# Variability in NGC 3201 giant stars and its impact on their spectroscopic
[Fe/H] determination

Á. Llancaqueo Albornoz Departamento de Astronomía, Casilla 160-C, Universidad
de Concepción, Concepción, Chile  S. Villanova Departamento de Astronomía,
Casilla 160-C, Universidad de Concepción, Concepción, Chile  C.C. Cortés
Departamento de Astronomía, Casilla 160-C, Universidad de Concepción,
Concepción, Chile  Departamento de Física, Facultad de Ciencias, Universidad
del Bío-Bío, Avenida Collao 1202, Casilla 15-C, Concepción, Chile  J.A.
Ahumada Observatorio Astronómico, Universidad Nacional de Córdoba, Laprida
854, 5000, Córdoba, Argentina.  C. Parisi Observatorio Astronómico,
Universidad Nacional de Córdoba, Laprida 854, 5000, Córdoba, Argentina.
Instituto de Astronomía Teórica y Experimental (CONICET-UNC), Laprida 854,
5000, Córdoba, Argentina.

(Received July 6, 2020; Revised November 12, 2020; Accepted November 29, 2020)

###### Abstract

We present the analysis of 34 light curves in $V$ and $I$ of 17 giant stars in
the globular cluster NGC 3201, to check if such stars are variable and if
their variability has some kind of impact on the iron abundance as obtained
from spectroscopic measurements. First, we computed the Generalized Lomb-
Scargle and Phase Dispersion Minimization periodograms on the sample to check
if the stars were variables. In this way, 7 stars of the sample were found to
be non-variable, 2 stars are considered as possible variables, and 8 stars
were found to be variable, with periods ranging from $0.0881\pm 0.0001$ to
$0.5418\pm 0.0027$ days. According to the literature, the variables have
distinct values of $\text{[FeI/H]}$: the 3 most metal-rich stars are in the
RGB stage, one has an $\text{[FeI/H]}=-1.37$ dex, while the other two have
$\text{[FeI/H]}=-1.31$ dex. The two most metal-poor variables have
$\text{[FeI/H]}=-1.61$ dex and $\text{[FeI/H]}=-1.62$ dex, and are AGB stars;
the remaining variables have $\text{[FeI/H]}=-1.44$, $-1.48$, and $-1.50$ dex,
the first two being RGB while the last is AGB star. On the other hand, stars
that appear to be non-variables have $-1.56\leq\text{[FeI/H]}\leq-1.40$. We
conclude that variability somehow affects the spectroscopic determination of
the iron content of giant stars in NGC 3201 increasing the iron spread of the
cluster. If variability is not taken into account, this spread could be
wrongly interpreted as due to an intrinsic iron spread affecting the stars of
the cluster.

globular clusters: individual (NGC 3201) — stars: abundances

††journal: AJ

##  1 Introduction

In several respects, Globular Clusters (GCs) have remained as one of the most
relevant astronomical objects of study during more than one hundred years.
They were at first thought to be examples of simple stellar populations, which
was later proven incorrect due to detailed chemical analyzes showing that
almost all GCs display inhomogeneities in their light element contents (e.g.,
Gratton et al., 2004; Carretta et al., 2009a, b). The most prominent feature
is the Na-O anticorrelation, with Ruprecht 106 (Villanova et al., 2013) being
the sole exception to date. But there are also GCs that display a metallicity
distribution or iron spread: $\omega$ Centauri (e.g., Origlia et al., 2003;
Pancino et al., 2011; Marino et al., 2011) and Terzan 5 (e.g., Origlia et al.,
2011, 2013), both displaying a multi-modal distribution spanning a range of
$\sim 1$ dex, and M54 (e.g., Bellazzini et al., 2008; Carretta et al., 2010;
Mucciarelli et al., 2017), although its metallicity distribution is not as
large as that in $\omega$ Cen and Terzan 5.

In recent years, several studies have been made with the objective to find
other GCs displaying intrinsic iron spread; however, they present smaller
inhomogeneities than $\omega$ Cen, Terzan 5, and M54. Examples of such
clusters are M22 (Marino et al., 2009), NGC 3201 (Simmerer et al., 2013), M2
(Yong et al., 2014), NGC 5286 (Marino et al., 2015), and M19 (Johnson et al.,
2015). Some of these works have been re-evaluated by other studies to either
confirm the intrinsic spread in the metallicities or conclude that those
findings were not correct at all.

NGC 3201 (C1015$-$461) is a nearby ($\sim 4.9$ kpc), low galactic latitude
($b=+8.64^{\circ}$) GC located at
$\alpha=10^{\mathrm{h}}~{}17^{\mathrm{m}}~{}36.^{\\!\\!{\mathrm{s}}}82$,
$\delta=-46^{\circ}~{}24^{{}^{\prime}}~{}44.^{\\!\\!{{}^{\prime\prime}}}9$
(J2000) (Harris, 1996, 2010 edition). Its $\text{[Fe/H]}$ content is
controversial, since different analyzes have yielded notably different
results. For example, the study of Gonzalez & Wallerstein (1998), that uses
CTIO high-resolution spectra, and that of Simmerer et al. (2013), which
employs high-resolution spectra from UVES-FLAMES@VLT and MIKE@Magellan, claim
that NGC 3201 has a spread in $\text{[Fe/H]}$ of $\sim 0.4$ dex. However,
Muñoz et al. (2013) do not find such spread, deriving $\text{[Fe/H]}=-1.53\pm
0.01$ dex, and also point out that 5 out of the 6 most metal-poor stars—with
$\text{[Fe/H]}\leq-1.59$ dex—from Simmerer et al. (2013) might be asymptotic
giant branch (AGB) stars instead of red giant branch (RGB) stars.

Mucciarelli et al. (2015) re-analyzed the spectra of 21 stars from the sample
of Simmerer et al. (2013) in the light of the results of Lapenna et al.
(2014), who proposed non-local thermodynamical equilibrium (NLTE) effects as
an explanation for the lower Fe abundances of AGB stars compared to those
observed in RGB stars. By adopting photometric gravities that use the Stefan-
Boltzmann equation, and measuring Fe I and Fe II lines individually, they
discovered that the spread found in Simmerer et al. (2013) was due to the
presence of AGB stars in the sample, and concluded that NGC 3201 does not have
an intrinsic iron spread. It should be noted, however, that for GCs with
metallicity similar to NGC 3201, NLTE corrections models from Bergemann et al.
(2012) and Lind et al. (2012) predict that the Fe I lines should be affected
in a similar way regardless the star is an RGB or AGB star, and thus there
should be another mechanism that could explain the behavior of the iron
content of the cluster.

In this context it is important to note that red giant stars are unstable
against radial pulsations; in fact, as stars evolve expanding and cooling into
the RG or AGB stages, they become pulsationally unstable, meaning
that—virtually—every star found in the upper right portion of the H-R diagram
is a variable. Moreover, pulsating red giants (PRGs) are complex objects,
mainly due to their variability being the result of a mix of pulsation and
convection (this not yet being a well understood process in astrophysics), and
because of the presence of extended atmospheric envelopes (Catelan & Smith,
2015). Olin Eggen published a series of papers during the 1970s (e.g., Eggen
1973, 1977) in which he developed a general classification for PRGs: large,
medium, and small-amplitude red variables (LARV, MARV, and SARV,
respectively), along with the $\sigma$ Librae variables, stars with amplitudes
less than $0.2$ mag in $V$. Pulsating red giants are characterized by having a
long period and a short amplitude.

As a first clue to a possible relation between variability and iron abundance
in red giants, recently Muñoz et al. (2018) analyzed the chemical abundance of
seven stars of NGC 6528, and found that one of the RGB stars of the sample
was: (1) more metal-poor, with $\mathrm{[Fe/H]}=-0.55$, than the other six
stars—having a mean value of $\mathrm{[Fe/H]}=-0.14\pm 0.03$), and that (2) it
was a variable star with a period of $0.26$ days and an infrared amplitude of
$0.05$ mag, using data from the VVV survey (Minniti et al., 2010; Saito et
al., 2012). These findings prompted us to ask ourselves the following
question: Can the variability of a star be related to its observed difference
in iron abundance, compared with that of the rest of the stars that are not
variable? To seek for an answer, we decided to revisit one of the GCs whose
iron content is debated, NGC 3201. As a further motivation, it is worth
pointing out that Layden & Sarajedini (2003) reported low-amplitude light
variations in several RGB stars in this cluster.

In this paper we present a study of thirty-four light curves (LCs) of
seventeen giant stars—17 in $V$ and 17 in $I$—of the globular cluster NGC
3201. First, we try to determine the possible variability of each star within
the limitations of our data. Then for the possible variables, we then searched
the literature regarding their $\text{[Fe/H]}$ content. The aim is to check if
these giant stars are variables or not and if such variability is somehow
related with their iron abundance as obtained from spectroscopic data using
standard data analysis. The layout of the paper is as follows. In Section § 2
we present the information regarding the data employed in the work. Section §
3 provides details of the methodology followed to analyze the sample. Section
§ 4 contains the main results obtained from our analysis. In Section § 5 we
present a discussion of our findings. Finally, in Section § 6 we summarize the
results and give the conclusions obtained from this work.

##  2 The Data

For this work we analyzed thirty-four light curves (17 for $V$ filter and 17
for $I$ filter) of seventeen giant stars from NGC 3201 provided by J. Ahumada,
previously published in Arellano Ferro et al. (2014). These stars were also
analyzed by Simmerer et al. (2013) and Mucciarelli et al. (2015), who
determined their $[\text{Fe/H}]$ abundance. Each light curve for the $V$
filter contains 142 epochs, and for the $I$ filter, four light curves (Stars
No 2, 5, 6 and 15) are based on 144 epochs, while the other thirteen contain
145 epochs. Figure 1 and Figure 2 show the light curves of all stars for $V$
and $I$ filter, respectively, and Table 1 displays a portion of the complete
content of each light curve.

Table 1: Sample time series for $V$ and $I$ filters of each star. Star | Filter | HJD | $M_{\mathrm{std}}$ | $\sigma_{M_{std}}$ | $\sigma_{int}$
---|---|---|---|---|---
|  | (d) | (mag) | (mag) | (mag)
No 1 | $V$ | 2456371.522859 | 13.93920 | 0.0078 | 0.00074
No 1 | $V$ | 2456371.525984 | 13.94512 | 0.0079 | 0.00063
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
No 1 | $I$ | 2456371.517222 | 12.62022 | 0.0092 | 0.00085
No 1 | $I$ | 2456371.519717 | 12.61734 | 0.0093 | 0.00080
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
No 2 | $V$ | 2456371.522859 | 12.89581 | 0.0090 | 0.00043
No 2 | $V$ | 2456371.525984 | 12.89554 | 0.0090 | 0.00035
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
No 2 | $I$ | 2456371.517222 | 11.36980 | 0.0095 | 0.00043
No 2 | $I$ | 2456371.519717 | 11.37122 | 0.0099 | 0.00041
Figure 1: Light Curves of the 17 stars analyzed in this work for $V$ filter.
The x-axis correspond to the Heliocentric Julian Days (HJD) and the y-axis
shows the standard magnitude $V$. We named each RGB as ‘Star No’ followed by a
number from 1 to 17 (e.g., Star No 1, Star No 2,…, Star No 17). The number is
written in the top right corner of each LC, all light curves are plotted with
the same scale. Figure 2: Light Curves of the 17 stars analyzed in this work
for $I$ filter. Same as in Fig 1, x-axis correspond to the Heliocentric Julian
Days (HJD) and the y-axis shows the standard magnitude $I$. The black ellipses
mark the 3 epoch that were affected by bad photometry and regarded as
outliers.

The data for all 17 giant stars were obtained on 2013 March 20–23 at the
Complejo Astronómico El Leoncito (CASLEO), San Juan, Argentina, using the
2.15-m telescope, carrying out Johnson-Kron-Cousins $V$ and $I$ observations.
The detector used was a Roper Scientific back-illuminated CCD of $2048\times
2048$ pixels with a scale of $0.15$ arcsec/pix and a Field of View of
approximately $5.1\times 5.1$ arcmin2. The data reduction and transformation
to the _VI_ standard magnitude system are described in Secs. 2.2 and 2.3 of
Arellano Ferro et al. (2014).

The finding chart of the stars that were analyzed in this work is in Figure 3.
It is worth noting that Stars No 8 and 10 are close to each other, while Stars
No 1, 6, 7, and 9 have close (although faint) neighbor stars. In spite of
this, contamination from these neighbours on the spectra used to obtain [Fe/H]
seems unlikely. This is because slit spectrographs like MIKE@Magellan can
easily avoid neighbour contamination just rotating the camera and in any case
during the reduction process the spectrum of a contaminating star can be
successfully subtracted from that of the target. Fiber spectrograph like UVES-
FLAMES@VLT have fibers that are only 1.2 arcsec in diameter and neighbours
stars visible in Figure 3 are not close enough to affect the incoming light.

Figure 3: Finding chart of the area of NGC 3201 where the 17 giant stars
analyzed in this work are indicated. These stars were studied previously by
Simmerer et al. (2013) and Mucciarelli et al. (2015). West is up, North is to
the left, and the shown area is $5.4^{\prime}\times 5.4^{\prime}$.

In order to identify our stars of interest, we matched them with the ones from
Mucciarelli et al. (2015) using both position (RA and DEC) and mean magnitudes
$\langle M_{V}\rangle$ and $\langle M_{I}\rangle$ for $V$ and $I$ filters,
respectively. We also matched our stars with those from the variable star
catalogue by Clement et al. (2001) using Tool for OPerations on Catalogues And
Tables (TOPCAT) to see if any of our stars were already catalogued as
variable, where we did not find any match at all. This is not surprising since
there are very few studies in literature about variability in globular cluster
giant branch stars. Table 2 shows the main information of the each stars and
Figure 4 shows the Color-Magnitude Diagram (CMD) of the cluster with the
position of the stars studied in this work marked with different symbols
depending if they were classified as variable, non-variable, or possible
variable, as explained in § 4.

Table 2: Basic information of the 17 giant stars studied in this work. Star | RA | DEC | $\langle M_{V}\rangle$ | $\langle M_{I}\rangle$ | $[\text{Fe/H}]$ | $[\text{FeI/H}]$ | $[\text{FeII/H}]$
---|---|---|---|---|---|---|---
No | (J2000) | (J2000) | (mag) | (mag) | (dex) | (dex) | (dex)
1 | $154.3433125$ | $-46.4197750$ | $13.94$ | $12.63$ | $-1.48$ | $-1.31\pm 0.03$ | $-1.29\pm 0.05$
2 | $154.3596208$ | $-46.4137333$ | $12.89$ | $11.37$ | $-1.42$ | $-1.37\pm 0.02$ | $-1.40\pm 0.05$
3 | $154.3617667$ | $-46.4310361$ | $13.54$ | $12.14$ | $-1.54$ | $-1.48\pm 0.02$ | $-1.43\pm 0.04$
4 | $154.3872500$ | $-46.4123778$ | $13.89$ | $12.63$ | $-1.79$ | $-1.62\pm 0.03$ | $-1.37\pm 0.04$
5 | $154.3946708$ | $-46.3968944$ | $12.82$ | $11.35$ | $-1.53$ | $-1.50\pm 0.02$ | $-1.40\pm 0.04$
6 | $154.4027208$ | $-46.4256111$ | $12.72$ | $11.18$ | $-1.45$ | $-1.47\pm 0.02$ | $-1.45\pm 0.05$
7 | $154.4065292$ | $-46.4010306$ | $13.53$ | $12.24$ | $-1.65$ | $-1.54\pm 0.02$ | $-1.39\pm 0.03$
8 | $154.4087875$ | $-46.4276361$ | $13.89$ | $12.62$ | $-1.71$ | $-1.61\pm 0.03$ | $-1.38\pm 0.04$
9 | $154.4095667$ | $-46.4021667$ | $13.78$ | $12.43$ | $-1.56$ | $-1.44\pm 0.02$ | $-1.44\pm 0.05$
10 | $154.4101042$ | $-46.4269639$ | $12.99$ | $11.55$ | $-1.64$ | $-1.56\pm 0.02$ | $-1.42\pm 0.04$
11 | $154.4196083$ | $-46.4122944$ | $13.61$ | $12.21$ | $-1.50$ | $-1.40\pm 0.02$ | $-1.42\pm 0.05$
12 | $154.4207125$ | $-46.3952389$ | $13.35$ | $11.92$ | $-1.54$ | $-1.43\pm 0.03$ | $-1.43\pm 0.04$
13 | $154.4288708$ | $-46.3798167$ | $13.58$ | $12.18$ | $-1.51$ | $-1.45\pm 0.02$ | $-1.45\pm 0.04$
14 | $154.4302625$ | $-46.4106583$ | $13.76$ | $12.39$ | $-1.52$ | $-1.50\pm 0.02$ | $-1.45\pm 0.04$
15 | $154.4340000$ | $-46.4239750$ | $12.71$ | $11.12$ | $-1.46$ | $-1.44\pm 0.03$ | $-1.44\pm 0.06$
16 | $154.4478125$ | $-46.3981083$ | $12.96$ | $11.46$ | $-1.59$ | $-1.54\pm 0.02$ | $-1.38\pm 0.04$
17 | $154.4543542$ | $-46.4162194$ | $13.82$ | $12.43$ | $-1.45$ | $-1.31\pm 0.03$ | $-1.29\pm 0.05$
Figure 4: CMD of NGC 3201 displaying the 17 giant stars studied in this work
with different markers based on their classification. Green stars are the
variable stars, blue squares represent the possible variables, and the red
crosses are the non-variables. Each star also has its corresponding ID.

##  3 Analysis of the Light Curves and Search for Variability

To check the possible variability of each star, we applied both the
Generalized Lomb-Scargle (GLS) (Zechmeister & Kürster, 2009) and the Phase
Dispersion Minimization (Stellingwerf, 1978) methods. We must first note that,
for all stars observed in the $I$ filter, there are 3 epochs taken in the
fourth night, which are indicated with an ellipse in Figure 2. These epochs
show evidence of having been affected by bad photometry, and were therefore
rejected for our analysis on the grounds that they are outliers.

Both analyzes were performed in Python using the
PyAstronomy111https://github.com/sczesla/PyAstronomy (PyA) package collection
(Czesla et al., 2019). The code requires as input: observation time, the data
(in this case magnitudes), and (optionally) the errors associated with them
(cols. 3, 4, and 5 of Table 1, respectively), along with a Starting Period
($P_{\mathrm{beg}}$) and an End Period ($P_{\mathrm{end}}$).

The frequency step utilized for the analysis is given by the equation:

${f_{\mathrm{step}}=1.0/\mathrm{(Time~{}Length)}/\mathrm{OFac},}$ (1)

with Time Length being HJD${}_{\mathrm{end}}-\mathrm{HJD}_{\mathrm{beg}}$, and
OFac is the Oversampling Factor, which has a value of 10. The GLS analysis
also carries out a False-Alarm-Probability (FAP), used to check which periods
are significant enough. The FAP is given by

$\mathrm{FAP}(P_{n})=1-{(1-\mathrm{Prob}(P>P_{n}))}^{M},$

where $M$ is the number of independent power values, computed internally by
the program and $\mathrm{Prob}()$ is the probability that a periodogram power
$P$ has to exceed a given value $P_{n}$. This function depends on the
periodogram and on the normalization.

After performing the GLS analysis, the program carried out the PDM analysis
for the star using the same parameters as before, viz., Starting Period, End
Period, and the Frequency Step. Figure 5 shows an example of the periodograms
obtained for Star No 8 with the GLS (top panels) and PDM (bottom panels)
methods, with the left and right panels being the results for $V$ and $I$
filters, respectively.

Figure 5: Resulting plots from the GLS (top panels) and PDM (bottom panels)
analysis performed with the PyAstronomy package collection for $V$ and $I$
filters (left and right panels, respectively). Both results are from the
initial period analysis of Star No8\. From the GLS analysis one can see the
different FAP level thresholds indicated with the different colored dashed
lines, with blue being 10%, yellow 5%, green 1% and red 0.1% probability. In
the PDM analysis the black dashed line indicates the minimum value of
$\Theta$. Note that the GLS analysis for $I$ filter is the only one that marks
a different period ($P=0.0811$ d) compared to the ones marked by GLS in $V$
filter and PDM for both filters.

In order to check if a giant of our sample can be considered as a candidate
variable star or not, we adopted the following criteria:

  1. 1.

The inequality $A\pm\sigma_{A}>\text{rms}_{\mathrm{mag}}$ must be satisfied
for both filters, with $A$, $\sigma_{A}$, and $\mathrm{rms}_{\mathrm{mag}}$
being the amplitude, its error, and the root-mean square of the magnitude
data, respectively. These parameters are given by the GLS analysis.

  2. 2.

The possible periods must display their peaks above the False Alarm
Probability (FAP) threshold of 0.1% in the GLS periodogram for both filters.

  3. 3.

The periods $P_{\text{GLS}}$ and $P_{\text{PDM}}$ must be similar for both
filters.

  4. 4.

The possible period must be present in both filters, and $P_{V}\approx P_{I}$.

  5. 5.

The visual inspection of the light curves must show that they maintain a
constant mean magnitude along the four nights of observation.

Adopting these criteria, the stars discarded as non-variables are Nos 6, 7,
10, 12, 14, and 15.

###  3.1 Significance Test

The purpose of this section is to present how we determined which giant could
be a candidate variable or not, as well as to determine the degree of
variability of each star through a significance test.

For this purpose we first calculate the magnitude difference $\Delta M$ for
each star, defined as

$\Delta M=M-\langle M\rangle,$ (2)

with $\langle M\rangle$ being the mean magnitude of the star. As in the
preliminary analysis, the three bad data points of the fourth night in the $I$
filter (indicated with ellipses) affected the measurement of $\langle
M_{I}\rangle$ and, in consequence, $\Delta M_{I}$, and therefore were
discarded.

After obtaining $\Delta M$ for all 17 giant stars, we first calculated
$\sigma_{M}=\sqrt{\frac{\sum{\Delta M^{2}}}{N-1}}$ and its error
$\mathrm{err}_{\sigma}=\frac{\sigma_{M}}{{\sqrt{2N}}}$ for the six objects
that were discarded as non-variable, using the data of all those stars
combined together for each filter, which we named $\sigma_{\text{NonVar.}}$
and $\mathrm{err}_{\sigma_{\text{NonVar.}}}$. This represents the variance of
a typical non-variable star. Then, we proceeded to calculate
$\sigma_{\mathrm{Star}}$ and $\mathrm{err}_{\sigma_{\mathrm{Star}}}$ for the
remaining eleven candidate variables, to determine their significance $S$ (for
each filter), given by

$S=\frac{\sigma_{\mathrm{Star}}-\sigma_{\text{NonVar.}}}{\sqrt{\mathrm{err}_{\sigma_{\mathrm{Star}}}^{2}+\mathrm{err}_{\sigma_{\text{NonVar.}}}^{2}}}.$
(3)

To find out whether or not the star could be a variable, we adopted the
following criteria based on the value of $S$ (Eq. 3):

  1. 1.

If $S<2$ for both filters, the star is not considered a variable.

  2. 2.

If $S>3$ for both filters, the star is considered a variable.

  3. 3.

If $S>3$ in one of the filters, the star is considered a possible variable.

  4. 4.

If $2<S<3$ in one or both filters, the star is considered a dubious variable
and requires further analysis for confirmation.

For stars in the cases (ii), (iii), and (iv), we re-performed the period
analysis described in the previous section by carefully adjusting the
different parameters from the PyAstronomy routines, while stars in the case
(i) were not re-analysed. Table 3 shows the values of $S$ derived for all
candidate variables along with the classification according to the above
criteria. One star—Nos 11—was rejected as variables due to its low values of
$S_{V}$ and $S_{I}$, which are below 2. Six stars—Nos 1, 2, 3, 5, 8, and
9—were regarded as variables, since $S_{V}>3$ and $S_{I}>3$ for all of them,
and their LCs from Figure 1 and Figure 2 show a clear change in their
magnitudes each night. Four stars were considered as possible variables (Nos
4, 13, 16, and 17), and were re-analyzed to see if they could be variables, or
candidates for future investigations. No star was classified as dubious
variable.

Table 3: Significance $S$ values for $V$ and $I$ filters for the eleven preliminary candidate variable stars and their classification based on the criteria from Section 3.1. Star | $S_{V}$ | $S_{I}$ | Classification
---|---|---|---
No 1 | 14.719 | 14.874 | Variable
No 2 | 11.695 | 8.311 | Variable
No 3 | 3.524 | 4.885 | Variable
No 4 | 0.490 | 3.900 | Possible
No 5 | 12.648 | 8.596 | Variable
No 8 | 11.013 | 7.833 | Variable
No 9 | 4.800 | 3.162 | Variable
No 11 | 0.000 | 0.000 | Non-Variable
No 13 | 2.454 | 0.990 | Possible
No 16 | 0.880 | 5.769 | Possible
No 17 | 0.000 | 7.570 | Possible

##  4 Analysis of the sample of giants

In this section we repeated the period analysis of the ten stars that were
considered variable or possible variable according to the results listed in
Table 3 (§ 3.1). The final classification of our targets is reported in Table
5. For the analysis of the variable stars, we have considered their
characteristics, position in the CMD (Figure 4) and comparing their light
curves with existing variable stars.

###  4.1 Variable Stars

In this section we will describe in detail each variable star found in this
work. We applied again to the stars No 1, 2, 3, 4, 5, 8, 9, 13, 16, and 17 the
period analysis described in § 3 and we could confirm that stars 4 and 17 are
also variable. Table 4 details the parameters for all the confirmed variable
stars, and Figures 6 shows their phased light curves for both filter and each
night is represented by a different color. We will also compare their LCs with
known variables taken as reference from the Optical Gravitational Lensing
Experiment (OGLE) Atlas of Variable Star Light
Curves222http://ogle.astrouw.edu.pl/atlas/index.html (Udalski et al., 1997;
Szymanski, 2005; Udalski et al., 2008) in order to attempt a classification.
We consider also Yepez et al. (2018) where the authors study variability in
the GC NGC 6934 and identify a kind of variable in the giant branch of the
cluster named semi regular (SR), which LC resemble some of the LCs of our
targets. We underline the fact that in some case the reference best match
variables have different parameters (mass, temperature, and gravity) compared
with our targets, and also are located at different evolutionary phase. In any
case this comparison can give us important hints about the physical mechanism
behind the variability of our targets. We notice first that confirmed variable
two kind of LCs. The first, represented by stars 1, 8 and 17, shows a sudden
rise in magnitude followed by a slow drop. The second, represented by stars 2,
3, 4, 5 and 9, shows the opposite behavior, a slow rise followed by a fast
drop.

Table 4: Periodicity parameters for all stars that presented a variable behavior. Star | $P_{V}$ | $\sigma_{P_{V}}$ | $A(V)$ | $\sigma_{A(V)}$ | $P_{I}$ | $\sigma_{P_{I}}$ | $A(I)$ | $\sigma_{A(I)}$ | $\langle M_{V}\rangle$ | $\langle M_{I}\rangle$
---|---|---|---|---|---|---|---|---|---|---
| (days) | (days) | (mag) | (mag) | (days) | (days) | (mag) | (mag) | (mag) | (mag)
No 1 | 0.5418 | 0.0027 | 0.0591 | 0.0025 | 0.5417 | 0.0023 | 0.0556 | 0.0021 | 13.9393 | 12.6293
No 2 | 0.5329 | 0.0026 | 0.0266 | 0.0012 | 0.5417 | 0.0043 | 0.0116 | 0.0008 | 12.8925 | 11.3689
No 3 | 0.3527 | 0.0037 | 0.0064 | 0.0007 | 0.3429 | 0.0034 | 0.0050 | 0.0006 | 13.5405 | 12.1394
No 4 | 0.1628 | 0.0008 | 0.0047 | 0.0005 | 0.1628 | 0.0007 | 0.0053 | 0.0006 | 13.8897 | 12.6326
No 5 | 0.4827 | 0.0024 | 0.0331 | 0.0013 | 0.4772 | 0.0034 | 0.0114 | 0.0009 | 12.8158 | 11.3473
No 8 | 0.0881 | 0.0001 | 0.0195 | 0.0012 | 0.0881 | 0.0002 | 0.0072 | 0.0004 | 13.8944 | 12.6193
No 9 | 0.5281 | 0.0054 | 0.0081 | 0.0006 | 0.5164 | 0.0067 | 0.0058 | 0.0005 | 13.7802 | 12.4272
No 17 | 0.3535 | 0.0035 | 0.0045 | 0.0005 | 0.3535 | 0.0021 | 0.0093 | 0.0007 | 13.8238 | 12.4290
Figure 6: Phased light curves of our 8 variable stars found in this work which
were plotted by phasing the LC of each star with their corresponding period
$P_{V}$ shown in Table 4. The top panel of each phased light curve correspond
to the phased curve in the $V$ filter and the bottom panel the phased curve in
the $I$ filter. The different colors indicate the data taken during each night
of observation. For the top panel Night 1 is green, Night 2 cyan, Night 3
yellow and Night 4 blue. For the bottom panel Night is 1 red, Night 2 magenta,
Night 3 orange and Night 4 purple.

Star No 1: This star has a period $P=0.5418\pm 0.0027$ days, with an amplitude
$A>0.055$ mag in both filters. It shows a clear variable behavior in $V$ and
$I$ (Figures 1 and 2) and with a clear phased light curve (Figure 6a). At
first, for $V$ filter the GLS analysis showed us a period of $P\sim 1.1135$
days if we considered a $P_{\mathrm{end}}=\text{TimeLength}$, but the PDM
analysis marked $P\sim 0.55996$ d, while in the $I$ filter both the GLS and
PDM analysis displayed a period between $0.5413$ and $0.55457$ d. The longest
period obtained from the GLS analysis of the $V$ filter is almost twice the
period obtained from the PDM analysis of the same filter and from the GLS and
PDM analysis of the $I$ filter, and it could have been a considered as the
actual period. However the phased light curve of this star displayed with such
period was not realistic, which lead us to discard it. Finally, after
restricting the $P_{\mathrm{end}}$ value to $P<1$ day, we obtained the period
listed in Table 4. It is worth noting that this star is positioned in the
lower part of the RGB (see Figure 4), and comparing its phased LC with known
variables, it resembles an RR Lyrae ab star and the period is similar to some
of the known RRab. However, its low amplitude does not agree with these type
of variables.

Star No 2: This star displays an evident magnitude difference, as shown in
both LCs (Figure 1 and Figure 2). For this case, the period analysis indicates
that this star has a period $P=0.5329\pm 0.0026$ days, but the phased LC shown
in Figure 6b does not display a complete curve. This is due to the diurnal
interruption of the observations. Figure 4 shows that this star is positioned
at the top of the RGB, and Figure 7 shows that its phased LC resembles that of
a SR variable like the V96 described by Yepez et al. (2018) in the RGB of the
GC NGC 6934 and are located in a similar position of the CMD near the tip.
However, since SR variables display large periods, in this particular case V96
has a period of $9.54$ days, we cannot attempt to classify this star as an SR
due to the discrepancy between these parameter. We do however note that the
period shown for this star is similar to those displayed in RR Lyrae stars.

Star No 3: The $V$ filter LC of this star shows a quite small difference in
magnitude, while in $I$ it is more evident, particularly during the 3rd night.
From the results obtained after a more precise analysis we found a period
$P=0.3527\pm 0.0037$. This star is positioned in the CMD in the middle part of
the RGB, and Figure 6c shows the phased light curve where a small but clear
variability is visible with its period resembling those of pulsating variables
(e.g., RR Lyrae stars). Figure 7, like in the case of Star No 2 shows that its
phased LC is similar to the one from V96 described by Yepez et al. (2018) in
the RGB of the GC NGC 6934. But both the location in the CMD as well as both
the period and amplitude do not agree with these kind of variables.

Star No 4: This particular star was also considered a possible variable
candidate due to the significance values from Table 3, with $S_{V}<2$, but
$S_{I}>3$. We therefore re-performed our period analysis with more caution and
detail. This star shows similar values of $P$ from the GLS method for both
filters, while for the PDM method the situation was unclear. However, after
restricting the P interval and taking caution that the periodogram does not
display the estimated period as the only possible one, both methods yielded
similar values of $P$, with the final result being $P=0.1628\pm 0.0008$ (Table
4). Figure 4 shows that this star is located in the lower part of the giant
branch. The phased light curve of this star is in Figure 6d and as in the
previous star, Figure 7 shows that its phased LC could resemble that of a SR
variable like the V96. However, like Star No 3, the period shown for this star
does not agree with the period of typical SR stars. We also note that the
phased LC of this star could also resemble that of an RR Lyrae type RRc
displaying a similar period to those kind of variables, but we went for the SR
variable due to a better resemblance of its phased LC.

Star No 5: This star shows a quite clear variable behavior as far as V filter
is concerned, while for the I filter the variability is not clear (see Figure
1 and 2). The variability analysis shows that this star have a period
$P=0.4827\pm 0.0024$ d; its phased light curve is reported in Figure 6e. We
can also see that this star, from its position in the CMD it displays a
similar case a Star No 2, located at the top of the giant branch and
displaying a period similar to those found in RR Lyrae variables, and Figure 7
shows that its phased LC is similar to that of V96 from Yepez et al. (2018)
but again its period and amplitude does not agree with these type of
variables.

Star No 8: This star displays a period $P=0.0881\pm 0.0001$ days, and was the
one that presented the most prominent variability along with Star No 1
although with a smaller amplitude; it was also the star on which we performed
all the parameter testing from the methodology presented in § 3, where we
tested how the PyAstronomy routines worked and adjusted the final parameters
in order to have precise results. Similar to the case of Star No 1, this
star’s position in the CMD is in the lower part of the RGB, with its phased LC
resembling more like a $\delta$ Scuti star, but with an amplitude lower than
those type of stars (see Figure 7). Figure 6f shows the phased light curve of
this star for $V$ and $I$ filters.

Star No 9: This star displayed in its $V$ filter LC a variable behavior. The
$I$ LC also displays a variability, although not as evident as in the V
filter. After performing our analysis, the most probable period for the $V$
filter was $P=0.5281\pm 0.0054$ days. The PDM method for the I filter
suggested also another period of $P\sim 0.3417$ days, that however is much
less probable that the first one. From Figure 4, we can see that this star is
positioned at the lower part of the AGB with its period being similar to those
found in typical RR Lyrae stars, and Figure 7 shows that its LC resembles that
of a SR variable like the V96 from Yepez et al. (2018), however as in other
stars that were compared with the SR V96, its period and amplitude do not
agree at all with this kind of variables. The left panel of Figure 6g displays
the phased light curve for this star.

Star No 17: Its LC for $V$ filter does not display an evident variable
behavior, but its LC in $I$ does show evident variability in magnitude, which
can be seen in the plot from the fifth row, first column of Figure 2. Due to
this feature and also to its significance $S_{I}$ value from Table 3, which is
higher that $3$, we re-performed our period analysis obtaining a period
$P=0.3535\pm 0.0035$ d and the amplitudes $A(V)$ and $A(I)$ shown in Table 4.
Also, based on the shape of its phased LC, like Stars No 1 and 8, its variable
behavior is similar to that of a pulsating variable, like a RR Lyrae or a
Cepheid. In this case we chose Cepheid as the best match (see Figure 7), but
again its low amplitude value makes it difficult to provide such
classification. The phased light curve of this star is shown in the right
panel of Figure 6h.

Figure 7: Comparison between our sample of our variables stars(top panels) and
variable stars from the OGLE catalogue of variable stars—an RR Lyrae ab, a
$\delta$ Scuti, and a Cepheid variable—and the SR variable V96 from Yepez et
al. (2018) (bottom panels). Both comparisons are made in $I$ filter, with each
star being compared with the known variable that resembles the most based on
its phased light curve from Figure 6 and from what was presented in Section
4.1.

###  4.2 Possible Candidates

In this section we present the two stars that, in spite of not showing a clear
variable behavior compared to the eight stars presented in the previous
Section, are considered as possible variable according to the results of our
period analysis and are good targets for future studies using a more complete
multi-epoch photometric approach.

Star No 13: This possible variable star is interesting since both LCs
displayed a variable behavior during the 3rd night (although quite small). The
GLS method derived a period $P=1.0614$ d but the PDM analysis showed a period
$P=0.6983$ d, even thought the second highest probability period for the GLS
analysis was at $P=0.5287$ (top panels of Figure 8). We adjusted the
PyAstronomy parameters to evaluate the other periods. It is worth to note that
there are two more periods above the 0.1% threshold at $P\sim 0.41564$ and
$P\sim 0.33818$, but the amplitudes derived by the GLS analysis were equal (if
not lower) than the mean magnitude error, so they were rejected. Figure 9
shows the phased light curves of the three different periods. Future
observations for this star could help to determine if this star is a real
variable and, if so, its variability period.

Figure 8: Resulting plots from the GLS (top panels) and PDM (bottom panels)
analysis performed with the for Star No 13\. Note that the GLS analysis marks
the highest period $P=1.0614$ d, while the PDM analysis mark a period
$P=0.6983$ d for both filters, meanwhile the second highest period for the GLS
analysis in both filters correspond to a period $P=0.5287$. Figure 9:
Different phased light curves of Star No 13, with each plot displaying a
different possible period period, with top left being $P=1.0614$ d, top right
$P=0.6983$ d and the bottom panel $P=0.5287$. Once again, the different colors
are the same as in Figure 6

Star No 16: This particular star was challenging since its LC for $V$ filter
did not present any magnitude variation, but for the $I$ filter it displayed a
peculiar behavior during the 3rd night (4th row and 4th column panel of Figure
2). We derived two different periods for $V$ and $I$ filters ($P_{V}=0.3981$ d
and $P_{I}=0.4174$ d, left and right panels of Figure 10, respectively), as in
the case of Star No 13, future observations for could help to determine if
this stars is a real variable and, if so, its variability period.

Figure 10: Left panels: Phased Light Curve of Star No 16 for $V$ (top panel)
and $I$ (bottom panel) filters, plotted by phasing the light curve with the
period obtained from our analysis in the $V$ filter $P_{V}=0.3981$ days. As in
previous figures, the different colors are the same as in Figure 6. Right
panels: Phased Light Curve of Star No 16 for $V$ (top panel) and $I$ (bottom
panel) filters, plotted by phasing the light curve with the period obtained
from our analysis in the in the $I$ filter $P_{I}=0.4174$ days. As in previous
figures, the different colors are the same as in Figure 6.

If the variability behaviour of star 13 and 16 is confirmed, a classification
could be possible for both stars in a similar manner as in the case of our
variable stars from § 4.1.

###  4.3 Variability and its possible impact on the spectroscopic [Fe/H]
determination

After finishing our period analysis of the variable stars, the next step was
to check if their variable behavior has an impact on the $\text{[Fe/H]}$
determination as obtained from spectra. First of all, we used the abundance
study of NGC 3201 by Mucciarelli et al. (2015), who measured the iron
abundance by adopting photometric values of $\log g$. They used photometric
gravities based on the results of Lapenna et al. (2014), who suggested that
NLTE effects could affect in AGB stars the abundances that are obtained from
Fe I lines, although the abundances from Fe II lines appear not to be
modified. This effect can be detected by assuming photometric values of $\log
g$ and measuring the abundances from Fe I and Fe II independently. They also
remarked that RGB stars do not display such effects. We point out here that
AGB and RGB stars in the sample have very similar atmospheric parameter. For
this reason, if they are affected by NLTE, the final Fe I abundances should
suffer the same change, independently of the evolutionary stage.

We grouped each star according to their classification given in § 3 and §
4—variable, non-variable and possible variable—and, with the [Fe I/H] values
shown in Table 2, we plotted the significance $S_{V}$ (top panels) and $S_{I}$
(bottom panels) against their [Fe I/H] for each group in the left panel of
Figure 11. In this figure we can see that: stars that have extreme [Fe I/H]
values are variables, i.e., the most metal-poor stars like Star No 8
($\text{[FeI/H]}=-1.61\pm 0.03$) and No 4 ($\text{[FeI/H]}=-1.62\pm 0.03$),
and the most metal-rich like stars No 1 and No 17 with
$\text{[FeI/H]}=-1.31\pm 0.03$, and Star No 2 with $\text{[FeI/H]}=-1.37\pm
0.02$ are located at the extremes of the [Fe I/H] distribution. However, the 3
remaining variable, i.e. stars No 3, 5, and 9, have a metallicity that well
agrees with that of the non-variable sample. This could indicate, as we will
infer in the discussion from the study of For et al. (2011), that the [Fe I/H]
metallicity as obtained from the spectra could depend on the phase at which
the spectrum is taken. Furthermore, we note that stars that are not variable,
and those which are possible variables (§ 4.2) have [Fe I/H] values that are
close to the mean value of [Fe I/H] of the total sample.

We also checked the $\text{[FeII/H]}$ content of each variable star to see if
it shows a similar behavior. The right panels of Figure 11 shows the
significance values of the variable stars, possible variables and non-
variables plotted against [Fe II/H]. From this figure we can see that variable
stars tend to have on average higher [Fe II/H] values compared with non-
variable and as that, as in the case of [Fe I/H], variable stars show a higher
spread degree in comparison with the possible and non-variables.

Figure 11: Left panels: Significance $S_{V}$ and $S_{I}$ values for each star
plotted against their respective $\text{[FeI/H]}$ measurements from
Mucciarelli et al. (2015). The green stars represent the variable stars, the
red squares represent the non-variables and the blue dots represent the
possible candidate variables. Right panels: Significance values for each star
plotted against their $\text{[FeII/H]}$ measurements from Mucciarelli et al.
(2015) plotted with the same x-axis scale as the left panel and the different
colors for each group of stars are the same as in the left panels.

As a second step we used the abundance from Simmerer et al. (2013) to see if
also in this case we obtain the same result. We point out the fact that
Simmerer et al. (2013) uses spectroscopic parameters and for this reason they
have $\text{[FeI/H]}=\text{[FeII/H]}$ and so a unique value of iron ([Fe/H])
for each star. Figure 12, shows the significance $S_{V}$ (top panel) and
$S_{I}$ (bottom panel) of our 17 giants plotted against their $\text{[Fe/H]}$.
From this plot we can see again that stars with extreme values of
$\text{[Fe/H]}$ (both metal-poor and metal-rich) are variable. However in this
case the most metal-poor variable are those that have a larger [Fe/H]
difference with respect the non-variable stars.

Table 5 shows the final classification of all 17 stars along with the iron
abundance measurements from Simmerer et al. (2013) and Mucciarelli et al.
(2015) and their significance $S_{V}$ and $S_{I}$. Note that for stars that
were classified as non-variable both significance values $S_{V}$ and $S_{I}$
were assumed to be zero. If we move to statistics, it is worth to report the
mean abundances and spreads with their errors of the three groups of stars
calculated for [Fe I/H], [Fe II/H] and [Fe/H]. These values can be found in
Table 6. We consider the mean abundances first. We see that mean [Fe I/H] from
Mucciarelli et al. (2015) agrees well for the three groups, especially for the
variable and the non-variable. For mean [Fe II/H] instead, variable stars show
a value that is 0.05 higher than the non-variables, beyond the statistics
errors. As far as Simmerer et al. (2013) is concerned, the agreement is well
within $1\sigma$ for all three groups.

The effect of variability appears when we consider the iron spread. For
Mucciarelli et al. (2015), [Fe I/H] spread for possible and non-variables is
of $0.06$ dex, while the spread for variables is much larger with a value of
$0.12$ dex. The same is true if we compare [Fe II/H] spreads for variable and
non-variables in Mucciarelli et al. (2015) and [Fe/H] spreads for the same
groups in Simmerer et al. (2013).

We conclude the variability has a negligible effect on mean iron metallicities
with a possible but small increase of mean [Fe II/H] compared with [Fe I/H]
when atmospheric parameters are obtained from photometry as in Mucciarelli et
al. (2015). On the other hand the effect on the iron spread is important. We
see in fact that variable stars have a much higher iron spread regardless the
method used to obtain the parameters. The spread for variable is up to twice
the values obtained for non-variable. This effect must be taken into account
in any study of intrinsic iron spread of stellar populations, especially in
globular clusters, otherwise the wrong conclusion could be that the cluster
hosts stars with different metallicities.

Figure 12: Top panel: plot displaying the significance $S_{V}$ values of the 17 stars against their $\text{[Fe/H]}$ abundance from Simmerer et al. (2013). Bottom panel: Significance $S_{I}$ values of the 17 giants plotted against their $\text{[Fe/H]}$. The colors for both panels are the same as in Figure 11. For the error bar we assumed an error on [Fe/H] of 0.05 dex, that is the typical error for purely spectroscopic measurement. Table 5: Final classification, iron abundance information and significance values of the 17 giant stars. Star | Classification | $\text{[Fe/H]}$ | $\text{[FeI/H]}$ | $\text{[FeII/H]}$ | $S_{V}$ | $S_{I}$
---|---|---|---|---|---|---
|  | (dex) | (dex) | (dex) |  |
1 | Variable | $-1.48$ | $-1.31\pm 0.03$ | $-1.29\pm 0.05$ | $14.719$ | $14.874$
2 | Variable | $-1.42$ | $-1.37\pm 0.02$ | $-1.40\pm 0.05$ | $11.695$ | $8.311$
3 | Variable | $-1.54$ | $-1.48\pm 0.02$ | $-1.43\pm 0.04$ | $3.524$ | $4.885$
4 | Variable | $-1.79$ | $-1.62\pm 0.03$ | $-1.37\pm 0.04$ | $0.490$ | $3.900$
5 | Variable | $-1.53$ | $-1.50\pm 0.02$ | $-1.40\pm 0.04$ | $12.648$ | $8.596$
6 | Non-Variable | $-1.45$ | $-1.47\pm 0.02$ | $-1.45\pm 0.05$ | $0.0$ | $0.0$
7 | Non-Variable | $-1.65$ | $-1.54\pm 0.02$ | $-1.39\pm 0.03$ | $0.0$ | $0.0$
8 | Variable | $-1.71$ | $-1.61\pm 0.03$ | $-1.38\pm 0.04$ | $11.013$ | $7.833$
9 | Variable | $-1.56$ | $-1.44\pm 0.02$ | $-1.44\pm 0.05$ | $4.800$ | $3.162$
10 | Non-Variable | $-1.64$ | $-1.56\pm 0.02$ | $-1.42\pm 0.04$ | $0.0$ | $0.0$
11 | Non-Variable | $-1.50$ | $-1.40\pm 0.02$ | $-1.42\pm 0.05$ | $0.0$ | $0.0$
12 | Non-Variable | $-1.54$ | $-1.43\pm 0.03$ | $-1.43\pm 0.04$ | $0.0$ | $0.0$
13 | Possible | $-1.51$ | $-1.45\pm 0.02$ | $-1.45\pm 0.04$ | $2.454$ | $0.990$
14 | Non-Variable | $-1.52$ | $-1.50\pm 0.02$ | $-1.45\pm 0.04$ | $0.0$ | $0.0$
15 | Non-Variable | $-1.46$ | $-1.44\pm 0.03$ | $-1.44\pm 0.06$ | $0.0$ | $0.0$
16 | Possible | $-1.59$ | $-1.54\pm 0.02$ | $-1.38\pm 0.04$ | $0.880$ | $5.769$
17 | Variable | $-1.45$ | $-1.31\pm 0.03$ | $-1.29\pm 0.05$ | $0.0$ | $7.570$
Table 6: Mean, $\sigma$ and $err_{\sigma}$ of the iron abundances for each group of stars. | Classification | $\bar{X}$ | $\sigma_{\bar{X}}$ | $\sigma$ | $err_{\sigma}$
---|---|---|---|---|---
|  | (dex) | (dex) | (dex) | (dex)
$\text{[FeI/H]}_{\text{Mucciarelli}}$ | Variable | $-1.46$ | $0.04$ | $0.12$ | $0.03$
| Possible | $-1.50$ | $0.03$ | $0.05$ | $0.02$
| Non-Variable | $-1.48$ | $0.02$ | $0.06$ | $0.01$
$\text{[FeII/H]}_{\text{Mucciarelli}}$ | Variable | $-1.37$ | $0.02$ | $0.06$ | $0.01$
| Possible | $-1.41$ | $0.03$ | $0.04$ | $0.02$
| Non-Variable | $-1.43$ | $0.01$ | $0.02$ | $0.01$
$\text{[Fe/H]}_{\text{Simmerer}}$ | Variable | $-1.56$ | $0.04$ | $0.13$ | $0.03$
| Possible | $-1.55$ | $0.04$ | $0.04$ | $0.02$
| Non-Variable | $-1.53$ | $0.03$ | $0.08$ | $0.02$

##  5 Discussion

At this point it is interesting to check where the different groups of stars
lie in the CMD of Figure 4. We see there that variable are distributed evenly
among RGB and AGB stars, with 3 objects being AGBs and 4 being RGBs. On the
other hand only 2 non-variable appear to be AGBs against 5 RGBs. In any case
we cannot reach any firm conclusion about a possible relation between
variability and the evolutionary phase due to the limited sample we have.

About the possible mechanism responsible of the variability, we note that the
LC of 3 of our targets (1, 8, and 17) resembles closely stars whose
variability is caused by radial or no-radial pulsations (RRLyrae, Delta-Scuti,
and Cepheid respectively). From that we deduce that pulsation is very likely
the cause of the variability of our stars and that it is related with the
spread in the spectroscopic [Fe/H] determination we observe.

At this point, it is also worth mentioning the possibility that the
variability we observe is not intrinsic to the RGB or AGB stars we are
studying, but it is due to an unresolved blending with other variables of the
cluster. In order to check this hypothesis we will analyze here the three
stars #1, #8, and #17 since they show a LC very similar to that of pulsating
variables (e.g., RR-Lyrae). Because of this and since the only known variables
bright enough to affect the $V$ and $I$ magnitude of a giant stars in NGC 3201
are RR-Lyrae (Arellano Ferro et al., 2014), we will consider only these object
as possible polluters of the LC. Figure 13 shows the position in the CMD of
our three stars (squares) and of the known RR-Lyrae (red circles). The aim is
know what the real position of the three targets would be if we remove the
contamination form the RR-Lyrae.

If we have two blended stars (stars $A$ and $B$) the equation to calculate the
magnitude of $A$ using the total magnitude ($A+B$) and the magnitude of $B$
(in our case a RR Lyrae) can be obtained from Pogson’s equation and is:

${m_{A}=m_{B}-2.5\log(10^{(-(m_{tot}-m_{B})/2.5)}-1)}$

Figure 13: CMD of NGC 3201 showing how photometric blending between a giant
star and a variable star could affect in the position of stars 1, 8, and 17
from this work. Red stars are all known RR Lyrae from the cluster, while our
stars are marked in different colors: Star No1 is blue, No 8 is magenta and No
17 is yellow.

We applied this equation to the $V$ and $I$ magnitudes of the three stars
using the $V$ and $I$ mean magnitudes of all the RR-Lyrae. The result is the
three strips of circles in Figure 13. As we can see, if the hypothesis of the
blending is true, the three targets would fall well out of the RGB of the
cluster. They could still be giants, but much more metal-rich than the value
obtained from spectroscopy ([Fe/H]$<-1.4$). They would also fall in a region
completely empty of stars and they will not be compatible with being members
of the clusters at odd with what found in all the spectroscopic studies we
mention in previous sections.

Another test we can do is to calculate the probability to find a blend between
a giant stars and a RR-Lyrae. In order to do that we first have to define the
area around a target where, if a RR-Lyrae falls, we have a unresolved blend.
Since the Full Width at Half Maximum (FWHM) of the observation was around $2$
arcsec, it is safe to assume a circular area with a radius of the same size.
Since the field of view (FOV) is 5.4’x5.4’ (see Figure 3) and that in Figure
13 we identify 35 RR-Lyrae, we find that that probability of a blend is:

$P=\frac{Area_{\text{circulararea}}}{Area_{\text{FOV}}}\cdot N_{\text{RR-
Lyrae}}=0.004.$

If we calculate then the probability to have three stars affected by a blend,
we end with total a probability of:

$P_{TOT}=P^{3}=0.00000007$

These numbers can vary if we change our inputs such as the FOV, the size of
the circular area, or if we consider that the RR-Lyrae are not uniformly
distributed in the FOV but they follow a King density profile. In any case the
probability we calculated is low enough to assure that, together with the CMD
position test, none of the LCs of our targets is the result of a blend between
a regular RGB or AGB and a variable star.

In order to further discuss the impact of variability on spectroscopic [Fe/H]
determinations, it is worth commenting here the paper by For et al. (2011),
where the abundances of 11 RR Lyrae ab-type variables were studied. They
analyzed more than 2300 high resolution spectra from the 2.5m du Pont
telescope distributed along the period curve of each variable. In Figure 14 we
overplot the [Fe/H] abundances of each variable, where each single measurement
of each variable was divided by the corresponding mean [Fe/H] abundance.
Figure 14 indicates that a variable can show a variation in [Fe I/H] up to
$\pm 0.2$ dex, depending on the phase. This means that a variable can have an
excursion up to $0.4$ dex in its [Fe/H] content. In addition the mean [Fe I/H]
behavior is quite regular with the phase, at least between Mid Phase $0$ and
$0.6$. At Mid Phase$=0$ the deviation from the mean iron abundance is around
zero. Then the deviation increases toward higher metallicities reaching its
maximum at Mid Phase$=0.2$. It remains constant until Mid Phase$=0.4$ and then
returns to zero at a Mid Phase$=0.6$. After that the behaviors is quite
chaotic with some variables showing a positive deviation of $0.1$ dex or more,
while others show a negative deviation of $-0.15$ dex or more. On the other
hand the maximum deviation in this plot is between $-0.3$ and $+0.3$ dex. Such
finding is important since it is an independent confirmation of the results
obtained in this work. It shows that in these kind of stars the
$\text{[FeI/H]}$ determination might be affected by variability. Furthermore,
it provides us with the possibility that the physical reason for the
variability of our stars is some kind of pulsations.

Figure 14: Plot displaying the Mid Phase in the x-axis and the
$\mathrm{[FeI/H]_{Star}}-\langle\mathrm{[FeI/H]_{Star}}\rangle$ in the y-axis
for the 11 different RR Lyrae stars from For et al. (2011).

Because of that we suggest that, if the variability is due to pulsations, the
[Fe/H] spread difference between variable and non-variable stars could
originate from the fact that the standard technique for chemical analysis uses
hydrostatic LTE atmospheric models, while pulsating stars have by definition
non-hydrostatic atmospheres. Hydrostatic LTE models work because they resemble
real atmospheres where, in most of the cases, the gas is static on the large
scale and the gas dense enough to allow the light coming from the deepest part
of the photosphere to be fully absorbed and re-processed by the different
atmospheric layers before leaving the stellar surface. This means that each
layer behaves like a black body and for this reason it can be treated by the
LTE approximation. This is not strictly true for the spectral lines formed at
the top of the photosphere, but it works for most of the lines that are formed
in the deeper parts of the atmosphere. In pulsating stars the atmosphere is
expanding and contracting radially or tangentially, and during the expansion
the gas become less dense and so its behavior deviates from the LTE
approximation. Also the pulsation itself could be have a direct impact on the
spectroscopic [Fe/H] determination in the sense that it could alter the
atmospheric structure and specifically how temperature and pressure vary with
optical depth. Since these variables are assumed to vary like it is predicted
in the atmospheric model, a mismatch between the predicted and the real
atmospheric structure can easily lead to an underestimation or overestimation
of the iron content.

##  6 Summary and Conclusions

We present in this paper the results of the variability analysis for 17 giant
stars in the GC NGC 3201 that were studied previously to obtain spectroscopic
metallicity $\text{[Fe/H]}$. The aim of our study is to check if variability
affects somehow the determination of the iron content of RGB and AGB stars in
globular clusters as obtained from the spectra using standard techniques. We
performed the Generalized Lomb-Scargle and the Phase Dispersion Minimization
methods along with a set of discarding criteria in order to check for
variables. Of the total sample of 17 giant stars, 8 targets were classified as
variable stars, 2 as possible variables, and 7 as non-variable.

After obtaining the main parameters (period $P$ and amplitude $A$) of the 8
variable stars for both $V$ and $I$ filters, we revised the results from
Mucciarelli et al. (2015) regarding their iron abundance. We also analyzed our
variable stars according to their period, amplitude, position in the CMD and
the shape of the light curve, comparing them with known variables, concluding
that most probable reason for the variability is radial or tangential
pulsation. We revised also the results from Simmerer et al. (2013) to see if
there is some difference depending on the different methods used to obtain the
metallicity. We conclude that stars having a larger spread in their [Fe/H]
estimates are variable, while non-variable have an observed iron content more
concentrated around the mean value of the cluster. This means that variable
stars can display extreme values of metallicity (both toward metal-richer and
metal-poorer ends), while stars that do not present variability display values
of their iron abundance closer to the mean abundance of the total sample.
Furthermore, this finding is consistent with the metal-poor variable RGB star
found in Muñoz et al. (2018). More data for both photometric and spectroscopic
analysis are needed in order to further confirm our results, but nevertheless
this is a step forward to determine the nature of the variable star and its
influence on metallicity. Finally, our result suggests that the spectroscopic
determination of the iron abundance of giant stars in NGC 3201 is affected in
some way by the variability of such stars and point toward the fact that this
cluster does not have an intrinsic iron spread.

SV gratefully acknowledges the support provided by Fondecyt reg. n. 1170518.
This work is based on data acquired at Complejo Astronómico El Leoncito,
operated under agreement between the Consejo Nacional de Investigaciones
Científicas y Técnicas de la República Argentina and the National Universities
of La Plata, Córdoba and San Juan. Furthermore, this work made use of the
PyAstronomy package collection and we are thankful at the PyA group for
answering our doubts regarding their packages. Finally, the authors gratefully
acknowledge the comments and suggestions of the anonymous referee that greatly
improved and strengthen this paper.

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# The Period of Ducci Cycles on $\mathbb{Z}_{2^{l}}$ for Tuples of Length
$2^{k}$

Mark L. Lewis  Department of Mathematical Sciences
Kent State University
Kent, OH 44242<EMAIL_ADDRESS>and  Shannon M. Tefft  Department of
Mathematical Sciences
Kent State University
Kent, OH 44242<EMAIL_ADDRESS>
(Date: January 2024)

###### Abstract.

Let the Ducci function $D:\mathbb{Z}_{m}^{n}\to\mathbb{Z}_{m}^{n}$ be defined
as
$D(x_{1},x_{2},...,x_{n})=(x_{1}+x_{2}\;\text{mod}\;m,x_{2}+x_{3}\;\text{mod}\;m,...,x_{n}+x_{1}\;\text{mod}\;m)$.
In this paper, we will prove that if $n,m$ are powers of $2$, then repeatedly
applying $D$ will eventually result in $(0,0,...,0)$, as well as establish an
upper bound for how many iterations it will take for this to happen.

###### Key words and phrases:

Ducci sequence, modular arithmetic, length, period, $n$-Number Game

###### 1991 Mathematics Subject Classification:

20D60, 11B83, 11B50

##  1\. Introduction

Consider a function $D:\mathbb{Z}_{m}^{n}\to\mathbb{Z}_{m}^{n}$ which we will
define as

$D(x_{1},x_{2},...,x_{n})=(x_{1}+x_{2}\;\text{mod}\;m,x_{2}+x_{3}\;\text{mod}\;m,...,x_{n}+x_{1}\;\text{mod}\;m)$

We call $D$ the Ducci function and for a given tuple
$\mathbf{u}\in\mathbb{Z}_{m}^{n}$, we say that the Ducci sequence of
$\mathbf{u}$ is the sequence $\\{D^{k}(\mathbf{u})\\}_{k=0}^{\infty}$ [1, 5,
7].
Looking at an example, consider $(3,1,3)\in\mathbb{Z}_{4}^{3}$ and the first
few terms in its Ducci sequence:
$(3,1,3),(0,0,2),(0,2,2),(2,0,2),(2,2,0),(0,2,2)$. One can see that if you
continue the Ducci sequence, it will cycle through the tuples
$(0,2,2),(2,0,2),$ $(2,2,0)$. We say that these three tuples are the Ducci
cycle of the Ducci sequence of $(3,1,3)$. More specifically, we say that a
Ducci cycle of a tuple $\mathbf{u}$ is the set $\\{\mathbf{v}\mid\exists
k\in\mathbb{Z}^{+},l\in\mathbb{Z}^{+}\cup\\{0\\}\ni\mathbf{v}=D^{l+k}(\mathbf{u})=D^{k}(\mathbf{u})\\}$
[1, 4, 8]. We define $\mathbf{Len(u)}$ to be the smallest $l$ that can be
found to satisfy the equation
$\mathbf{v}=D^{l+k}(\mathbf{u})=D^{k}(\mathbf{u})$ from our definition for
some $k\in\mathbb{Z}^{+},\mathbf{v}\in\mathbb{Z}_{m}^{n}$ [1]. We also define
$\mathbf{Per(u)}$ to be the smallest $k$ that satisfies the same equation.
Notice that $Per(\mathbf{u})$ is also the number of distinct tuples in the
Ducci cycle of $\mathbf{u}$. If a tuple is in the Ducci cycle of the Ducci
sequence for some tuple $\mathbf{u}$, we may simply say that is in a Ducci
cycle. Because $\mathbb{Z}_{m}^{n}$ is finite, every Ducci sequence will enter
a cycle.
Define $K(\mathbb{Z}_{m}^{n})$ to be the set of all tuples in
$\mathbb{Z}_{m}^{n}$ that are in the Ducci cycle for the Ducci sequence of
some $\mathbf{u}\in\mathbb{Z}_{m}^{n}$ [1]. We will prove the following
theorem in Section 3:

######  Theorem 1.

$K(\mathbb{Z}_{m}^{n})$ is a subgroup of $\mathbb{Z}_{m}^{n}$.

If for a tuple $\mathbf{u}$ there exists a tuple $\mathbf{v}$ such that
$D(\mathbf{v})=\mathbf{u}$, we say that $\mathbf{v}$ is a predecessor of
$\mathbf{u}$ [1, 7].
It is not always the case that the Ducci sequence will end in a cycle
containing more than one tuple. Notice that the Ducci sequence of
$(0,0,...,0)$ consists only of itself and that it satisfies
$Per(\mathbf{u})=1$. In addition to this, $(0,0,...,0)$ is the only tuple that
satisfies $D(\mathbf{u})=\mathbf{u}$ [1], which means that only $(0,0,...,0)$
and tuples whose Ducci cycle is $\\{(0,0,..,0)\\}$ has a period of $1$. We say
that a Ducci sequence vanishes when its cycle is $\\{(0,0,...,0)\\}$ [1, 7].
An important Ducci sequence is the one corresponding to
$(0,0,...,0,1)\in\mathbb{Z}_{m}^{n}$. We call the Ducci sequence of this tuple
the basic Ducci sequence of $\mathbb{Z}_{m}^{n}$ [1, 5, 7]. In a similar way
to the literature, we denote $L_{m}(n)=Len(0,0,...,0,1)$ and
$P_{m}(n)=Per(0,0,...,0,1)$ [5, 7]. These are significant because for every
$\mathbf{u}\in\mathbb{Z}_{m}^{n}$, $L_{m}(n)\geq Len(\mathbf{u})$ and
$Per(\mathbf{u})|P_{m}(n)$ [1].

In this paper, we focus on the case where $m=2^{l}$ and $n=2^{k}$ for integers
$k,l\geq 1$. The goal of this paper is to prove the following theorem:

######  Theorem 2.

Let $n=2^{k}$, $m=2^{l}$ for integers $l,k\geq 1$. Then
$L_{m}(n)=(l+1)2^{k-1}$ and $P_{m}(n)=1$.

As a result of this theorem, all Ducci sequences vanish and this gives an
upper bound for how long it takes for a Ducci sequence in
$\mathbb{Z}_{2^{l}}^{2^{k}}$ to vanish. It also yields the following
corollary:

######  Corollary 3.

$K(\mathbb{Z}_{2^{l}}^{2^{k}})$ is the trivial subgroup.

The work in this paper was done while the second author was a Ph.D. student at
Kent State University under the advisement of the first author and will appear
as part of the second author’s dissertation.

##  2\. Background

There are many different versions of the Ducci function. The most common Ducci
function is $\bar{D}:(\mathbb{Z}^{+})^{n}\to(\mathbb{Z}^{+})^{n}$ being
defined as
$\bar{D}(x_{1},x_{2},...,x_{n})=(|x_{1}-x_{2}|,|x_{2}-x_{3}|,...,|x_{n}-x_{1}|)$
[5, 6, 7, 8] or similarly defined on $\mathbb{Z}^{n}$ [1]. Other sources
define it on $\mathbb{R}^{n}$ [2, 4, 9].
For the case on $\mathbb{Z}^{n}$ or $(\mathbb{Z}^{+})^{n}$, it has been proven
that all Ducci sequences will enter a cycle [5, 6, 7, 8, 9]. When looking at
the tuples in a Ducci cycle, all of the entries belong to $\\{0,c\\}$ for some
$c\in\mathbb{Z}^{+}$ [8, 9]. Because $D(\lambda\mathbf{u})=\lambda
D(\mathbf{u})$ when $\lambda\in\mathbb{Z}$ and
$\mathbf{u}\in\mathbb{Z}_{m}^{n}$, this means that for this case, one only
needs to worry about when Ducci is defined on $\mathbb{Z}_{2}$ when examining
Ducci cycles and periods [1, 5, 7, 8].
It has been proven that if $n$ is a power of $2$, then the Ducci sequence will
eventually vanish for all tuples in $\mathbb{Z}^{n}$ or $(\mathbb{Z}^{+})^{n}$
[5, 6]. If $n$ is not a power of $2$, then in general, this does not happen
for all tuples [6]. This finding then leads to the question that if you define
Ducci on $\mathbb{Z}_{m}^{n}$ as we did before, are there other $m$ for which
all tuples will vanish besides $m=2$? Our goal is to prove that the answer is
yes when $m=2^{l}$ for some $l\in\mathbb{Z}^{+}$.
Returning to our case of Ducci defined on $\mathbb{Z}_{m}^{n}$, we will start
by discussing more about our example of the Ducci cycle of $(3,1,3)$ to
provide a better visualization of what Ducci sequences in $\mathbb{Z}_{m}^{n}$
look like. We can do this by creating a transition graph that maps all of the
Ducci sequences of $\mathbb{Z}_{4}^{3}$ and direct our focus to the connected
component containing $(3,1,3)$:

$(0,2,2)$$(0,0,2)$$(3,1,3)$$(1,3,1)$$(2,2,0)$$(0,2,0)$$(3,1,1)$$(1,3,3)$$(2,0,2)$$(2,0,0)$$(1,1,3)$$(3,3,1)$

Using our definitions from before, one can see that $Len(3,1,3)=2$,
$Len(0,0,2)=1$, and $Len(0,2,2)=0$. Although $(1,3,1)$ is not in the Ducci
sequence of $(3,1,3)$, we can see that the Ducci sequences of $(3,1,3)$ and
$(1,3,1)$ have the same Ducci cycle, as well as every other tuple, in this
connected component. In addition to this, all of these tuples have period $3$.
Notice that $(0,0,2)$ has two predecessors, $(3,1,3)$ and $(1,3,1)$. But not
all tuples have predecessors; both $(3,1,3)$ and $(1,3,1)$ are examples of
this.
We can also look at the transition graph for $\mathbb{Z}_{4}^{2}$ to give an
example where all Ducci sequences vanish:

$(0,0)$$(1,3)$$(3,1)$$(2,2)$$(0,2)$$(2,0$$(3,3)$$(0,3)$$(3,0)$$(1,2)$$(2,1)$$(1,1)$$(0,1)$$(1,0)$$(2,3)$$(3,2)$

For this graph, all tuples in $\mathbb{Z}_{4}^{2}$ are part of the same
component. Here, we can see that $L_{2}(4)=3$, but not all tuples without
predecessors have this length. For example, $(2,3)$ and $(1,3)$ both do not
have predecessors, but $Len(2,3)=3$ and $Len(1,3)=1$.
Notice that all tuples in $\mathbb{Z}_{4}^{2}$ have 4 predecessors. This is
not a coincidence, in fact

######  Theorem 4.

Let $n$ be even. If a tuple $x=(l_{1},l_{2},...,l_{n})\in\mathbb{Z}_{m}^{n}$
has a predecessor $(k_{1},k_{2},...,k_{n})$, then
$(k_{1}+1,k_{2}-1,...,k_{n-1}+1,k_{n}-1)$ is also a predecessor of $x$.
Moreover, if a tuple $x$ has a predecessor, it has exactly $m$ predecessors.

###### Proof.

Let $l_{i},k_{i}\in\mathbb{Z}_{m}$. Clearly, if $(k_{1},k_{2},...,k_{n})$ is a
predecessor to $(l_{1},l_{2},...,l_{n})$, then so is
$(k_{1}+1,k_{2}-1,...,k_{n-1}+1,k_{n}-1)$. We can then continue to use this
method on $(k_{1}+1,k_{2}-1,...,k_{n-1}+1,k_{n}-1)$ until we see that
$(k_{1}+j,k_{2}-j,...,k_{n}-j)$ is also a predecessor to
$(l_{1},l_{2},...,l_{n})$ for $j\in\mathbb{Z}_{m}$. This will give us at least
m predecessors to $(l_{1},l_{2},...,l_{n})$, each beginning with a different
element in $\mathbb{Z}_{m}$.
We now to confirm that these are all the predecessors of
$(l_{1},l_{2},...,l_{n})$. If we consider two tuples that are predecessors to
$(l_{1},l_{2},...,l_{n})$ that both begin with the same element, say
$(k_{1},k_{2},...,k_{n})$ and $(k_{1},k_{2}^{\prime},...,k_{n}^{\prime})$,
then we have the relations

$k_{1}+k_{2}\equiv l_{1}\;\text{mod}\;m$

and

$k_{1}+k_{2}^{\prime}\equiv l_{1}\;\text{mod}\;m$

which gives us that $k_{2}=k_{2}^{\prime}$. We can repeat this to get
$k_{i}=k_{i}^{\prime}$ for every $i,1\leq i\leq n$ and to see that
$(l_{1},l_{2},...,l_{n})$ only has one unique predecessor beginning with
$k_{1}$ so it has exactly $m$ predecessors. ∎

Let’s now draw our attention back to $D$ specifically. Because

$D((x_{1},x_{2},...,x_{n})+(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime}))=D(x_{1}+x_{1}^{\prime},x_{2}+x_{2}^{\prime},...,x_{n}+x_{n}^{\prime})$
$=(x_{1}+x_{2}+x_{1}^{\prime}+x_{2}^{\prime},x_{2}+x_{3}+x_{2}^{\prime}+x_{3}^{\prime},...,x_{n}+x_{1}+x_{n}^{\prime}+x_{1}^{\prime})$
$=(x_{1}+x_{2},x_{2}+x_{3},...,x_{n}+x_{1})+(x_{1}^{\prime}+x_{2}^{\prime},x_{2}^{\prime}+x_{3}^{\prime},...,x_{n}^{\prime}+x_{1}^{\prime})$
$=D(x_{1},x_{2},...,x_{n})+D(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime})$

it is true that $D\in End(\mathbb{Z}_{m}^{n})$. Notice also that it remains
true that $D(\lambda\mathbf{u})=\lambda D(\mathbf{u})$ when
$\lambda\in\mathbb{Z}_{m}$ and $\mathbf{u}\in\mathbb{Z}_{m}^{n}$. Since this
means $D^{k}(\lambda\mathbf{u})=\lambda D^{k}(\mathbf{u})$ for
$k\in\mathbb{Z}$, the Ducci sequence for $\lambda\mathbf{u}$ is $\\{\lambda
D^{k}(\mathbf{u})\\}_{k=0}^{\infty}$ and if $\mathbf{u}\in
K(\mathbb{Z}_{m}^{n})$, then so is $\lambda\mathbf{u}$.
Define $H:\mathbb{Z}_{m}^{n}\to\mathbb{Z}_{m}^{n}$ as

$H(x_{1},x_{2},...,x_{n})=(x_{2},x_{3},...,x_{n},x_{1})$

$H\in End(\mathbb{Z}_{m}^{n})$ as well and $D=I+H$ where $I$ is the identity
endomorphism. Note that

$H(D(x_{1},x_{2},..,x_{n}))=H(x_{1}+x_{2},x_{2}+x_{3},...,x_{n}+x_{1})$
$=(x_{2}+x_{3},x_{3}+x_{4},...,x_{n}+x_{1},x_{2}+x_{1})$
$=D(x_{2},x_{3},..,x_{n},x_{1})$ $=D(H(x_{1},x_{2},...,x_{n})$

and therefore $H,D$ commute. This means that if
$\\{D^{r}(\mathbf{u})\\}_{r\geq 0}$ is the Ducci sequence for
$\mathbf{u}\in\mathbb{Z}_{m}^{n}$, then $\\{H^{k}(D^{r}(\mathbf{u}))\\}_{r\geq
0}$ is the Ducci sequence for $H^{k}(\mathbf{u})$ when $0\leq k\leq n-1$.
Consequently, if $\mathbf{u}\in K(\mathbb{Z}_{m}^{n})$, then so is
$H^{k}(\mathbf{u})$.

##  3\. Findings on Ducci for $n,m$ Arbitrary

We can now prove Theorem 1 from Section 1. Note that this theorem was stated
as a fact in [1], but a proof is provided here for completeness.

###### Proof of Theorem 1.

We start by showing that
$D^{l}((x_{1},x_{2},...,x_{n})+(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime}))=D^{l}(x_{1},x_{2},...,x_{n})+D^{l}(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime})$
by induction for $l\in\mathbb{Z}^{+}$. This is true for $l=1$ because $D$ is
an endomorphism.

Assume that is true for $l-1$. Then

$D^{l}((x_{1},x_{2},...,x_{n})+(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime}))$
$=D(D^{l-1}((x_{1},x_{2},...,x_{n})+(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime}))$
$=D(D^{l-1}(x_{1},x_{2},...,x_{n})+D^{l-1}(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime}))$
$=D(D^{l-1}(x_{1},x_{2},...,x_{n}))+D(D^{l-1}(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime}))$
$=D^{l}(x_{1},x_{2},...,x_{n})+D^{l}(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime})$

Now suppose that
$(x_{1},x_{2},...,x_{n}),(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime})\in
K(\mathbb{Z}_{m}^{n})$. If $p=P_{m}(n)$, then
$D^{p}(x_{1},x_{2},...,x_{n})=(x_{1},x_{2},...,x_{n})$ and
$D^{p}(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime})=(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime})$.
Therefore,

$D^{p}((x_{1},x_{2},...,x_{n})+(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime}))$
$=D^{p}(x_{1},x_{2},...,x_{n})+D^{p}(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime})$
$=((x_{1},x_{2},...,x_{n})+(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime}))$

and thus
$((x_{1},x_{2},...,x_{n})+(x_{1}^{\prime},x_{2}^{\prime},...,x_{n}^{\prime}))\in
K(\mathbb{Z}_{m}^{n})$. It follows then that
$K(\mathbb{Z}_{m}^{n})\leq\mathbb{Z}_{m}^{n}$ ∎

When examining a Ducci sequence, it is useful to be able to know what
$D^{r}(\mathbf{u})$ is given $\mathbf{u}\in\mathbb{Z}_{m}^{n}$. To do this,
define $a_{r,s}\in\mathbb{Z}$ with $r,s$ integers, $r\geq 0,1\leq s\leq n$ so
that $D^{r}(0,0,...,0,1)=(a_{r,n},a_{r,n-1},...,a_{r,1})$. Notice that for any
tuple $(x_{1},x_{2},...,x_{n})\in\mathbb{Z}_{m}^{n}$, we can write it as

$(x_{1},x_{2},...,x_{n})=\sum_{s=1}^{n}x_{s}H^{-s}(0,0,..,0,1)$

which gives

$D^{r}(x_{1},x_{2},...,x_{n})=\sum_{s=1}^{n}x_{s}H^{-s}(a_{r,n},a_{r,n-1},...,a_{r,1})$
$=\sum_{s=1}^{n}x_{s}(a_{r,s},a_{r,s-1},...,a_{r,s+1})$

Using this, we can now define $a_{r,s}$ as the coefficient on $x_{s-i+1}$ in
the $i$th coordinate of $D^{r}(x_{1},x_{2},...,x_{n})$, with

$a_{0,s}=\begin{cases}0&s\neq 1\\\ 1&s=1\end{cases}$

We now aim to prove a few more observations about $a_{r,s}$.

######  Theorem 5.

Let $r\geq 1$, $1\leq s\leq n$

  1. (1)

$a_{r,s}=a_{r-1,s}+a_{r-1,s-1}$.

  2. (2)

For $r<n$, $a_{r,s}=\binom{r}{s-1}$ .

  3. (3)

$a_{r+t,s}=\sum_{i=1}^{n}a_{t,i}a_{r,s-i+1}$ where $t\geq 0$.

###### Proof.

We will prove all of these via induction.
(1): Letting $i=1$, we only need to consider $a_{r,s}$ as the coefficient on
$x_{s}$ in the first entry of $D^{r}(x_{1},x_{2},...,x_{n})$. Note that by how
$a_{r,s}$ is defined,

$D^{r}(x_{1},x_{2},...,x_{n})$
$=D(a_{r-1,1}x_{1}+a_{r-1,2}x_{2}+\cdots+a_{r,n}x_{n},a_{r-1,1}x_{2}+a_{r-1,2}x_{3}+\cdots+a_{r,n}x_{1},...,a_{r-1,1}x_{n}+a_{r-1,2}x_{1}+\cdots+a_{r,n}x_{n-1})$
$=((a_{r-1,1}+a_{r-1,n})x_{1}+(a_{r-1,2}+a_{r-1,1})x_{2}+\cdots+(a_{r,n}+a_{r,n-1})x_{n},...,$
$(a_{r-1,1}+a_{r-1,n})x_{n}+(a_{r-1,2}+a_{r-1,1})x_{1}+\cdots+(a_{r,n}+a_{r,n-1})x_{n-1})$

by definition,$a_{r,s}$ is the coefficient on $x_{s}$ in the first entry,
which the above tells us is also $a_{r-1,s}+a_{r-1,s-1}$ and (1) follows.
(2): We prove this via induction, noting that the basis case follows from (1)
and

$a_{0,s}=\begin{cases}0&s\neq 1\\\ 1&s=1\end{cases}$

Note that if $s-1>r$, we use the standard notation that this means
$a_{r,s}=\binom{r}{s-1}=0$. Assume $a_{r-1,s}=\binom{r-1}{s-1}$ when $r<n$. By
(1), we have that for $1<s\leq n$,

$a_{r,s}=a_{r-1,s}+a_{r-1,s-1}$ $=\binom{r-1}{s-1}+\binom{r-1}{s-2}$
$=\binom{r}{s-1}$

For $s=1$, we have

$a_{r,1}=a_{r,1}+a_{r,n}$ $=\binom{r-1}{0}+\binom{r-1}{n-1}$ $=1$
$=\binom{r}{0}$

as long as $r<n$.
(3): We prove this via induction on $t$. The basis case includes $t=1,2$.
Basis:

$a_{r,s}=a_{r-1,s}+a_{r-1,s-1}$ $=a_{r-2,s}+2a_{r-2,s-1}+a_{r-2,s-2}$
$=a_{2,1}a_{r-2,s}+a_{2,2}a_{r-2,s-1}+a_{2,3}a_{r-2,s-2}$

Inductive: Assume that for $t^{\prime}<t$,
$a_{r+t^{\prime},s}=\sum_{i=1}^{n}a_{t^{\prime},i}a_{r,s-i+1}$. Then take
$t^{\prime}=t-1$

$a_{r+t,s}=\sum_{i=1}^{n}a_{t-1,i}a_{r+1,s-i+1}$
$=\sum_{i=1}^{n}a_{t-1,i}(a_{r,s-i+1}+a_{r,s-i})$
$=\sum_{i=1}^{n}a_{t-1,i}a_{r,s-i+1}+\sum_{i=1}^{n}a_{t-1,i}a_{r,s-i}$

Note that for the sum $\sum_{i=1}^{n}a_{t-1,i}a_{r,s-i}$ above, we have

$\sum_{i=1}^{n}a_{t-1,i}a_{r,s-i}=\sum_{i=2}^{n+1}a_{t-1,i-1}a_{r,s-i+1}$
$=\sum_{i=1}^{n}a_{t-1,i-1}a_{r,s-i+1}$

because the $s$ coordinate of $a_{r,s}$ is reduced modulo $n$. Therefore,

$a_{r+t,s}=\sum_{i=1}^{n}a_{t-1,i}a_{r,s-i+1}+\sum_{i=1}^{n}a_{t-1,i-1}a_{r,s-i+1}$
$=\sum_{i=1}^{n}(a_{t-1,i}+a_{t-1,i-1})a_{r,s-i+1}$
$=\sum_{i=1}^{n}a_{t,i}a_{r,s-i+1}$

∎

######  Corollary 6.

$a_{n,1}=\begin{cases}\binom{n}{s-1}&s\neq 1\\\ 2&s=1\end{cases}$

###### Proof.

Let $s>1$. Then

$a_{n,s}=a_{n-1,s}+a_{n-1,s-1}$ $=\binom{n-1}{s-1}+\binom{n-1}{s-2}$
$=\binom{n}{s-1}$

For $s=1$,

$a_{n,1}=a_{n-1,1}+a_{n-1,n}$ $=\binom{n-1}{0}+\binom{n-1}{n-1}$ $=2$

∎

######  Lemma 7.

$a_{r,s}=a_{r,r-s+2}$

###### Proof.

We work by induction on $r$.
Basis: We take advantage of Theorem 5 to see that when $r<n$,
$a_{r,s}=\binom{r}{s-1}=\binom{r}{r-s+1}=a_{r,r-s+2}$
Inductive: Suppose that the theorem is true for $r-1$. Then we have
$a_{r-1,s}=a_{r-1,r-s+1}$ and $a_{r-1,s-1}=a_{r-1,r-s+2}$. Then

$a_{r,s}=a_{r-1,s}+a_{r-1,s-1}$ $=a_{r-1,r-s+1}+a_{r-1,r-s+2}$
$=a_{r-1,r-s+2}$

∎

##  4\. All Tuples Vanish in $\mathbb{Z}_{2^{l}}^{2^{k}}$

Recall that it is our main goal to prove Theorem 2. In order to do this, we
will first need a number of lemmas. We will start with a few lemmas that
explore the value of certain binomial coefficients. We believe that lemmas
8-10, 12 are known but we are including the following proofs for the sake of
completeness.

######  Lemma 8.

When $k\geq 2$

$\binom{2^{k}}{2^{k-1}}\equiv 2\;\text{mod}\;4$

###### Proof.

By the Chu-Vandermonde identity [9],

$\binom{2^{k}}{2^{k-1}}=\sum_{j=0}^{2^{k-1}}\binom{2^{k-1}}{j}^{2}=\binom{2^{k-1}}{0}^{2}+\binom{2^{k-1}}{2^{k-1}}^{2}+\sum_{j=1}^{2^{k-1}-1}\binom{2^{k-1}}{j}^{2}$
$\equiv 2\;\text{mod}\;4$

∎

######  Lemma 9.

When $t\neq 0,2^{k-1},2^{k}$ and $k\geq 2$, then

$\binom{2^{k}}{t}\equiv 0\;\text{mod}\;4$

###### Proof.

First to address the cases when $t\in\\{0,2^{k-1},2^{k}\\}$, we know
$\binom{2^{k}}{0}=\binom{2^{k}}{2^{k}}=1$ and $\binom{2^{k}}{2^{k-1}}\equiv
2\;\text{mod}\;4$. We prove the rest of the lemma by induction on $k$.
Basis $\mathbf{k=1}$:

$\binom{4}{1}=\binom{4}{3}=4\equiv 0\;\text{mod}\;4$

Inductive: Assume that $\binom{2^{k-1}}{t}\equiv 0\;\text{mod}\;4$ when $t\neq
0,2^{k-2},2^{k-1}$. We start with when $0<t<2^{k-2}$ and use the Chu-
Vandermonde identity to get

$\binom{2^{k}}{t}=\sum_{j=0}^{t}\binom{2^{k-1}}{j}\binom{2^{k-1}}{t-j}=\binom{2^{k-1}}{0}\binom{2^{k-1}}{t}+\binom{2^{k-1}}{t}\binom{2^{k-1}}{0}+\sum_{j=1}^{t-1}\binom{2^{k-1}}{j}\binom{2^{k-1}}{t-j}\equiv
0\;\text{mod}\;4$

by induction. Next we take $2^{k-2}<t<2^{k-1}$

$\binom{2^{k}}{t}=\binom{2^{k-1}}{0}\binom{2^{k-1}}{t}+\binom{2^{k-1}}{t}\binom{2^{k-1}}{0}+\binom{2^{k-1}}{2^{k-2}}\binom{2^{k-1}}{t-2^{k-2}}+\binom{2^{k-1}}{t-2^{k-2}}\binom{2^{k-1}}{2^{k-2}}$
$+\sum_{\begin{subarray}{c}j=1\\\ j\neq
2^{k-2}\end{subarray}}^{t-1}\binom{2^{k-1}}{j}\binom{2^{k-1}}{t-j}\equiv
0\;\text{mod}\;4$

by induction. Now consider $t=2^{k-2}$

$\binom{2^{k}}{2^{k-2}}=\binom{2^{k-1}}{0}\binom{2^{k-1}}{2^{k-2}}+\binom{2^{k-1}}{2^{k-2}}\binom{2^{k-1}}{0}+\sum_{j=1}^{2^{k-2}-1}\binom{2^{k-1}}{j}\binom{2^{k-1}}{t-j}\equiv
0\;\text{mod}\;4$

by induction and using Lemma 8.
Now suppose that $2^{k-1}<t<2^{k}$. Then $0<2^{k}-t<2^{k-1}$ so
$\binom{2^{k}}{t}=\binom{2^{k}}{2^{k}-t}\equiv 0\;\text{mod}\;4$ and the lemma
follows.

∎

######  Lemma 10.

For $k\geq 2$,

$\binom{2^{k}}{2^{k-1}}\equiv 6\;\text{mod}\;8$

###### Proof.

Basis $\mathbf{k=2}$

$\binom{4}{2}=6\equiv 6\;\text{mod}\;8$

Inductive: Assume $\binom{2^{k-1}}{2^{k-2}}\equiv 6\;\text{mod}\;8$. By the
Chu-Vandermonde Identity,

$\binom{2^{k}}{2^{k-1}}=\sum_{j=0}^{2^{k-1}}\binom{2^{k-1}}{j}^{2}$
$=\binom{2^{k-1}}{0}^{2}+\binom{2^{k-1}}{2^{k-1}}^{2}+\binom{2^{k-1}}{2^{k-2}}^{2}+\sum_{j=1}^{2^{k-1}-1}\binom{2^{k-1}}{j}^{2}$
$\equiv 6\;\text{mod}\;8$

by induction and Lemma 9. ∎

Lemma 11 follows from Lucas’s Theorem, which can be found in many books,
including [10]

######  Lemma 11.

$\binom{2^{k}-1}{t}$ is odd for every $0\leq t\leq 2^{k}-1$.

######  Lemma 12.

For $k\geq 2$

$\binom{2^{k}-1}{2^{k-1}}\equiv 3\;\text{mod}\;4$

###### Proof.

We prove this via induction.
Basis $k=2$: $\binom{3}{2}=3$.
Inductive: Assume that $\binom{2^{k-1}-1}{2^{k-2}}\equiv 3\;\text{mod}\;4$. By
the Chu-Vandermonde Identity, we have

$\binom{2^{k}-1}{2^{k-1}}=\sum_{j=0}^{2^{k-1}}\binom{2^{k-1}}{j}\binom{2^{k-1}-1}{2^{k-1}-j}$
$=\binom{2^{k-1}}{0}\binom{2^{k-1}-1}{2^{k-1}}+\binom{2^{k-1}}{2^{k-1}}\binom{2^{k-1}-1}{0}+\binom{2^{k-1}}{2^{k-2}}\binom{2^{k-1}-1}{2^{k-2}}+\sum_{\begin{subarray}{c}j=1\\\
j\neq
2^{k-2}\end{subarray}}^{2^{k-1}-1}\binom{2^{k-1}}{j}\binom{2^{k-1}-1}{2^{k-1}-j}$
$\equiv 1+(2*3)\text{mod}\;4$ $\equiv 3\;\text{mod}\;4$

and the lemma follows. ∎

We now direct our attention back to the $a_{r,s}$ coefficients and examine
what happens at certain values of $r$. We first note that for $n=2^{k}$,
$a_{r,s+2^{k-1}}=a_{r,s-2^{k-1}}$.

######  Lemma 13.

Let $l\geq 1$ and $n=2^{k}$. Then for every $1\leq s\leq 2^{k}$,

$a_{l2^{k-1},s}+a_{l2^{k-1},s-2^{k-1}}\equiv 0\;\text{mod}\;2^{l}$

###### Proof.

We prove this by induction on $l$.
Basis $\mathbf{l=1}$: We want to show
$a_{2^{k-1},s}+a_{2^{k-1},s-2^{k-1}}\equiv 0\;\text{mod}\;4$. Then for $s\neq
1,2^{k-1}+1$,

$a_{2^{k-1},s}+a_{2^{k-1},s-2^{k-1}}=\binom{2^{k-1}}{s-1}+\binom{2^{k-1}}{s-2^{k-1}-1}\equiv
0\;\text{mod}\;2$

because both of these binomial coefficients are even. For $s=1,2^{k-1}+1$,

$a_{2^{k-1},1}+a_{2^{k-1},1+2^{k-1}}=\binom{2^{k-1}}{0}+\binom{2^{k-1}}{2^{k-1}}\equiv
0\;\text{mod}\;2$

and the basis case follows.

Inductive: Assume $a_{(l-1)2^{k-1},s}+a_{(l-1)2^{k-1},s-2^{k-1}}\equiv
0\;\text{mod}\;2^{l-1}$. Then

$a_{l2^{k-1},s}+a_{l2^{k-1},s-2^{k-1}}=$
$\sum_{i=1}^{2^{k}}[a_{(l-1)2^{k-1},i}a_{2^{k-1},s-i+1}+a_{(l-1)2^{k-1},i}a_{2^{k-1},s-2^{k-1}-i+1}]$

$=\sum_{i=1}^{2^{k-1}}[a_{(l-1)2^{k-1},i}a_{2^{k-1},s-i+1}+a_{(l-1)2^{k-1},i}a_{2^{k-1},s-2^{k-1}-i+1}]$
$+\sum_{i=2^{k-1}+1}^{2^{k}}[a_{(l-1)2^{k-1},i}a_{2^{k-1},s-i+1}+a_{(l-1)2^{k-1},i}a_{2^{k-1},s-2^{k-1}-i+1}]$
$=\sum_{i=1}^{2^{k-1}}[a_{(l-1)2^{k-1},i}a_{2^{k-1},s-i+1}+a_{(l-1)2^{k-1},i}a_{2^{k-1},s-2^{k-1}-i+1}]$
$+\sum_{i=1}^{2^{k-1}}[a_{(l-1)2^{k-1},i-2^{k-1}}a_{2^{k-1},s-i+2^{k-1}+1}+a_{(l-1)2^{k-1},i-2^{k-1}}a_{2^{k-1},s-i+1}]$
$=\sum_{i=1}^{2^{k-1}}(a_{2^{k-1},s-i+1}+a_{2^{k-1},s-i+2^{k-1}+1})(a_{(l-1)2^{k-1},i}+a_{(l-1)2^{k-1},i-2^{k-1}})$

By induction, since $(a_{(l-1)2^{k-1},i}+a_{(l-1)2^{k-1},i-2^{k-1}})\equiv
0\;\text{mod}\;2^{l-1}$, the term in the sum will be $\equiv
0\;\text{mod}\;2^{l}$ if both $a_{2^{k-1},i}$ and $a_{2^{k-1},i-2^{k-1}}$ are
even. The only time this does not happen is when $i=s$ and $i=s+2^{k-1}$. In
which case, the term is

$(a_{2^{k-1},1}+a_{2^{k-1},2^{k-1}+1})(a_{(l-1)2^{k-1},1}+a_{(l-1)2^{k-1},2^{k-1}+1})$
$2(a_{(l-1)2^{k-1},1}+a_{(l-1)2^{k-1},2^{k-1}+1})$ $\equiv
0\;\text{mod}\;2^{l}$

and the lemma follows. ∎

######  Lemma 14.

Let $n=2^{k}$. For $l\geq 3$,

$a_{(l-1)2^{k-1},l2^{k-2}+1}+a_{(l-1)2^{k-1},l2^{k-2}-2^{k-1}+1}\equiv
0\;\text{mod}\;2^{l}$

###### Proof.

Note

$a_{(l-1)2^{k-1},l2^{k-2}+1}+a_{(l-1)2^{k-1},l2^{k-2}-2^{k-1}+1}$
$=\sum_{i=1}^{2^{k}-1}a_{2^{k-1},i}a_{(l-2)2^{k-1},l2^{k-2}+2-i}+a_{2^{k-1},i}a_{(l-2)2^{k-1},l2^{k-2}+2^{k-1}+2-i}$
$=\sum_{i=1}^{2^{k-1}}a_{2^{k-1},i}a_{(l-2)2^{k-1},l2^{k-2}+2-i}+a_{2^{k-1},i}a_{(l-2)2^{k-1},l2^{k-2}+2^{k-1}+2-i}$
$+\sum_{i=2^{k-1}+1}^{2^{k}}a_{2^{k-1},i-2^{k-1}}a_{(l-2)2^{k-1},l2^{k-2}++2^{k-1}2-i}+a_{2^{k-1},i-2^{k-1}}a_{(l-2)2^{k-1},l2^{k-2}+2-i}$
$=\sum_{i=1}^{2^{k-1}}(a_{2^{k-1},i}+a_{2^{k-1},i-2^{k-1}})(a_{(l-2)2^{k-1},l2^{k-2}+2-i}+a_{(l-2)2^{k-1},l2^{k-2}+2^{k-1}+2-i})$

Note that By Lemma 13, the second part of the product in the term is $\equiv
0\;\text{mod}\;2^{l-2}$. Consequently, if the first part is $\equiv
0\;\text{mod}\;4$, then the whole term will be $\equiv 0\;\text{mod}\;2^{l}$.
Note that if $i\neq 1,2^{k-2}+1,2^{k-1}+1,2^{k-1}+2^{k-2}+1$, then both
$a_{2^{k-1},i}$ and $a_{2^{k-1},i-2^{k-1}}$ will be $\equiv 0\;\text{mod}\;4$.
Using this, we are left with

$(a_{2^{k-1},1}+a_{2^{k-1},2^{k-1}+1})(a_{(l-2)2^{k-1},l2^{k-2}+1}+a_{(l-2)2^{k-1}l2^{k-2}+2^{k-1}+1})$
$+(a_{2^{k-1},2^{k-1}+1}+a_{2^{k-1},1})(a_{(l-2)2^{k-1},l2^{k-2}+2^{k-1}+1}+a_{(l-2)2^{k-1},l2^{k-2}+1})$
$+(a_{2^{k-1},2^{k-1}+1}+a_{2^{k-1},2^{k-2}+2^{k-1}+1})(a_{(l-2)2^{k-1},(l-1)2^{k-2}+1}+a_{(l-2)2^{k-1},(l-1)2^{k-2}+2^{k-1}+1})$
$+(a_{2^{k-1},2^{k-2}+2^{k-1}+1}+a_{2^{k-1},2^{k-2}+1})(a_{(l-2)2^{k-1},(l-1)2^{k-2}+2^{k-1}+1}+a_{(l-2)2^{k-1},(l-1)2^{k-2}+1})$

which is $\equiv 0\;\text{mod}\;2^{l}$ by Lemma 13 when $l-2\geq 1$. Hence,
the lemma follows.

∎

###### Proof of Theorem 2.

We will prove this via induction on $l$ with two basis cases. However, we will
often need to use theorems that rely on $k\geq 2$ in our proof, so we will
first prove our theorem for $k=1$ or that $L_{2^{l}}(2)=l+1$ and that
$P_{2^{l}}(2)=1$ for every $l>0$.
Notice that $D^{r}(0,1)=(2^{r-1},2^{r-1})$. This means that
$D^{l+1}(0,1)=(0,0)$ and that $P_{2^{l}}(2)=1$. Since we also have that
$D^{l}(0,1)=(2^{l-1},2^{l-1})\not\equiv(0,0)\;\text{mod}\;2^{l}$,
$L_{2^{l}}(2)=l+1$ follows.
Basis $l=1$: We first need to prove $a_{2^{k},s}\equiv 0\;\text{mod}\;2$ for
all $1\leq s\leq n$. Recall that $a_{2^{k},1}=2$ and for $s\neq 1$,
$a_{2^{k},s}=\binom{2^{k}}{s-1}\equiv 0\;\text{mod}\;2$ which gives us
$P_{2}(2^{k})=1$. Note also that for $r<2^{k}$, $a_{r,1}=1\not\equiv
0\;\text{mod}\;2$. Therefore, this is the first time we have that
$a_{r,s}\equiv 0\;\text{mod}\;2$ for all $s$, so $L_{2}(2^{k})=2^{k}$
Basis $l=2$: We want to show that $L_{4}(2^{k})=3*2^{k-1}$ and
$P_{4}(2^{k})=1$. We begin by showing that $a_{3*2^{k-1},s}\equiv
0\;\text{mod}\;4$ for every s.

$a_{3*2^{k-1},s}=\sum_{i=1}^{2^{k}}a_{2^{k-1},i}a_{2^{k},s-i+1}$
$=a_{2^{k-1},2^{k-1}+1}a_{2^{k},s-2^{k-1}}+a_{2^{k-1},1}a_{2^{k},s}+\sum_{\begin{subarray}{c}i=2\\\
i\neq 2^{k-1}+1\end{subarray}}a_{2^{k-1},i}a_{2^{k},s-i+1}$ $\equiv
a_{2^{k},s-2^{k-1}}+a_{2^{k},s}\;\text{mod}\;4$

because $a_{2^{k-1},i}\equiv 0\;\text{mod}\;2$ for $i\neq 1,2^{k-1}+1$ and
$a_{2^{k},s-i+1}\equiv 0\;\text{mod}\;2$ as shown in our first basis case. We
then have that this is

$\equiv 0\;\text{mod}\;4$

because of Lemma 13. Therefore we have that $P_{4}(2^{k})=1$ and
$L_{4}(2^{k})\leq 3*2^{k-1}$. So we now need to show that there exists $s$
such that $a_{3*2^{k-1}-1,s}\not\equiv 0\;\text{mod}\;4$.

$a_{3*2^{k-1}-1,s}=\sum_{i=1}^{2^{k}}a_{2^{k-1},i}a_{2^{k}-1,s-i+1}$
$=a_{2^{k-1},1}a_{2^{k}-1,s}+a_{2^{k-1},2^{k-1}+1}a_{2^{k}-1,s-2^{k-1}}+a_{2^{k-1},2^{k-2}+1}a_{2^{k}-1,s-2^{k-2}}+\sum_{i\in
J}a_{2^{k-1},i}a_{2^{k}-1,s-i+1}$

Where $J=\\{2\leq i\leq 2^{k}\;|\;i\neq 2^{k-1},2^{k-2}+1\\}$. Notice
$a_{2^{k-1},i}\equiv 0\;\text{mod}\;4$ for every $i\in J$ so the sum over
$i\in J$ is $\equiv 0\;\text{mod}\;4$. If we also take $s=1$, we are left with

$\equiv
a_{2^{k-1},1}a_{2^{k}-1,1}+a_{2^{k-1},2^{k-1}+1}a_{2^{k}-1,2^{k-1}+1}+a_{2^{k-1},2^{k-2}+1}a_{2^{k}-1,2^{k-2}+1}\;\text{mod}\;4$
$\equiv 1+a_{2^{k}-1,2^{k-1}+1}+2\;\text{mod}\;4$

because $a_{2^{k-1},2^{k-2}+1}\equiv 2\;\text{mod}\;4$. From Lemma 12, this is
$\equiv 2\;\text{mod}\;4$ and $a_{3*2^{k-1}-1,1}\not\equiv 0\;\text{mod}\;4$,
so $L_{4}(2^{k})=3*2^{k-1}$

Inductive: Assume that $L_{2^{l-1}}(2^{k})=l2^{k-1}$ and
$P_{2^{l-1}}(2^{k})=1$. This implies that $a_{l2^{k-1},s}\equiv
0\;\text{mod}\;2^{l-1}$ for every $s$.

$a_{(l+1)2^{k-1},s}=\sum_{i=1}^{2^{k}}a_{l2^{k-1},i}a_{2^{k-1},s-i+1}$
$=a_{l2^{k-1},s}a_{2^{k-1},1}+a_{l2^{k-1},s-2^{k-1}}a_{2^{k-1},2^{k-1}+1}+\sum_{i\in
J}a_{l2^{k-1},i}a_{2^{k-1},s-i+1}$ $\equiv
a_{l2^{k-1},s}+a_{l2^{k-1},s-2^{k-1}}\;\text{mod}\;2^{l}$

By induction where $J=\\{1\leq i\leq 2^{k}\;|\;i\neq s,s-2^{k-1}\\}$ and this
is

$\equiv 0\;\text{mod}\;2^{l}$

by Lemma 13. This gives us that $P_{m}(n)=1$ and $L_{m}(n)\leq(l+1)2^{k-1}$.
Now we need to prove that $L_{m}(n)>(l+1)2^{k-1}$. Note that because
$P_{m}(n)=1$, it suffices to show that there exists $s$ such that
$a_{(l+1)2^{k-1},s}\not\equiv 0\;\text{mod}\;2^{l}$. We start by breaking down
$a_{(l+1)2^{k-1},s}$ as follows:

$a_{(l+1)2^{k-1},s}=\sum_{i=1}^{2^{k}}a_{l2^{k-1},i}a_{2^{k-1}-1,s-i+1}$

Because $a_{r,s}=a_{r,r-s+2}$, most coefficients will have another coefficient
that it is equal to. The case where they would not is when $s\equiv
r-s+2\;\text{mod}\;2^{k}$. For our case then, it would be when

$i=l2^{k-1}-i+2$ $i=l2^{k-2}+1$

and

$i=l2^{k-1}-i+2^{k}+2$ $i=l2^{k-2}+2^{k-1}+1$

we will separate these out from our sum so we can view $a_{(l+1)2^{k-1},s}$
like

$a_{(l+1)2^{k-1},s}=a_{l2^{k-1},l2^{k-2}+1}a_{2^{k-1}-1,s-l2^{k-2}+a_{l2^{k}-2},l2^{k-2}+2^{k-1}+1}a_{2^{k-1}-1,s-l2^{k-2}-2^{k-1}}$
$+\sum_{i\in
M}(a_{l2^{k-1},i}a_{2^{k-1},s-i+1}+a_{l2^{k-1},l2^{k-1}-i+2}a_{2^{k-1},s-l2^{k-1}+i-1})$

where $M$ is defined to preserve equality. We now take $s=l2^{k-2}+1$ so we
get

$a_{(l+1)2^{k-1}-1,l2^{k-2}+1}=a_{l2^{k-1},l2^{k-2}+1}a_{2^{k-1}-1,1}+a_{l2^{k-2},l2^{k-2}+2^{k-1}+1}a_{2^{k-1}-1,2^{k-1}+1}$
$+\sum_{i\in
M}(a_{l2^{k-1},i}a_{2^{k-1},l2^{k-2}-i+2}+a_{l2^{k-1},l2^{k-1}-i+2}a_{2^{k-1},l2^{k-2}-l2^{k-1}+i})$

Taking a look at each piece,
$a_{l2^{k-2},l2^{k-2}+2^{k-1}+1}a_{2^{k-1}-1,2^{k-1}+1}=0$ because
$a_{2^{k-1}-1,2^{k-1}+1}=0$. For the sum, this is equal to

$\sum_{i\in
M}a_{l2^{k-1},i}(a_{2^{k-1}-1,l2^{k-2}-i+2}+a_{2^{k-1}-1,l2^{k-2}-l2^{k-1}+i})\equiv
0\;\text{mod}\;2^{l}$

because $a_{l2^{k-1},i}=a_{l2^{k-1},l2^{k-1}-i+2}$ and by Lemma 11,
$a_{2^{k-1}-1,t}$ is odd for all $t$, so
$a_{2^{k-1}-1,l2^{k-2}-i+2}+a_{2^{k-1}-1,l2^{k-2}-l2^{k-1}+i}$ is even. Since
$a_{l2^{k-1},i}\equiv 0\;\text{mod}\;2^{l-1}$, each individual term is $\equiv
0\;\text{mod}\;2^{l}$ so the sum is as well. This leaves us with

$a_{(l+1)2^{k-1},l2^{k-2}+1}\equiv a_{l2^{k-1},l2^{k-2}+1}\;\text{mod}\;2^{l}$

So if we can prove that $a_{l2^{k-1},l2^{k-2}+1}\not\equiv
0\;\text{mod}\;2^{l}$, then the theorem will follow. Since we already know
that $a_{l2^{k-1},l2^{k-2}+1}\equiv 0\;\text{mod}\;2^{l-1}$, we need to prove
the following claim:
Claim: $a_{l2^{k-1},l2^{k-2}+1}\equiv 2^{l-1}\;\text{mod}\;2^{l}$.
We prove this claim via induction $l$. Our basis case is for $l=2$:
$a_{2^{k},2^{k-1}+1}\equiv 2\;\text{mod}\;4$ by Lemma 8 and the basis case
follows.
Assume now that $a_{(l-1)2^{k-1},(l-1)2^{k-2}+1}\equiv
2^{l-2}\;\text{mod}\;2^{l-1}$.

$a_{l2^{k-1},l2^{k-2}+1}=\sum_{i=1}^{2^{k}}a_{(l-1)2^{k-1},i}a_{2^{k-1},l2^{k-2}-i+2}$
$=a_{(l-1)2^{k-1},(l-1)2^{k-2}+1}a_{2^{k-1},2^{k-2}+1}+a_{(l-1)2^{k-1},l2^{k-2}+1}a_{2^{k-1},1}$
$+a_{(l-1)2^{k-1},l2^{k-2}-2^{k-1}+1}a_{2^{k-1},2^{k-1}+1}+\sum_{i\in
J^{*}}a_{(l-1)2^{k-1},i}a_{2^{k-1},l2^{k-2}-i+2}$

Where $J^{*}=\\{1\leq i\leq
2^{k}\;|\;i\neq(l-1)2^{k-2}+1,l2^{k-2}+1,l2^{k-2}-2^{k-1}+1\\}$. Looking at
each piece, $a_{(l-1)2^{k-1},(l-1)2^{k-2}+1}a_{2^{k-1},2^{k-2}+1}\equiv
2^{l-1}\;\text{mod}\;2^{l}$ by induction.

$a_{(l-1)2^{k-1},l2^{k-2}+1}a_{2^{k-1},1}+a_{(l-1)2^{k-1},l2^{k-2}-2^{k-1}+1}a_{2^{k-1},2^{k-1}+1}$
$=a_{(l-1)2^{k-1},l2^{k-2}+1}+a_{(l-1)2^{k-1},l2^{k-2}-2^{k-1}+1}\equiv
0\;\text{mod}\;2^{l}$

By Lemma 14.

$\sum_{i\in J^{*}}a_{(l-1)2^{k-1},i}a_{2^{k-1},l2^{k-2}-i+2}\equiv
0\;\text{mod}\;2^{l}$

because $a_{(l-1)2^{k-1},i}\equiv 0\;\text{mod}\;2^{l-2}$ and
$a_{2^{k-1},l2^{k-2}-i+2}\equiv 0\;\text{mod}\;4$ by Lemma 9. The claim
follows.
Therefore, $a_{(l+1)2^{k-1}-1,l2^{k-2}+1}\equiv 2^{l-1}\;\text{mod}\;2^{l}$
and $L_{m}(n)>(l+1)2^{k-1}-1$. Hence, $L_{m}(n)=(l+1)2^{k-1}$ and the theorem
follows. ∎

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