$ few hundred GeV) and wide baseline
range (few km $-12950$ km). It gives both neutrino and anti-neutrino, which
behave oppositely with matter. This helps to detect the sign( $\Delta
m_{32}^{2}$), the value of $\theta_{13}$ as well as the octant of
$\theta_{23}$. One can exploit this feature to measure the precision of these
parameters and the mass ordering at the magnetized ICAL detector at INO. It
should be noted here that the non-magnetized detectors, like water Cherenkov
detector, can also contribute in this study since the cross section, the
$y(=(E_{\nu}-E_{\rm lepton})/E_{\nu})$ dependence of the cross section are
different for $\nu$ and $\bar{\nu}$. The water detectors may also be able to
distinguish statistically $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ due to different
capture rates and lifetimes of the charged muons in water.

However, one of the crucial problems in neutrino physics experiments is the
wide resolutions of $E$ and $L$ between true neutrinos and reconstructed
neutrinos, which smears the oscillation effect to some significant extent.
This arises mainly due to interaction kinematics. The un-observable product
particles, un-measurable momentum of recoiled nucleus are the main sources of
this huge uncertainty in reconstructed neutrino momentum. These are strongly
neutrino energy dependent.

Due to the above complications, the method of extraction of the results from
the data is not straightforward. The results depend crucially on the way of
the analysis and particularly on the type of binning of the data. This fact is
well-known from the analysis of atmospheric neutrino data of SK experiment
[21, 22]. In this paper we consider the reconstructed energy and the direction
of an event only from the muon generating it by the neutrino event generator
Nuance-v3[27]. The addition of hadrons to the muon, which might increase the
reconstructed neutrino energy resolution, is not considered here for
conservative estimation of the sensitivity. It would be realistic in case of
GEANT-based studies since the number of hits produced by the hadron shower
strongly depends on the iron thickness. However, INO can also detect the
neutral current events. Though it is expected that these will not have any
directional information, the energy dependency of the averaged oscillation
over all directions can also contribute to the total $\chi^{2}$ separately in
the sensitivity studies. Here we have studied the atmospheric neutrino
oscillation by binning the data in different ways and finally compared the
results. These are discussed in the next sections.

##  2 The $\chi^{2}$ analysis

Now we will describe a general expression for $\chi^{2}$, the method for
generation of the theoretical data, the estimated systematic uncertainties,
and finally the ways of binning of the data. The number of events falls very
rapidly with the increase of energy and the statistics is very poor at high
energy. However, the contribution to the sensitivities of the oscillation
parameters is significant from these high energy events. To incorporate these
events at high energy, the $\chi^{2}$ value is calculated according to Poisson
probability distribution. For all types of binning, we define a general
expression of $\chi^{2}$ as

$\displaystyle\chi^{2}$ $\displaystyle=$
$\displaystyle\sum_{I=1}^{N}\left[2\left\\{N^{p}_{I}-N^{o}_{I}\right\\}-2N^{o}_{I}\ln\left(\frac{N^{p}_{I}}{N^{o}_{I}}\right)\right]+\sum_{k=1}^{n_{s}}{\xi_{k}}^{2}$
(6)

with

$\displaystyle N^{p}_{I}$ $\displaystyle=$ $\displaystyle\sum_{i,j=n_{c}^{\rm
low},n_{E}^{\rm low}}^{n_{c}^{\rm high},n_{E}^{\rm
high}}N^{p}_{ij}\left(1+\sum_{k=1}^{n_{s}}f^{k}_{ij}\cdot\xi^{k}\right),$
$\displaystyle{\rm and}$ (7) $\displaystyle N^{o}_{I}$ $\displaystyle=$
$\displaystyle\sum_{i,j=n_{c}^{\rm low},n_{E}^{\rm low}}^{n_{c}^{\rm
high},n_{E}^{\rm high}}N^{o}_{ij}$ (8)

The $N^{o}_{ij}$ ($N^{p}_{ij}$) is considered as the number of observed
(predicted) events in the $ij$th grid in the plane of $\log E-\cos\theta_{\rm
zenith}$. Here we consider the data for 1 Mton.year exposure of the detector.
The $f^{k}_{ij}$ is the systematic error of $N^{p}_{ij}$ due to the $k$th
uncertainty. The ${\xi_{k}}$ is the pull variable for the $k$th systematic
error. We consider $n_{s}=5$. Here we have considered 30 bins of $\log E$ and
300 bins of $\cos\theta_{\rm zenith}$ for both $N^{p}_{ij}$ and $N^{o}_{ij}$.
However, it should be noted here that in calculation of the oscillated flux we
consider 200 bins of $\log E$ and 300 bins of $\cos\theta_{\rm zenith}$ to
find the accurate oscillation pattern. We consider the $E$ range $0.8-50$ GeV
and $\cos\theta_{\rm zenith}$ range $-1$ to $+1$. It should be noted here that
the energy and angular resolutions between the muons and the neutrinos of the
events differ significantly for neutrinos and anti-neutrinos due to their
different ways of interactions.

To generate the theoretical data $N^{p}_{ij}$ for the chi-square analysis, we
first generate 500 years un-oscillated data for 1 Mton detector. From this
data we find the energy-angle correlated resolutions (see figs. 1) in 30 bins
of energy (in log scale) and 10 bins of cosine of zenith angle ($-1$ to $+1$).
For a given $E_{\nu}$, we calculate the efficiency of having $E_{\mu}\geq$ 0.8
GeV (threshold of the detector). For each set of oscillation parameters, we
integrate the oscillated atmospheric neutrino flux folding the total CC cross
section, the exposure time, the target mass, the efficiency and the resolution
function to obtain the predicted data in the reconstructed $\log
E-\cos\theta_{\rm zenith}$ grid 444One can do this in an another way. This is
generating the theoretical data directly for each set of oscillation
parameters. To ensure that the statistical error is negligible, one needs
first to generate a huge number of events. For example, one may generate
events for 500 Mton.year exposure of the detector for each set of oscillation
parameters. Then to obtain the theoretical data, one needs to normalize the
data to 1 Mton.year exposure of the detector dividing the events of each
energy and zenith angle bin by 500 since the experimental data is considered
for 1 Mton.year exposure. This would be the more straightforward method. But
the marginalization study with this method is almost an undoable job in normal
CPU. However, an exactly equivalent result is obtained here using the energy-
angle correlated resolution function.. We use the CC cross section of
Nuance-v3 [27] and the Honda flux of 3-dimensional scheme [28].

The atmospheric neutrino flux is not known precisely. There are huge
uncertainties in its estimation. We may divide them into two categories: I)
overall uncertainties (which are flat with respect to energy and zenith
angle), and II) tilt uncertainties (which are function of energy and/or zenith
angle). These have been estimated as the following [21]:

  1. 1.

The energy dependence uncertainty which arises due to the uncertainty in
spectral indices, can be expressed as:

$\Phi_{\delta_{E}}(E)=\Phi_{0}(E)\left(\frac{E}{E_{0}}\right)^{\delta_{E}}\approx\Phi_{0}(E)\left[1+\delta_{E}\log_{10}\frac{E}{E_{0}}\right].$
(9)

The uncertainty of ${\delta_{E}}=$5% and $E_{0}=2$ GeV is considered.

  2. 2.

Again, the flux uncertainty as a function of zenith angle can be expressed as

$\Phi_{\delta_{z}}(\cos\theta_{z})\approx\Phi_{0}(\cos\theta_{z})\left[1+\delta_{z}|\cos\theta_{z}|\right].$
(10)

The uncertainty of $\delta_{z}$ is considered to be $2\%$.

  3. 3.

A flux normalization uncertainty of 20%.

  4. 4.

An over all uncertainty of 10% in neutrino cross section.

  5. 5.

An overall 5% uncertainty for this analysis.

We consider three types of binning:

  * •

Type I: The events are binned in the grid of $\log E-\log L$ plane. We use
total number of $\log E$ bins $n_{E}$ = 30 (0.8 $-$ 50 GeV) and the number of
$\log L$ bins as a function of of the energy. We consider $n_{L}=2\times
14,~{}2\times 18,~{}2\times 22,~{}2\times 26,$ and $~{}2\times 30$ for
$E=0.8-1.2,~{}1.2-2.4,~{}2.4-3.6,~{}3.6-4.8,~{}{\rm and}~{}>4.8$ GeV,
respectively. For the down-going events the binning is done by replacing
‘$\log L$’ by $`-\log L$’. The factor ‘2’ is to consider both up and down
going cases.

  * •

Type II: The events are binned in the grid of $\log E-\cos\theta_{\rm zenith}$
plane with exactly in the same fashion of type I. The only difference is that
the binning is done in $\cos\theta_{\rm zenith}$ instead of $\log L$.

  * •

Type III: The events are binned in 100 $\log(L/E)$ bins and replacing
‘$\log(L/E)$’ by ‘$-\log(L/E)$’ for down-going events.

For the up-going neutrino, $L$ is the distance traveled by the neutrino from
the detector to the source at the atmosphere. In case of down-going neutrinos
the distance traveled from the source to detector is negligible for getting an
appreciable oscillation. However, these events help to minimize the systematic
uncertainties when considered in the $\chi^{2}$ analysis. The flux for a fixed
$E$ is strongly dependent on the zenith angle. So, for the down-going
neutrinos, we mapped the zenith angle into $L$ considering the mirror $L$.
This is the same $L$ if the neutrino comes from exactly opposite direction. It
should be noted here that the angular error makes a much smaller error to $L$
when the tracks are near vertical. It increases gradually when the tracks are
slanted and very rapidly when they are near horizontal.

For each set of oscillation parameters we calculate the $\chi^{2}$ in two
stages. First we used $\xi_{k}$ such that
$\frac{\delta\chi^{2}}{\delta\xi_{k}}=0$, which can be obtained solving the
equations [29]. Then we calculate the final $\chi^{2}$ with these $\xi_{k}$
values. Finally, we find the minimum from these $\chi^{2}$ with respect to all
oscillation parameters 555Here we consider all uncertainties as a function of
reconstructed neutrino energy and direction. We assumed that the tilt
uncertainties will not be changed too much due to reconstruction. However, on
the other hand, if any tilt uncertainty arises in reconstructed neutrino
events from the reconstruction method or kinematics of scattering, these are
then accommodated in $\chi^{2}$. We first incorporate all uncertainties in
$\log E-\cos\theta_{\rm zenith}$ bins. Then we re-bin the data in the form
what we want, e.g.; $\log E-\log L$ bins. It should be noted that we first
binned the data into a large number of $\cos\theta_{\rm zenith}$ bins compared
to number of $L$ bins to get proper binning in $\log L$..

Figure 1: The sample energy-angle correlated resolution plots for neutrino
(left column) and anti-neutrino (right column) for the bins of
$E_{\nu}=0.85-0.98$ GeV with $\cos\theta_{\rm zenith}=-0.40$ to $-0.20$ (upper
row) and $E_{\nu}=6.84-7.86$ GeV and $\cos\theta_{\rm zenith}=0$ to $0.20$
(lower row). The data are obtained from the simulation of 500 MTon.year
exposure of ICAL considering no oscillation. Figure 2: The oscillation
probability of $\nu_{\mu}\rightarrow\nu_{\mu}$. We choose $\Delta
m_{32}^{2}=-2.5\times 10^{-3}$eV2, $\theta_{23}=45^{\circ}$ and
$\theta_{13}=0^{\circ}$. Figure 3: The typical distribution of
$\Delta\chi^{2}$ with $\Delta m^{2}_{32}$. We choose the input of $\Delta
m_{32}^{2}=+2.5\times 10^{-3}$eV2, $\theta_{23}=42^{\circ}$ and
$\theta_{13}=7.5^{\circ}$.

##  3 Result

In this section, we first discuss the results qualitatively in a very general
way for all analysis techniques. Then we compare the results for different
techniques. In all cases a global scan is carried out over the three
oscillation parameters $\Delta m_{32}^{2},~{}\theta_{23}$ and $\theta_{13}$
for both normal and inverted hierarchies with neutrinos and anti-neutrinos
separately. We have considered the range of $\Delta m_{32}^{2}=2.0-3.0\times
10^{-3}$eV2, $\theta_{23}=37^{\circ}-54^{\circ}$, and
$\theta_{13}=0^{\circ}-12.5^{\circ}$. We have fixed other parameters $\Delta
m_{21}^{2}$ and $\theta_{12}$ at their best-fit values and $\delta_{\rm
CP}=0$. The 2-dimensional 68%, 90%, 99% confidence level allowed parameter
spaces (APSs) are obtained by considering $\chi^{2}=\chi^{2}_{\rm
min}+2.48,~{}4.83,~{}9.43$. To obtain the APS in $\theta_{13}-\Delta
m_{32}^{2}$ ($\Delta m_{32}^{2}-\theta_{23}$) plane, we marginalize the
$\chi^{2}$ over $\theta_{23}~{}(\theta_{13})$ over its whole range.

The experiment indicates that the value of $\theta_{13}$ is very small
compared to $\theta_{23}$ [30]. So, the atmospheric neutrino oscillation is
mainly governed by two flavor oscillation
$\nu_{\mu}~{}(\bar{\nu}_{\mu})~{}\leftrightarrow\nu_{\tau}~{}(\bar{\nu}_{\tau})$.
This constrains $\sin^{2}2\theta_{23}$ and $|\Delta m_{32}^{2}|$. Now, there
appears a degeneracy in $\theta_{23}$ whether it is larger or smaller than
$45^{\circ}$ due to the $\sin^{2}2\theta_{23}$ dependence of oscillation
probability. However, when the matter effect comes into the play, the
effective value of $\theta_{13}$ becomes large and a resonance occurs in
$\nu_{\mu}~{}(\bar{\nu}_{\mu})~{}\leftrightarrow\nu_{e}~{}(\bar{\nu}_{e})$
oscillation. This breaks the above $\theta_{23}$ degeneracy.

The difference in oscillation probability between two $\theta_{13}$ values for
neutrinos with NH and for anti-neutrinos with IH becomes significant when
matter effect comes in the picture (see eq. 3). We have plotted the APS in
$\theta_{13}-\Delta m_{32}^{2}$ plane considering both neutrinos and anti-
neutrinos (i.e. with $\chi^{2}_{\rm
total}=\chi^{2}_{\nu}+\chi^{2}_{\bar{\nu}}$) for different sets of input
parameters at 68%, 90% and 99% CL in fig 4, 5 and 6, respectively for each
type of binning of the data. We see that the matter effect significantly
constrains $\theta_{13}$ over the present limit, which is a very stimulating
result for atmospheric neutrino oscillation analysis.

Again, for the APS in $\Delta m^{2}_{32}-\theta_{23}$ plane, $\theta_{13}$ is
marginalized over the present allowed range. The APSs are shown in fig. 7, 8
and 9 at 68%, 90% and 99% CL, respectively for each type of binning of the
data. If the value of $\theta_{13}$ is nonzero, the matter effect plays a role
in determination of the octant of $\theta_{23}$ as discussed previously and
also constrains the $\theta_{23}$ range (compare its range for zero and non-
zero values of $\theta_{13}$). We find that for some combinations of
($\theta_{13},~{}\theta_{23}$), the octant determination is possible.

Now we will compare the APSs coming from different analysis method. From the
APSs it is clear that the $L/E$ analysis gives very poor results compared to
the other two methods. It happens due to the mixing of events from different
$E$ and $L$ resolutions since the resolution widths are strongly energy
dependent. It should be noted here that we have not used any selection
criteria for the events, which might improve the results.

Now we will compare the positive and the negative sides of the rest two
methods. We find a relatively stronger upper bound of $\Delta m^{2}_{32}$ in
case of binning in the grids of $\log E-\log L$ plane than the other case.
This is very important since it comes from the events with high $E$ and low
$L$ values. The $L$ resolution is very poor at low $L$ and the statistics is
low at high $E$. However, a stronger bound is obtained for this special type
of binning. We will explain it with the oscillation probability in vacuum,
which is a sinusoidal function of $L/E$. For a fixed $L$, the distance between
two consecutive peaks in $E$ increases rapidly with $E$. Again, if we compare
the distances between two consecutive peaks in $E$ for two fixed values of
$L$, it is larger for smaller $L$ value. Therefore, as one goes to smaller $L$
values, this distance in $E$ increases rapidly. So, these two consecutive
peaks of the oscillation in $E$ can be resolved with much better resolution as
one goes gradually from larger $L$ values to lower $L$ values. This is
pictorially illustrated in fig. 2. To get the reflection of this fact in
$\chi^{2}$, the finer binning at lower $L$ is essential. Though the angular
resolution is worsened at lower $L$, but the rapid increase of $E$ resolution
between two peaks wins the competition here. This is the main advantage of
this type of binning. So, we binned the data in a two dimensional grids of
$\log L-\log E$ plane 666This captures the oscillation effect well in
$\chi^{2}$ analysis without mixing events from different $E$ and $L$
resolutions.. In type II this behavior is not taken into account in the
binning of the data. However, there is a disadvantage in type I that the bin
size at high $L$ values is very large compared to type II, which gives weaker
lower bound on $\Delta m_{32}^{2}$. So, the combination of type I and II (type
I at the lower range of $L$ and type II for the rest) is a better choice than
the individual cases. However, this is not studied in this paper, but is
reflected when we compare two results. This is also demonstrated in terms of
$\Delta\chi^{2}$ for a typical set of parameters in fig. 3. It should be noted
here that the contrast between two methods would be prominent when the number
of bins in $L$ or $\cos\theta_{\rm zenith}$ will be relatively lowered than
that used in this paper.

For a quantitative assessment of the result, we define the precision $P$ of a
parameter $t$ as:

$P=2\left(\frac{t^{\rm max}-t^{\rm min}}{t^{\rm max}+t^{\rm min}}\right)$ (11)

We see, one can achieve the precision of $\Delta m_{32}^{2}\approx$
$4.8-7.5\%~{}(5.4-8.0\%),~{}6.9-10.9\%~{}(8.0-12.6\%)~{}{\rm
and}~{}9.6-15.7\%~{}(10.9-17.6\%)$, at 68%, 90% and 99% CL, respectively in
type I (II) method. For the input with bi-maximal mixing of $\theta_{23}$, we
find its precision in terms of $\sin^{2}\theta_{23}\approx$
$14.3-31.8\%~{}(16.9-36.9\%),~{}21.6-36.8\%~{}(22.4-41.7\%),{~{}\rm
and}~{}28.5-42.1\%~{}(27.9-45.9\%)$ at 68%, 90% and 99% CL, respectively in
type I (II) method. The precision of $\theta_{13}$ is strongly dependent on
its input value. For $\theta_{13}=0$, we find its upper bound $\approx
6.4^{\circ}~{}(8.0^{\circ}),~{}8.0^{\circ}~{}(9.5^{\circ})$ and
$10.1^{\circ}~{}(11.5^{\circ})$ at 68%, 90% and 99% CL, respectively in type I
(II) methods. The both lower and upper bounds are also possible for some
combinations of ($\theta_{23},\theta_{13}$) and it happens mainly for
$\theta_{23}\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$>$}\hss}{\lower
2.15277pt\hbox{$\sim$}}}45^{\circ}$.

##  4 Conclusion

In this paper we have binned the atmospheric data in three ways: i) in the
grids of $\log E-\log L$ plane, ii) in the grids of $\log E-\cos\theta_{\rm
zenith}$ plane, and iii) in the bins of $\log(L/E)$. We have performed a
marginalized $\chi^{2}$ study over $\Delta m_{32}^{2},~{}\theta_{13}$ and
$\theta_{23}$ for neutrinos and anti-neutrinos separately for each method.
Finally, we find that in spite of very poor resolutions at low $L$, which is
the main problem as $\Delta m_{32}^{2}$ goes to the upper range, one can
obtain a relatively stronger upper bound in case of binning in $\log E-\log L$
plane compared to the binning in $\log E-\cos\theta_{\rm zenith}$ plane.
However, it is also found from both analysis that considerable precisions of
$\theta_{13}$ and $\Delta m_{32}^{2}$ can be achieved and the octant
discrimination can also be possible for some combinations of
($\theta_{23},\theta_{13}$).

Figure 4: The 68% CL allowed regions in $\theta_{13}-\Delta m_{32}^{2}$ plane
for type I (top) type II (middle) and type III (bottom) binning of the data
with the input of
$\theta_{23}=40^{\circ},42^{\circ},45^{\circ},48^{\circ},50^{\circ}$ with
$\theta_{13}=0^{\circ}$ (first column), $5^{\circ}$ (second column),
$7.5^{\circ}$ (third column) from neutrinos with NH and $7.5^{\circ}$ (fourth
column) from anti-neutrinos with IH.

Figure 5: The same plots of fig. 4 but with 90% CL.

Figure 6: The same plots of fig. 4 but with 99% CL.

Figure 7: The 68% CL allowed regions in $\theta_{23}-\Delta m_{32}^{2}$ plane
for type I (top) type II (middle) and type III (bottom) binning of the data
with the input of
$\theta_{23}=40^{\circ},42^{\circ},45^{\circ},48^{\circ},50^{\circ}$ with
$\theta_{13}=0^{\circ}$ (first column), $5^{\circ}$ (second column),
$7.5^{\circ}$ (third column) from neutrinos with NH and $7.5^{\circ}$ (fourth
column) from anti-neutrinos with IH.

Figure 8: The same plots of fig. 7 but with 90% CL.

Figure 9: The same plots of fig. 7 but with 99% CL.

Acknowledgments: I would like to acknowledge the excellent cluster
computational facility of HRI, which makes the work possible. The research has
been supported by the funds from Neutrino Physics project at HRI.

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# Examining the Values Reflected by Children during AI Problem Formulation

Utkarsh Dwivedi, Salma Elsayed-Ali, Elizabeth Bonsignore & Hernisa Kacorri
<EMAIL_ADDRESS>University of Maryland, College
ParkCollege ParkMarylandUSA

(2023)

###### Abstract.

Understanding how children design and what they value in AI interfaces that
allow them to explicitly train their models such as teachable machines, could
help increase such activities’ impact and guide the design of future
technologies. In a co-design session using a modified storyboard, a team of 5
children (aged 7-13 years) and adult co-designers, engaged in AI problem
formulation activities where they imagine their own teachable machines. Our
findings, leveraging an established psychological value framework (the Rokeach
Value Survey), illuminate how children conceptualize and embed their values in
AI systems that they themselves devise to support their everyday activities.
Specifically, we find that children’s proposed ideas require advanced system
intelligence, e.g. emotion detection and understanding the social
relationships of a user. The underlying models could be trained under multiple
modalities and any errors would be fixed by adding more data or by
anticipating negative examples. Children’s ideas showed they cared about
family and expected machines to understand their social context before making
decisions.

participatory machine learning, machine teaching, values, cooperative inquiry,
co-design

††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: ACM CHI Workshop on Child-Centred AI Design:
Definition, Operation, and Considerations; April 23–28, 2023; Hamburg,
Germany††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06

##  1\. Introduction

Figure 1. Our co-design study comprises: a) Modified Big Paper with a
structured storyboard that b) child-adult pairs use to frame and discuss their
ideas and then c) present their teachable machines while a researcher
summarizes their input.

Gaining insight into children’s values is a longstanding challenge in Child-
Computer Interaction (CCI) (Elsayed-Ali et al., 2020; Spiel et al., 2018; Van
Mechelen et al., 2014; Skovbjerg et al., 2016; Druin, 2002; Iversen and Leong,
2012; Iversen et al., 2012). Aligned with these efforts, in this work we
involve children as active participants in designing future technology to
reveal how they conceptualize and embed their values in a variety of AI system
that they envision will support their everyday lives. The technology children
are called to design is a teachable machine111A term first coined in 1978
(Andreae and Andreae, 1978) and more recently used by Google (Lab, 2017).,
where “the user is a willing participant in the adaptation process and
actively provides feedback to the machine to guide its learning” (Patel and
Roy, 1998). Teachable machines are playing an essential role in designing
activities that can spark curiosity and promote children’s understanding of
how AI works, its limitations, how bias creeps in, and its potential
implications for our society (Dwivedi et al., 2021; Vartiainen et al., 2021;
Hitron et al., 2019). To elicit children’s interests and values we don’t ask
them to tinker with the input or output of existing teachable machines e.g.,
recognize objects (Vartiainen et al., 2020; Dwivedi et al., 2021) or respond
to gestures (Hitron et al., 2019). Instead, we invite them to imagine their
own. Specifically, we invite a university-based intergenerational design team
of 5 children (aged 8-13 years) and 5 adults, who have previously engaged in
typical black-box tinkering teachable machines activities (e.g., similar to
Dwivedi et al. (Dwivedi et al., 2021) and Vartiainen et al. (Vartiainen et
al., 2021)). As shown in Figure 1, we employ a structured “Big Paper”
technique (Fails et al., 2012). Structured storyboards allow us to explore how
children define the context (i.e. everyday use cases) as well as input and
outputs for their personal AI-infused technologies. More so, they explicitly
elicit children’s notions of potential system failures their machine might
encounter, along with their proposed approach for error recovery.

We used the Rokeach Value Survey (Rokeach, 1973), an established psychological
value framework, as an analytical framework to examine the values reflected in
children’s designs while triangulating data from 5 “Big Paper” storyboards, 2
hours of session recordings, and 2 sets of field notes. We contribute evidence
that participatory machine learning approaches that engage children in AI
problem formulation can offer designers and researchers a means for re-
centering children in the design of AI-infused technologies and for gaining
insights into children’s values in AI. Specifically, our exploratory study:
(i) provides empirical results on how children (8-13 years old) formulate the
everyday issues their AI systems will address; and (ii) provides an approach
for inferring human values in participatory machine learning projects by using
Rokeach’s value survey (Rokeach, 1973) as an analytical lens for interpreting
the personal values in children’s designs.

##  2\. Related Work

Here, we summarize two strands of related work that supported our efforts to
co-design the everyday AI systems that children envisioned.

###  2.1. Problem Formulation via Participatory Machine Learning

There is a recent push for participatory techniques in machine learning (see
Sloane et al. (Sloane et al., 2020) for a detailed review). The term
‘participatory machine learning’ is often used when participatory techniques
are included in the problem formulation phase (e.g., (Google, 2021; Martin et
al., 2020)). Martin et al. (Martin et al., 2020) state that problem
formulation, the focus of our co-design work, is a vital step in any machine
learning application. They define it as“a step that involves determining the
strategic goals driving the interventions and translating those strategic
goals into tractable machine learning problems.” Practical guidelines have
emerged to help developers formulate their problems in machine learning
(Google, 2022; Amazon, 2022). However, these guidelines typically do not
mention or recommend stakeholder involvement. Most prior work refers to adults
as stakeholders; very few studies have engaged children in problem formulation
via participatory machine learning (e.g. (Woodward et al., 2018; Vartiainen et
al., 2021; Druga et al., 2022)).

###  2.2. Children’s Values in Design

Technology is inherently value-laden (Eriksson et al., 2022; Verbeek, 2011;
Iversen and Leong, 2012; Van Mechelen et al., 2014) and computing research on
values has largely focused on values of ethical import (Muller et al., 1997;
Friedman, 1996) such as privacy and security. Few of these studies focus on
personal values and they largely emphasize the values of adults and
researchers rather than children (Skovbjerg et al., 2016; Yarosh et al., 2011;
Kawas et al., 2020; Van Mechelen et al., 2014). Skovbjerg et al. (Skovbjerg et
al., 2016) recommend that Child-Computer Interaction (CCI) researchers
acknowledge and discuss the embedded values in systems from the outset. In
case of AI, important initial design implications can emerge in the problem
formulation phase, raising profoundly different ethical concerns, such as
possible threats to fairness and civil rights (Passi and Barocas, 2019). Thus,
it is critical to examine and explicitly foreground children’s personal values
in the design of AI-infused technology, and technology more broadly (Elsayed-
Ali et al., 2020; Druin, 2002; Yarosh et al., 2011; Skovbjerg et al., 2016;
Spiel et al., 2018). Social psychologist Milton Rokeach defines values as
enduring beliefs and personal standards that guide and determine actions,
attitudes, ideologies, judgments, justifications, and presentations of self
(Rokeach, 1973). In his established value survey (RVS), he identified 36
values: 18 instrumental reflecting preferred behaviors and 18 terminal
reflecting preferred end-states of existence (Rokeach, 1973; Iversen and
Leong, 2012; Voida and Mynatt, 2005; Elsayed-Ali et al., 2020). We use these
as an analytical tool to examine the perceived values reflected in children’s
designs.

##  3\. Insights

We developed an understanding of the instrumental and terminal values in
Rokeach’s framework (Rokeach, 1973) from past work by He et al. (He et al.,
2010) who used the RVS to classify persuasive energy feedback technologies.
Similarly, we sought to employ the RVS as an instrument for analyzing
children’s values. Using Rokeach’s 36 values and their definitions, two coders
found patterns in the data from 5 “Big Paper” storyboards, 2 hours of session
recordings, and 2 sets of field notes. They individually coded the data,
discussed classifications, resolved disagreements, and reached consensus.

###  3.1. Instrumental Values in Children’s Designs

Rokeach describes instrumental values as preferred modes of behavior. For
example, the value of “capability” entails competency and efficiency. Of the
18 instrumental values, we observed 5 in children’s designs of their everyday
AI systems. All of the children’s designs appealed to the following
instrumental values: “capability,” “logic,” “helpfulness,” and
“responsibility.” Children described how their teachable machines would assist
them in accomplishing a specific task, often explicitly using the word “help”
to describe their machines. For example, Penny (a pseudonym) stated, “My
machine is a wearable sensor, like a bracelet, and it’s a bracelet for
cheerleaders to help them stay safe.” Not only would Penny’s safety bracelet
ensure the welfare of the wearer, but also anyone within a close distance to
the wearer. She continued, “it tells people to get off the mat because it
measures the people who are on the mat first and it alerts them to get off the
mat to back up, and it measures if someone is still on or not.”

###  3.2. Terminal Values in Children’s Designs

Rokeach describes terminal values as preferred end-states. For example, the
value of “family security” would entail taking care of loved ones. Of the 18
values, we observed 4 that were present in children’s designs. We classify
children’s designs as appealing to the following terminal values: “family
security,” “a comfortable life,” “inner harmony,” and “an exciting life.” For
example, Alan’s teachable machine reflected Rokeach’s value “inner harmony,”
which means to have freedom from inner conflict. Alan designed an emotion
detection machine in order to identify and alleviate times when someone may be
experiencing negative emotions like sadness or anger, as well as detect and
reward times when someone is happy or excited.

##  4\. Conclusion

Our exploratory co-design session offers insights into the everyday issues
children imagine that AI systems can support (i.e., children’s AI problem
formulation). Moreover, we provide an approach for inferring human values in
participatory machine learning projects by using Rokeach’s value survey [22]
as an analytical lens for interpreting the personal values in children’s
designs. We hope to engage other workshop participants in future
collaborations that may benefit from these insights, or provide constructive
critiques of their potential utility in child-centered AI system design.

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# Densities for sets of natural numbers vanishing on a given family

Rafał Filipów  Institute of Mathematics
Faculty of Mathematics, Physics and Informatics
University of Gdańsk
ul. Wita Stwosza 57
80-308 Gdańsk
Poland<EMAIL_ADDRESS>http://mat.ug.edu.pl/ rfilipow and  Jacek
Tryba  Institute of Mathematics
Faculty of Mathematics, Physics and Informatics
University of Gdańsk
ul. Wita Stwosza 57
80-308 Gdańsk
Poland<EMAIL_ADDRESS>
###### Abstract.

Abstract upper densities are monotone and subadditive functions from the power
set of positive integers into the unit real interval that generalize the upper
densities used in number theory, including the upper asymptotic density, the
upper Banach density, and the upper logarithmic density.

At the open problem session of the Workshop “Densities and their application”,
held at St. Étienne in July 2013, G. Grekos asked a question whether there is
a “nice” abstract upper density, whose the family of null sets is precisely a
given ideal of subsets of $\mathbb{N}$, where “nice” would mean the properties
of the familiar densities consider in number theory.

In 2018, M. Di Nasso and R. Jin (Acta Arith. 185 (2018), no. 4) showed that
the answer is positive for the summable ideals (for instance, the family of
finite sets and the family of sequences whose series of reciprocals converge)
when “nice” density means translation invariant and rich density (i.e. density
which is _onto_ the unit interval).

In this paper we extend their result to all ideals with the Baire property.

###### Key words and phrases:

density of sets of integers, abstract upper density, submeasure, ideal of
sets, Baire property, maximal almost disjoint family, ideal convergence.

###### 2010 Mathematics Subject Classification:

Primary: 11B05. Secondary: 03E05.

The second author has been supported by the grant MNiD-539-5100-B348-19.

###### Contents

  1. 1 Abstract densities and ideals
  2. 2 Rich densities
  3. 3 Rich and translation invariant densities

##  1\. Abstract densities and ideals

######  Definition 1.1.

An _abstract upper density_ on $\mathbb{N}$ is a function
$\delta:\mathcal{P}(\mathbb{N})\to[0,1]$ that satisfies the following
properties:

  1. (1)

$\delta(\mathbb{N})=1$,

  2. (2)

if $F\subseteq\mathbb{N}$ is finite then $\delta(F)=0$,

  3. (3)

if $A\subseteq B$ then $\delta(A)\leq\delta(B)$,

  4. (4)

$\delta(A\cup B)\leq\delta(A)+\delta(B)$.

######  Notation.

For $A\subseteq\mathbb{N}$ and $k\in\mathbb{Z}$ we write $A+k$ to denote the
set $(A+k)\cap\mathbb{N}$.

######  Definition 1.2.

An abstract upper density $\delta$ is

  1. (1)

_translation invariant_ if $\delta(A+k)=\delta(A)$ for every
$A\subseteq\mathbb{N}$ and $k\in\mathbb{Z}$;

  2. (2)

_rich_ if for every $r\in[0,1]$ there is $A\subseteq\mathbb{N}$ with
$\delta(A)=r$.

######  Definition 1.3.

An _ideal on $\mathbb{N}$_ (in short _ideal_) is a family
$\mathcal{I}\subseteq\mathcal{P}(\mathbb{N})$ that satisfies the following
properties:

  1. (1)

if $A,B\in\mathcal{I}$ then $A\cup B\in\mathcal{I}$,

  2. (2)

if $A\subseteq B$ and $B\in\mathcal{I}$ then $A\in\mathcal{I}$,

  3. (3)

$\mathcal{I}$ contains all finite subsets of $\mathbb{N}$,

  4. (4)

$\mathbb{N}\notin\mathcal{I}$.

An ideal $\mathcal{I}$ is _translation invariant_ if $A+k\in\mathcal{I}$ for
every $A\in\mathcal{I}$ and $k\in\mathbb{Z}$. For an ideal $\mathcal{I}$, we
write $\mathcal{I}^{+}=\\{A\subseteq\mathbb{N}:A\notin\mathcal{I}\\}$ and call
it the _coideal of $\mathcal{I}$_, and we write
$\mathcal{I}^{*}=\\{A\subseteq\mathbb{N}:\mathbb{N}\setminus
A\in\mathcal{I}\\}$ and call it the _dual filter of $\mathcal{I}$_.

######  Remark.

We also consider ideals on any infinite countable set by identifying this set
with $\mathbb{N}$ via a fixed bijection.

######  Example 1.4.

  1. (1)

The family $\mathrm{Fin}=\\{A\subseteq\mathbb{N}:A\text{ is finite}\\}$ is a
translation invariant ideal.

  2. (2)

Let $f:\mathbb{N}\to[0,\infty)$ be a non-increasing function such that
$\sum_{n=1}^{\infty}f(n)=\infty$. The family
$\mathcal{I}_{f}=\\{A\subseteq\mathbb{N}:\sum_{n\in A}f(n)<\infty\\}$ is a
translation invariant ideal. We call it the _summable ideal_ determined by
$f$.

  3. (3)

The family
$\mathcal{I}_{d}=\\{A\subseteq\mathbb{N}:\limsup_{n\to\infty}|A\cap\\{1,\dots,n\\}|/n=0\\}$
of all sets of asymptotic density zero is a translation invariant ideal. We
call it the _asymptotic density zero ideal_.

######  Example 1.5.

  1. (1)

Let $\delta$ be an abstract upper density. The family
$\mathcal{Z}_{\delta}=\\{A\subseteq\mathbb{N}:\delta(A)=0\\}$ is an ideal. If
$\delta$ is translation invariant, so is the ideal $\mathcal{Z}_{\delta}$.

  2. (2)

Let $\mathcal{I}$ be an ideal. The function
$\delta:\mathcal{P}(\mathbb{N})\to[0,1]$ given by

$\delta(A)=\begin{cases}0&\text{if $A\in\mathcal{I}$},\\\
1&\text{otherwise}\end{cases}$

is an abstract upper density and $\mathcal{I}=\mathcal{Z}_{\delta}$. If
$\mathcal{I}$ is translation invariant, so is $\delta$.

An ideal $\mathcal{I}$ is _maximal_ if $\mathcal{I}\subseteq\mathcal{J}$
implies $\mathcal{I}=\mathcal{J}$ for every ideal $\mathcal{J}$ (i.e.
$\mathcal{I}$ is a maximal element with respect to inclusion among all
ideals). It is known that an ideal $\mathcal{I}$ is maximal if and only if
$A\in\mathcal{I}$ or $\mathbb{N}\setminus A\in\mathcal{I}$ for every
$A\subseteq\mathbb{N}$ (see e.g. [6, Lemma 7.4, p. 74]).

A translation invariant ideal $\mathcal{I}$ is _T-maximal_ if
$\mathcal{I}\subseteq\mathcal{J}$ implies $\mathcal{I}=\mathcal{J}$ for every
translation invariant ideal $\mathcal{J}$ (i.e. $\mathcal{I}$ is a maximal
element with respect to inclusion among all translation invariant ideals).

######  Remark.

By identifying sets of natural numbers with their characteristic functions, we
equip $\mathcal{P}(\mathbb{N})$ with the topology of the Cantor space
$\\{0,1\\}^{\mathbb{N}}$ and therefore we can assign topological complexity to
ideals on $\mathbb{N}$. In particular, an ideal $\mathcal{I}$ has the Baire
property if $\mathcal{I}$ has the Baire property as a subset of the Cantor
space. (Recall that a subset $B$ of a topological space $X$ has the _Baire
property_ if there is an open set $U\subseteq X$ such that the set
$(B\setminus U)\cup(U\setminus B)$ is of first category.)

######  Theorem 1.6 (Sierpiński, see e.g. [1, Theorem 4.1.1]).

Let $\mathcal{I}$ be an ideal on $\mathbb{N}$. Then $\mathcal{I}$ is of first
category or $\mathcal{I}$ does not have the Baire property.

######  Theorem 1.7 (Talagrand [11, Théorème 21] (see also [1, Theorem
4.1.2])).

An ideal $\mathcal{I}$ on $\mathbb{N}$ has the Baire property if and only if
there is an increasing sequence $k_{1}<k_{2}<\dots$ such that if there are
infinitely many $n$ with $[k_{n},k_{n+1})\cap\mathbb{N}\subseteq A$, then
$A\in\mathcal{I}^{+}$.

######  Example 1.8.

  1. (1)

It is known that the ideals $\mathrm{Fin}$, $\mathcal{I}_{f}$ and
$\mathcal{I}_{d}$ have the Baire property, in fact they are Borel ideals (see
e.g. [4]).

  2. (2)

Maximal ideals do not have the Baire property (see e.g. [1, Theorem 4.1.1]).

######  Theorem 1.9 (Plewik [10, Theorem 1]).

The intersection of a countable family of ideals without the Baire property is
an ideal without the Baire property. In particular, the intersection of
countable family of maximal ideals is an ideal without the Baire property.

######  Definition 1.10.

Let $\mathcal{I}$ be an ideal on $\mathbb{N}$. We say that a family
$\mathcal{A}\subseteq\mathcal{P}(\mathbb{N})$ is

  1. (1)

_$\mathcal{I}$ almost disjoint family_ (in short _$\mathcal{I}$ -AD family_)
if

    1. (a)

$\mathcal{A}\subseteq\mathcal{I}^{+}$,

    2. (b)

$A\cap B\in\mathcal{I}$ for any distinct $A,B\in\mathcal{A}$.

  2. (2)

_$\mathcal{I}$ translation almost disjoint family_ (in short _$\mathcal{I}$
-TAD family_) if

    1. (a)

$\mathcal{A}\subseteq\mathcal{I}^{+}$,

    2. (b)

$A\cap(B+k)\in\mathcal{I}$ for any distinct $A,B\in\mathcal{A}$ and
$k\in\mathbb{Z}$.

##  2\. Rich densities

######  Theorem 2.1.

Let $\mathcal{I}$ be an ideal on $\mathbb{N}$. If there exists an
$\mathcal{I}$-AD family of cardinality $\mathfrak{c}$, then there is a rich
abstract upper density $\delta$ such that $\mathcal{Z}_{\delta}=\mathcal{I}$.

###### Proof.

Using Zorn’s lemma, it is not difficult to extend $\mathcal{I}$-AD family to a
maximal (with respect to the inclusion) $\mathcal{I}$-AD family. Let
$\mathcal{A}$ be a maximal $\mathcal{I}$-AD family of cardinality
$\mathfrak{c}$. Let $\mathcal{A}=\\{A_{\alpha}:\alpha<\mathfrak{c}\\}$ and
$(0,1)=\\{r_{\alpha}:\alpha<\mathfrak{c}\\}$. We define
$\delta:\mathcal{P}(\mathbb{N})\to[0,1]$ by

$\delta(A)=\sup\\{r_{\alpha}:A_{\alpha}\cap A\in\mathcal{I}^{+}\\}$

with the convention that $\sup\emptyset=0$.

First, we show that $\delta$ is an abstract upper density.

For every $\alpha<\mathfrak{c}$ we have
$A_{\alpha}\cap\mathbb{N}=A_{\alpha}\in\mathcal{I}^{+}$, so
$\delta(\mathbb{N})=\sup(0,1)=1$.

For every $\alpha<\mathfrak{c}$ and a finite set $F\subseteq\mathbb{N}$, we
have $A_{\alpha}\cap F\subseteq F\in\mathcal{I}$, so
$\delta(F)=\sup\emptyset=0$.

If $A\subseteq B$, then $A_{\alpha}\cap A\in\mathcal{I}^{+}$ implies
$A_{\alpha}\cap B\in\mathcal{I}^{+}$. Thus $\delta(A)\leq\delta(B)$.

If $A_{\alpha}\cap(A\cup B)\in\mathcal{I}^{+}$, then $A_{\alpha}\cap
A\in\mathcal{I}^{+}$ or $A_{\alpha}\cap B\in\mathcal{I}^{+}$. Thus
$\delta(A\cup B)\leq\delta(A)+\delta(B)$.

Now we show that $\delta$ is rich.

It is enough to show that $\delta(A_{\alpha})=r_{\alpha}$ for every
$\alpha<\mathfrak{c}$. Since $\mathcal{A}$ is an $\mathcal{I}$-AD family,
$A_{\beta}\cap A_{\alpha}\in\mathcal{I}$ for every $\alpha\neq\beta$. On the
other hand, $A_{\alpha}\cap A_{\alpha}=A_{\alpha}\in\mathcal{I}^{+}$. Thus
$\delta(A_{\alpha})=\sup\\{r_{\alpha}\\}=r_{\alpha}$.

Finally, we show that $\mathcal{Z}_{\delta}=\mathcal{I}$.

If $A\in\mathcal{I}$, then $A_{\alpha}\cap A\subseteq A\in\mathcal{I}$ for
every $\alpha<\mathfrak{c}$. Thus $\delta(A)=\sup\emptyset=0$.

If $A\notin\mathcal{I}$, then, by maximality of $\mathcal{A}$, there is
$\alpha<\mathfrak{c}$ with $A_{\alpha}\cap A\in\mathcal{I}^{+}$. Thus
$\delta(A)\geq r_{\alpha}>0$. ∎

######  Theorem 2.2.

Let $\mathcal{I}$ be an ideal on $\mathbb{N}$ with the Baire property. There
is a rich abstract upper density $\delta$ such that
$\mathcal{Z}_{\delta}=\mathcal{I}$.

###### Proof.

Since it is known (see [5] or [7]) that there is an $\mathcal{I}$-AD family of
cardinality $\mathfrak{c}$ for every ideal with the Baire property, Theorem
2.1 finishes the proof. ∎

######  Definition 2.3 (Oliver [9]).

Let $\mathcal{I}$ be an ideal on $\mathbb{N}$. We say that a partition
$\\{P_{n}:n\in\mathbb{N}\\}$ of $\mathbb{N}$ is an _$\omega$ -partition of
$\mathbb{N}$ with respect to $\mathcal{I}$_ if

  1. (1)

$P_{n}\not\in\mathcal{I}$ for all $n\in\mathbb{N}$,

  2. (2)

for every $A\subseteq\mathbb{N}$, if $A\cap P_{n}\in\mathcal{I}$ for every
$n\in\mathbb{N}$, then $A\in\mathcal{I}$.

######  Theorem 2.4.

If there exists an $\omega$-partition $\\{P_{n}:n\in\mathbb{N}\\}$ of
$\mathbb{N}$ with respect to $\mathcal{I}$, then there is a rich abstract
upper density $\delta$ with $\mathcal{Z}_{\delta}=\mathcal{I}$.

###### Proof.

It is easy to see that

$\mathcal{I}=\\{A\subseteq\mathbb{N}:A\cap P_{n}\in\mathcal{I}\text{ for every
$n\in\mathbb{N}$}\\}$

and

$\delta(A)=\sum_{A\cap P_{n}\notin\mathcal{I}}\frac{1}{2^{n}}$

is an abstract upper density with $\mathcal{Z}_{\delta}=\mathcal{I}$.

To see that $\delta$ is rich it is enough to notice that for every $r\in(0,1)$
there is $B\subseteq\mathbb{N}$ with $\sum_{n\in B}1/2^{n}=r$ and then
$\delta(\bigcup\\{P_{n}:n\in B\\})=r$. ∎

######  Theorem 2.5.

There exists an ideal $\mathcal{I}$ for which there is a rich abstract upper
density $\delta$ with $\mathcal{Z}_{\delta}=\mathcal{I}$, but there is no
$\mathcal{I}$-AD family of cardinality $\mathfrak{c}$.

###### Proof.

Let $\\{P_{n}:n\in\mathbb{N}\\}$ be a partition of $\mathbb{N}$ such that
$P_{n}$ is infinite for all $n$. Let $\mathcal{I}_{n}$ be a maximal ideal on
$P_{n}$ for every $n\in\mathbb{N}$. It is easy to see that

$\mathcal{I}=\\{A\subseteq\mathbb{N}:A\cap P_{n}\in\mathcal{I}_{n}\text{ for
every $n\in\mathbb{N}$}\\}$

is an ideal on $\mathbb{N}$. Since $\\{P_{n}:n\in\mathbb{N}\\}$ is an
$\omega$-partition of $\mathbb{N}$ with respect to $\mathcal{I}$, there is a
rich abstract upper density $\delta$ with $\mathcal{Z}_{\delta}=\mathcal{I}$
(by Theorem 2.4).

Lastly, we show that there is no $\mathcal{I}$-AD family of cardinality
$\mathfrak{c}$ (in fact we show that there is no uncountable $\mathcal{I}$-AD
family). Let $\mathcal{A}\subseteq\mathcal{I}^{+}$ be uncountable. For every
$A\in\mathcal{A}$ there is at least one $n_{A}\in\mathbb{N}$ with $A\cap
P_{n_{A}}\in\mathcal{I}_{n_{A}}^{+}$. Since $\mathcal{A}$ is uncountable,
there are distinct $A,B\in\mathcal{A}$ with $n_{A}=n_{B}$. Let $n=n_{A}$.
Since $\mathcal{I}_{n}$ is a maximal ideal on $P_{n}$,
$\mathcal{I}_{n}^{+}=\mathcal{I}_{n}^{*}$. Thus, $(A\cap P_{n})\cap(B\cap
P_{n})\in\mathcal{I}_{n}^{*}$, so $(A\cap B)\cap
P_{n}\in\mathcal{I}_{n}^{*}=\mathcal{I}_{n}^{+}$. Hence $A\cap
B\notin\mathcal{I}$, so $\mathcal{A}$ is not an $\mathcal{I}$-AD family. ∎

######  Theorem 2.6.

There exists an ideal $\mathcal{I}$ without the Baire property for which there
is $\mathcal{I}$-AD family of cardinality $\mathfrak{c}$.

###### Proof.

Let $\\{P_{n}:n\in\mathbb{N}\\}$ be a partition of $\mathbb{N}$ such that
$P_{n}$ is infinite for all $n$. Let $\mathcal{I}_{n}$ be a maximal ideal on
$P_{n}$ for every $n\in\mathbb{N}$. It is easy to see that

$\mathcal{I}=\\{A\subseteq\mathbb{N}:A\cap P_{n}\in\mathcal{I}_{n}\text{ for
all but finitely many $n\in\mathbb{N}$}\\}$

is an ideal on $\mathbb{N}$.

Let $\mathcal{A}$ be $\mathrm{Fin}$-AD family of cardinality $\mathfrak{c}$
(there is one, see e.g. [6, Lemma 9.21]). For $A\in\mathcal{A}$, we define
$C_{A}=\bigcup\\{P_{n}:n\in A\\}$. Then $C_{A}\notin\mathcal{I}$ for every
$A\in\mathcal{A}$, and for distinct $A,B\in\mathcal{A}$ we have $C_{A}\cap
C_{B}=\bigcup\\{P_{n}:n\in A\cap B\\}\in\mathcal{I}$. Thus,
$\\{C_{A}:A\in\mathcal{A}\\}$ is $\mathcal{I}$-AD family of cardinality
$\mathfrak{c}$.

Lastly, we show that $\mathcal{I}$ does not have the Baire property. For every
$n\in\mathbb{N}$ we define $\mathcal{J}_{n}=\\{A\subseteq\mathbb{N}:A\cap
P_{n}\in\mathcal{I}_{n}\\}$. It is easy to see that $\mathcal{J}_{n}$ is a
maximal ideal on $\mathbb{N}$ for every $n\in\mathbb{N}$ and

$\mathcal{I}=\bigcup_{n\in\mathbb{N}}\bigcap_{k\geq n}\mathcal{J}_{k}.$

By Theorem 1.9, the ideal $\bigcap_{k\geq n}\mathcal{J}_{k}$ does not have the
Baire property for every $k\in\mathbb{N}$. In particular, $\bigcap_{k\geq
1}\mathcal{J}_{k}$ does not have the Baire property, so it is of second
category. Thus, $\mathcal{I}$ is of second category as a superset of
$\bigcap_{k\geq 1}\mathcal{J}_{k}$. Then, by Theorem 1.6, $\mathcal{I}$ does
not have the Baire property. ∎

######  Theorem 2.7.

Let $\mathcal{I}$ be a maximal ideal. If $\delta$ is an abstract upper density
$\delta$ with $\mathcal{Z}_{\delta}=\mathcal{I}$ then $\delta$ takes only two
values: zero and one. In particular, there is no rich abstract upper density
$\delta$ with $\mathcal{Z}_{\delta}=\mathcal{I}$.

###### Proof.

Let $A\subseteq\mathbb{N}$. If $A\in\mathcal{I}$ then $\delta(A)=0$. If
$A\notin\mathcal{I}$, then $\mathbb{N}\setminus A\in\mathcal{I}$, hence
$1=\delta(\mathbb{N})\leq\delta(A)+\delta(\mathbb{N}\setminus
A)=\delta(A)+0=\delta(A)\leq 1$, thus $\delta(A)=1$. ∎

######  Theorem 2.8.

There exists a non-maximal ideal $\mathcal{I}$ for which there is no rich
abstract upper density $\delta$ with $\mathcal{Z}_{\delta}=\mathcal{I}$.

###### Proof.

Let $N\geq 2$. Let $\\{P_{n}:n\leq N\\}$ be a partition of $\mathbb{N}$ such
that $P_{n}$ is infinite for all $n\leq N$. Let $\mathcal{I}_{n}$ be a maximal
ideal on $P_{n}$ for every $n\in\mathbb{N}$. It is easy to see that

$\mathcal{I}=\\{A\subseteq\mathbb{N}:A\cap P_{n}\in\mathcal{I}_{n}\text{ for
every $n\leq N$}\\}$

is an ideal on $\mathbb{N}$. Since both $P_{1}\notin\mathcal{I}$ and
$\mathbb{N}\setminus P_{1}\notin\mathcal{I}$, $\mathcal{I}$ is not a maximal
ideal.

Let $\delta$ be any abstract upper density such that
$\mathcal{Z}_{\delta}=\mathcal{I}$.

Let $n\leq N$ and $A\subseteq P_{n}$. If $A\in\mathcal{I}_{n}$ then
$\delta(A)=0$. If $A\notin\mathcal{I}_{n}$, then $P_{n}\setminus
A\in\mathcal{I}_{n}$, hence $\delta(P_{n})\leq\delta(A)+\delta(P_{n}\setminus
A)=\delta(A)+0=\delta(A)\leq\delta(P_{n})$, hence $\delta(A)=\delta(P_{n})$.

Let $A\notin\mathcal{I}$. There is at least one $n\leq N$ with $A\cap
P_{n}\notin\mathcal{I}_{n}$, so $\delta(A)\geq\delta(A\cap
P_{n})=\delta(P_{n})$.

Thus, $\delta(A)\geq\min\\{\delta(P_{n}):n\leq N\\}>0$ for every
$A\notin\mathcal{I}$, hence $\delta$ is not rich. ∎

The following diagram summarizes all known relationships between ideals
$\mathcal{I}$ and existence of rich abstract upper density with
$\mathcal{I}=\mathcal{Z}_{\delta}$.

There is a rich densityThere is no rich densityAD-ideals(Thm. 2.1)Ideals with
Baire property(Thm. 2.2)(Thm. 2.6)(Thm. 2.5)Maximalideals(Thm. 2.7)(Thm. 2.8)

##  3\. Rich and translation invariant densities

######  Theorem 3.1.

Let $\mathcal{I}$ be a translation invariant ideal on $\mathbb{N}$. If there
exists an $\mathcal{I}$-TAD family of cardinality $\mathfrak{c}$, then there
is a translation invariant and rich abstract upper density $\delta$ such that
$\mathcal{Z}_{\delta}=\mathcal{I}$.

###### Proof.

Using Zorn’s lemma, it is not difficult to extend $\mathcal{I}$-TAD family to
a maximal (with respect to the inclusion) $\mathcal{I}$-TAD family. Let
$\mathcal{A}$ be a maximal $\mathcal{I}$-TAD family of cardinality
$\mathfrak{c}$. Let $\mathcal{A}=\\{A_{\alpha}:\alpha<\mathfrak{c}\\}$ and
$(0,1)=\\{r_{\alpha}:\alpha<\mathfrak{c}\\}$. We define
$\delta:\mathcal{P}(\mathbb{N})\to[0,1]$ by

$\delta(A)=\sup\\{r_{\alpha}:A_{\alpha}\cap(A+k)\in\mathcal{I}^{+}\text{ for
some $k\in\mathbb{Z}$}\\}$

with the convention that $\sup\emptyset=0$.

First, we show that $\delta$ is an abstract upper density.

For every $\alpha<\mathfrak{c}$ we have
$A_{\alpha}\cap(\mathbb{N}+0)=A_{\alpha}\in\mathcal{I}^{+}$, so
$\delta(\mathbb{N})=\sup(0,1)=1$.

For every $\alpha<\mathfrak{c}$, $k\in\mathbb{Z}$ and a finite set
$F\subseteq\mathbb{N}$, we have $A_{\alpha}\cap(F+k)\subseteq
F+k\in\mathcal{I}$, so $\delta(F)=\sup\emptyset=0$.

If $A\subseteq B$, then $A_{\alpha}\cap(A+k)\in\mathcal{I}^{+}$ implies
$A_{\alpha}\cap(B+k)\in\mathcal{I}^{+}$. Thus $\delta(A)\leq\delta(B)$.

If $A_{\alpha}\cap((A\cup B)+k)\in\mathcal{I}^{+}$, then
$A_{\alpha}\cap(A+k)\in\mathcal{I}^{+}$ or
$A_{\alpha}\cap(B+k)\in\mathcal{I}^{+}$. Thus $\delta(A\cup
B)\leq\delta(A)+\delta(B)$.

Second, we show that $\delta$ is translation invariant.

If $A_{\alpha}\cap((A+m)+k)\in\mathcal{I}^{+}$ then
$A_{\alpha}\cap(A+(m+k))\in\mathcal{I}^{+}$ and vice versa. Thus
$\delta(A+m)=\delta(A)$.

Now we show that $\delta$ is rich.

It is enough to show that $\delta(A_{\alpha})=r_{\alpha}$ for every
$\alpha<\mathfrak{c}$. Since $\mathcal{A}$ is an $\mathcal{I}$-TAD family,
$A_{\beta}\cap(A_{\alpha}+k)\in\mathcal{I}$ for every $k\in\mathbb{Z}$ and
$\alpha\neq\beta$. On the other hand,
$A_{\alpha}\cap(A_{\alpha}+0)=A_{\alpha}\in\mathcal{I}^{+}$. Thus
$\delta(A_{\alpha})=\sup\\{r_{\alpha}\\}=r_{\alpha}$.

Finally, we show that $\mathcal{Z}_{\delta}=\mathcal{I}$.

If $A\in\mathcal{I}$, then $A_{\alpha}\cap(A+k)\subseteq A+k\in\mathcal{I}$
for every $\alpha<\mathfrak{c}$ and $k\in\mathbb{Z}$. Thus
$\delta(A)=\sup\emptyset=0$.

If $A\notin\mathcal{I}$, then, by maximality of $\mathcal{A}$, there is
$\alpha<\mathfrak{c}$ and $k\in\mathbb{Z}$ with
$A_{\alpha}\cap(A+k)\in\mathcal{I}^{+}$. Thus $\delta(A)\geq r_{\alpha}>0$. ∎

######  Theorem 3.2.

Let $\mathcal{I}$ be an ideal on $\mathbb{N}$ with the Baire property. There
is an $\mathcal{I}$-TAD family of cardinality $\mathfrak{c}$.

###### Proof.

By Theorem 1.7 there is an increasing sequence $k_{1}<k_{2}<\dots$ such that
if there are infinitely many $n$ with $[k_{n},k_{n+1})\cap\mathbb{N}\subseteq
A$, then $A\in\mathcal{I}^{+}$. Without loss of generality we can assume that
$\lim_{n\to\infty}(k_{n+1}-k_{n})=\infty$.

Let us denote $I_{n}=[k_{n},k_{n+1})\cap\mathbb{N}$ and
$|I_{n}|=k_{n+1}-k_{n}$ for every $n\in\mathbb{N}$.

Let $\mathcal{A}\subseteq\mathcal{P}(\mathbb{N})$ be a $\mathrm{Fin}$-AD
family of cardinality $\mathfrak{c}$. For every $A\in\mathcal{A}$ we define

$C_{A}=\bigcup_{n\in A}I_{2n}.$

We claim that $\mathcal{C}=\\{C_{A}:A\in\mathcal{A}\\}$ is the required
family.

First, we show that $C_{A}\in\mathcal{I}^{+}$ for every $A\in\mathcal{A}$.

If $A\in\mathcal{A}$, then $A$ is infinite, so $C_{A}$ contains infinitely
many intervals $[k_{n},k_{n+1})\cap\mathbb{N}$, thus
$C_{A}\in\mathcal{I}^{+}$.

Second, we show that $\mathcal{C}$ is of cardinality $\mathfrak{c}$.

If $A,B\in\mathcal{A}$ are distinct, then there is $n\in(A\setminus
B)\cup(B\setminus A)$, so $I_{2n}\subseteq(C_{A}\setminus
C_{B})\cup(C_{B}\setminus C_{A})$. Thus
$|\mathcal{C}|=|\mathcal{A}|=\mathfrak{c}$.

Finally, we show that $\mathcal{C}$ is $\mathcal{I}$-TAD.

Let $A,B\in\mathcal{A}$ be distinct and $k\in\mathbb{Z}$. We have to show that
$C_{A}\cap(C_{B}+k)\in\mathcal{I}$.

Let $N$ be such that $|I_{n}|>|k|$ for every $n>N$. Then $I_{2n}+k\subseteq
I_{2n-1}\cup I_{2n}\cup I_{2n+1}$ for every $n>N$. Hence $(I_{2n}+k)\cap
I_{2m}=\emptyset$ for any $n>N$, $m\in\mathbb{N}$, $n\neq m$.

Since

$C_{A}\cap(C_{B}+k)=\bigcup_{m\in A}I_{2m}\cap\left(\bigcup_{n\in
B}I_{2n}+k\right)=\bigcup_{m\in A}\bigcup_{n\in B}(I_{2m}\cap(I_{2n}+k))$

and $A\cap B$ is finite, we obtain that $C_{A}\cap(C_{B}+k)$ is contained in
finitely many intervals $I_{2m}$. Thus
$C_{A}\cap(C_{B}+k)\in\mathrm{Fin}\subseteq\mathcal{I}$. ∎

######  Theorem 3.3.

Let $\mathcal{I}$ be a translation invariant ideal on $\mathbb{N}$ with the
Baire property. There is a translation invariant and rich abstract upper
density $\delta$ such that $\mathcal{Z}_{\delta}=\mathcal{I}$.

###### Proof.

Follows from Theorems 3.1 and 3.2. ∎

######  Corollary 3.4 (Di Nasso and Jin [3, Theorems 2.1 and 2.2]).

Let $\mathcal{I}$ be a summable ideal. There is a translation invariant and
rich abstract upper density $\delta$ such that
$\mathcal{Z}_{\delta}=\mathcal{I}$.

###### Proof.

It is not difficult to check that summable ideals are translation invariant,
and it is known that summable ideals have the Baire property (see e.g. [4,
Example 1.2.3(c)]). Thus Theorem 3.3 finishes the proof. ∎

######  Question 1.

Does there exist a translation invariant ideal $\mathcal{I}$ without the Baire
property such that there is an $\mathcal{I}$-TAD family of cardinality
$\mathfrak{c}$?

######  Definition 3.5.

Let $\mathcal{I}$ be an ideal on $\mathbb{N}$. We say that a sequence of reals
$\langle x_{n}\rangle$ is _$\mathcal{I}$ -convergent to $L$_ if
$\\{n\in\mathbb{N}:|x_{n}-L|\geq\varepsilon\\}\in\mathcal{I}$ for every
$\varepsilon>0$. The number $L$ is called _an $\mathcal{I}$-limit of $\langle
x_{n}\rangle$_ and we denote it by $\lim^{\mathcal{I}}x_{n}$ .

######  Theorem 3.6.

There exists a translation invariant ideal $\mathcal{I}$ for which there is a
rich translation invariant abstract upper density $\delta$ with
$\mathcal{Z}_{\delta}=\mathcal{I}$, but there is no $\mathcal{I}$-TAD family
of cardinality $\mathfrak{c}$.

###### Proof.

Let $\mathcal{J}$ be a maximal ideal on $\mathbb{N}$. It is known (see e.g.
[2, p. 3314]) that

$\delta(A)={\lim}^{\mathcal{J}}\frac{|A\cap\\{1,\dots,n\\}|}{n}$

is a translation invariant finitely additive measure (in particular, $\delta$
is a translation invariant abstract upper density).

Moreover, $\delta$ is rich. Indeed, it is enough to observe (see e.g. [8, p.
216]) that for every $r\in[0,1]$ there is a set $A\subseteq\mathbb{N}$ such
that the asymptotic density of $A$ equals $r$ i.e.
$r=\lim_{n\to\infty}\frac{|A\cap\\{1,\dots,n\\}|}{n}$.

Let $\mathcal{I}=\mathcal{Z}_{\delta}$. Since $\delta$ is a translation
invariant abstract upper density, $\mathcal{I}$ is a translation invariant
ideal. To finish the proof we have to show that there is no $\mathcal{I}$-AD
family of cardinality $\mathfrak{c}$ (in fact we show that there is no
uncountable $\mathcal{I}$-AD family). Suppose
$\mathcal{A}\subseteq\mathcal{I}^{+}$ is an uncountable $\mathcal{I}$-AD
family. For every $A\in\mathcal{A}$, let $n_{A}\in\mathbb{N}$ be such that
$\delta(A)\geq 1/n_{A}$. Since $\mathcal{A}$ is uncountable, there are
infinitely many sets $A_{1},A_{2},\dots\in\mathcal{A}$ with
$n_{A_{i}}=n_{A_{j}}$ for every $i,j\in\mathbb{N}$. Let $n=n_{A_{1}}$. Since
$A_{j}\cap A_{j}\in\mathcal{I}$ for distinct $i,j\in\mathbb{N}$,
$\delta(A_{i}\cap A_{j})=0$. Since $\delta$ is a finitely additive measure,
$\delta(A_{i}\cup A_{j})=\delta(A_{i})+\delta(A_{j})-\delta(A_{i}\cap
A_{j})=\delta(A_{i})+\delta(A_{j})$ for distinct $i,j\in\mathbb{N}$. Finally,
$1\geq\delta(A_{1}\cup\dots\cup
A_{n+1})=\delta(A_{1})+\dots+\delta(A_{n+1})\geq(n+1)/n>1$, a contradiction. ∎

We do not know if there is any translation invariant ideal such that there is
no rich translation invariant abstract upper density $\delta$ with
$\mathcal{Z}_{\delta}=\mathcal{I}$.

######  Question 2.

Does there exist a translation invariant ideal $\mathcal{I}$ such that there
is no rich translation invariant abstract upper density $\delta$ with
$\mathcal{Z}_{\delta}=\mathcal{I}$?

Keeping Theorem 2.7 in mind, it is natural to examine T-maximal ideals in the
quest for a solution of the above question.

######  Question 3.

Does there exist a T-maximal ideal $\mathcal{I}$ such there is a rich
translation invariant abstract upper density $\delta$ with
$\mathcal{Z}_{\delta}=\mathcal{I}$?

In Theorems 3.7 and 3.11, we show that if the above question had positive
answer, then the required density would be different from densities
constructed in the proofs of Theorems 3.1 and 3.6.

######  Theorem 3.7.

If $\mathcal{I}$ is a T-maximal ideal, then for any $A,B\notin\mathcal{I}$
there is $k$ with $(A+k)\cap B\notin\mathcal{I}$. In particular, there is no
$\mathcal{I}$-TAD family of cardinality $\mathfrak{c}$.

###### Proof.

Let $A,B\notin\mathcal{I}$. Since $\mathcal{I}$ is maximal among all
translation invariant ideals, there are $k_{1},\dots,k_{n}\in\mathbb{Z}$ with
$(A+k_{1})\cup\dots\cup(A+k_{n})\in\mathcal{I}^{*}$. Thus,
$B\cap[(A+k_{1})\cup\dots\cup(A+k_{n})]\notin\mathcal{I}$. Hence, there is
$i\leq n$ with $B\cap(A+k_{i})\notin\mathcal{I}$. ∎

######  Proposition 3.8.

Let $\mathcal{I}$ be an ideal on $\mathbb{N}$.

  1. (1)

The family

$t(\mathcal{I})=\\{A\subseteq\mathbb{N}:A+k\in\mathcal{I}\text{ for every
$k\in\mathbb{Z}$}\\}$

is a translation invariant ideal.

  2. (2)

If $\mathcal{I}$ does not have the Baire property then $t(\mathcal{I})$ does
not have the Baire property.

###### Proof.

(1) Straightforward. (2) It follows from Theorem 1.9, because
$\mathcal{I}+k=\\{A\subseteq\mathbb{N}:A-k\in\mathcal{I}\\}$ is the ideal
without the Baire property for every $k\in\mathbb{Z}$ and
$t(\mathcal{I})=\bigcap\\{\mathcal{I}+k:k\in\mathbb{Z}\\}$. ∎

######  Theorem 3.9.

Let $\delta$ be defined as in the proof of Theorem 3.6. Then
$\mathcal{Z}_{\delta}\neq t(\mathcal{I})$ for any maximal ideal $\mathcal{I}$.

###### Proof.

Suppose that $\mathcal{Z}_{\delta}=t(\mathcal{I})$ for some maximal ideal
$\mathcal{I}$. Since $t(\mathcal{I})\subseteq\mathcal{I}$ and one could easily
notice that $\mathcal{I}_{d}\subseteq\mathcal{Z}_{\delta}$, we get
$\mathcal{I}_{d}\subseteq\mathcal{I}$. Let $A=\bigcup_{n\in\mathbb{N}}I_{2n}$,
where $I_{n}=(2^{n},2^{n+1}]\cap\mathbb{N}$. Then
$\limsup_{n\to\infty}\frac{|A\cap\\{1,\dots,n\\}|}{n}=\frac{2}{3}$ and
$\liminf_{n\to\infty}\frac{|A\cap\\{1,\dots,n\\}|}{n}=\frac{1}{3}$ as
$\frac{|I_{2n}|}{|I_{2n-1}\cup I_{2n}|}=\frac{2}{3}$ and
$\frac{|I_{2n}|}{|I_{2n}\cup I_{2n+1}|}=\frac{1}{3}$. Therefore,
$\delta(A)\in\left[\frac{1}{3},\frac{2}{3}\right]$, thus
$A\not\in\mathcal{Z}_{\delta}$.

Now, since $\mathcal{I}$ is a maximal ideal, either $A\in\mathcal{I}$ or
$\mathbb{N}\setminus A\in\mathcal{I}$. Assume that $A\in\mathcal{I}$ (the
other case is identical since $\mathbb{N}\setminus
A\not\in\mathcal{Z}_{\delta}$ too). By the supposition that
$\mathcal{Z}_{\delta}=t(\mathcal{I})$, we know that $A\not\in t(\mathcal{I})$,
so there exists $k\in\mathbb{Z}$ such that $A+k\not\in\mathcal{I}$. It follows
that $B=A+k\setminus A\not\in\mathcal{I}$. However, for all but finitely many
$n\in\mathbb{N}$ we have

$\frac{|B\cap I_{n}|}{|I_{n}|}\leq\frac{k}{2^{n}},$

which tends to $0$ as $n$ tends to infinity. Thus,
$B\in\mathcal{I}_{d}\subseteq\mathcal{I}$, a contradiction with
$B\not\in\mathcal{I}$. ∎

######  Theorem 3.10.

  1. (1)

If $\mathcal{J}$ is a T-maximal ideal then $\mathcal{J}=t(\mathcal{I})$ for
any maximal ideal $\mathcal{I}\supseteq\mathcal{J}$.

  2. (2)

There is a maximal ideal $\mathcal{I}$ such that $t(\mathcal{I})$ is not a
T-maximal ideal.

  3. (3)

Any T-maximal ideal does not have the Baire property.

###### Proof.

(1) ($\subseteq$) If $A\in\mathcal{J}$, then
$A+k\in\mathcal{J}\subseteq\mathcal{I}$ for every $k\in\mathbb{Z}$. Thus,
$A\in t(\mathcal{I})$.

($\supseteq$) Since $\mathcal{J}\subseteq t(\mathcal{I})$, $t(\mathcal{I})$ is
translation invariant and $\mathcal{J}$ is maximal among all translation
invariant ideals, $\mathcal{J}=t(\mathcal{I})$.

(2) Let $B=\\{b_{n}:n\in\mathbb{N}\\}$ be such that the sequence $\langle
b_{n+1}-b_{n}:n\in\mathbb{N}\rangle$ is increasing and diverging to infinity.
Let $\mathcal{I}$ be a maximal ideal with $\mathbb{N}\setminus
B\in\mathcal{I}$. If we show that $t(\mathcal{I})\cup\\{B\\}$ can be extended
to a translation invariant ideal, $t(\mathcal{I})$ will not be a T-maximal
ideal.

To prove that $t(\mathcal{I})\cup\\{B\\}$ can be extended to a translation
invariant ideal, we have to show that
$\mathbb{N}\setminus[(B+k_{1})\cup\dots\cup(B+k_{n})]\notin t(\mathcal{I})$
for any $k_{1},\dots,k_{n}\in\mathbb{Z}$.

Let $k_{1},\dots,k_{n}\in\mathbb{Z}$. Let
$k\in\mathbb{Z}\setminus\\{-k_{1},\dots,-k_{n}\\}$. Since $|(B+l)\cap B|\leq
1$ for every $l\neq 0$, $\left([(B+k_{1})\cup\dots\cup(B+k_{n})]+k\right)\cap
B$ is finite. Since $\mathbb{N}\setminus B\in\mathcal{I}$,
$[(B+k_{1})\cup\dots\cup(B+k_{n})]+k\in\mathcal{I}$. Then
$\mathbb{N}\setminus([(B+k_{1})\cup\dots\cup(B+k_{n})]+k)\notin\mathcal{I}$,
and consequently
$(\mathbb{N}\setminus[(B+k_{1})\cup\dots\cup(B+k_{n})])+k\notin\mathcal{I}$
which means that $\mathbb{N}\setminus[(B+k_{1})\cup\dots\cup(B+k_{n})]\notin
t(\mathcal{I})$.

(3) Follows from (1) and Proposition 3.8(2). ∎

######  Theorem 3.11.

Let $\delta$ be defined as in the proof of Theorem 3.6. Then
$\mathcal{Z}_{\delta}$ is not a T-maximal ideal.

###### Proof.

It follows from Theorems 3.9 and 3.10(1). ∎

The following diagram summarizes all (un)known relationships between
translation invariant ideals $\mathcal{I}$ and existence of rich translation
invariant abstract upper density with $\mathcal{I}=\mathcal{Z}_{\delta}$.

There is a rich translationinvariant densityThere is no rich translation
invariant densityTAD-ideals(Thm. 3.1)Ideals with Baire property(Thm. 3.3)?(Q.
1)(Thm. 3.6)?(Q. 2)

## Acknowledgment

The authors would like to thank Piotr Szuca for fruitful discussion.

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