# Superhumps in the cataclysmic variable BG Tri

S. Y. Stefanov,1,2 G. Latev,1,3 S. Boeva,1 M. Moyseev1,2
1Institute of Astronomy and National Astronomical Observatory, Bulgarian
Academy of Sciences, Tsarigradsko Shose 72, BG-1784 Sofia, Bulgaria
2Department of Astronomy, Sofia University "St. Kliment Ohridski", James
Bourchier 5, BG-1164 Sofia, Bulgaria
3Astronomical Association Sofia, Tsar Asen 49, BG-1463 Sofia, Bulgaria

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a detailed photometric study of the bright cataclysmic variable BG
Tri using ground-based observations mainly from Rozhen Observatory, ASAS-SN,
TESS, and WASP sky surveys. We report the discovery of a negative superhump
with $P_{-sh}=0.1515(2)$ days and a co-existing superorbital variation with
$P=3.94(53)$ days in data from 2019 and 2020. A positive superhump with
$P_{+sh}=0.1727(14)$ days is also discovered in data from 2006. The obtained
negative superhump deficit $\varepsilon_{-}=0.044(1)$ and the positive
superhump excess $\varepsilon_{+}=0.090(9)$ give us an independent photometric
evaluation of the mass ratio $(q)$ of the system, which we find to be
$q_{-}=0.37(2)$ and $q_{+}=0.40(5)$ respectively. We also present a study of
the quasi-periodic oscillations (QPOs) and stochastic variability (flickering)
in BG Tri. The light curves show a rich mixture of simultaneously overlapping
quasi-periods ranging from 5 to 25 minutes. The multi-color $(UBVRI)$
photometric observations from Rozhen Observatory reveal the typical increase
of the flickering amplitudes to the shorter wavelengths. The recently
introduced $A_{60}$ amplitude of the flickering light source in all studied
photometric bands is systematically lower when the negative superhump is gone
in season 2021.

###### keywords:

stars: activity – (stars:) binaries: close – (stars:) novae, cataclysmic
variables – stars: individual: BG Tri

††pubyear: 2022††pagerange: Superhumps in the cataclysmic variable BG
Tri–Superhumps in the cataclysmic variable BG Tri

##  1 Introduction

Cataclysmic variables (CVs) are close binary systems consisting of a Roche-
lobe filling secondary, orbiting an accreting white dwarf primary. The
secondaries in CVs are usually a low mass main-sequence star (most often a red
dwarf, losing matter through the inner Lagrange point $(L_{1})$. The
transferred matter forms an accretion disc around the white dwarf, which is
the primary source of the system’s luminosity (Warner, 2003; Hellier, 2001).

Due to the Roche lobe geometry, the secondary star has a teardrop shape with
the apex pointing towards the primary. This, in combination with the orbital
motion of the system, results in the so-called ellipsoidal modulations
(fainter for low-inclination binaries). Detecting these is a direct way to
determine the orbital period of the cataclysmic variables (Hellier, 2001).
Along with the orbital periodicity, some CVs display periodic changes in
brightness with periods close to the orbital period called superhumps. This
phenomenon is commonly seen and comprehensively studied in SU UMa stars -
subclass of dwarf novae CVs (e.g., Hirose & Osaki, 1990; Kato et al., 2009,
2017). It is believed that a precessing accretion disc can cause such
photometric behaviour. Superhumps can either be positive (apsidal - a few
percent longer than the orbital period $(P_{orb})$, or negative (nodal) - a
few percent shorter. Superhumps can also appear in other sub-classes of CVs.
Reports for 15 nova-likes (NLs - a subclass of CVs in the permanently high
photometric state) with positive and 11 NLs with negative superhumps can be
found in the literature (e.g., Pavlenko et al., 2020; Ritter & Kolb, 2005;
Castro Segura et al., 2021; Iłkiewicz et al., 2021). The existence of both
periodicities has been observed in some systems - TT Ari (Belova et al., 2013;
Kraicheva et al., 1999), AQ Men (Iłkiewicz et al., 2021), and LS Cam (Rawat et
al., 2022; Stefanov, 2021). Positive superhumps are believed to appear when
tidal stresses from the secondary affect the disc and cause apsidal precession
in the prograde direction (e.g., Lubow, 1991). Negative superhumps can be
explained by a retrograde nodal precession of a tilted accretion disc. The
stream impacts the face of the disc and the retrograde precession causes
periods shorter than Porb (e.g., Wood et al., 2009; Montgomery, 2009).

BG Tri was discovered and classified as a cataclysmic variable by Khruslov
(2008). Although the system is a bright CV (11.9 mag in $V$), it is poorly
studied. Other mentions in the literature include a cross-association with
X-ray sources by Haakonsen & Rutledge (2009). In Makarov (2017), BG Tri is
shown to have significant excess in near UV. In 2018 the star dropped in
brightness by 2.5 mag in V and was classified as a VY Scl subclass by Kato
(2018). The variables in this subclass of nova-like CVs are predominately in
high photometric state except for rare brightness drops up to 6 magnitudes
lasting from a few weeks to a few years.

The first comprehensive study of BG Tri was published by Hernández et al.
(2021), where the system is shown to have the spectroscopic characteristics of
RW Sex-type and similar nova-like variables. These variables a have high mass
transfer rates and bright accretion discs. The authors estimate the
inclination angle as $i=25(5)^{\circ}$, the mass of the primary $0.8M_{\sun}$,
the mass of the secondary $0.3M_{\sun}$, and the binary separation -
$1.23R_{\sun}$. The orbital period is measured to be 3.8028(24) hours.

Here we present photometric data of BG Tri from ground-based observations and
sky surveys. We report the detection of superhump variability for the first
time in BG Tri. In addition, we analyse the long and short-term variability -
superorbital periodicity, quasi-periodic oscillations (QPOs), and flickering.

##  2 Observations, data reduction, and photometry

Our first photometric data were gathered during the 2020 summer school of
astronomy and astrophysics "Beli Brezi" with a 25 cm Newton telescope
($FOV=51^{\prime}\times 39^{\prime}$) equipped with a QHY9 CCD ($5.4\mu m$
$3358\times 2536$ pix) camera and standard Johnson-Cousins $UBVRI$ filters.
Later, in the period from August 2020 to November 2021 several observations in
the $UBVRI$ bands were carried out at Rozhen National Astronomical Observatory
(NAO) in Bulgaria on the 50/70 cm Schmidt telescope ($FOV=73^{\prime}\times
73^{\prime}$) equipped with FLI PL16803 CCD ($9\mu m$ square $4096\times 4096$
pix) and the 2.0 m RCC telescope ($FOV=10^{\prime}\times 10^{\prime}$) with
two-channel focal reducer FoReRo2 (Jockers et al., 2000), attached to the
Ritchey–Chrétien focus and equipped with two ANDOR iKON-L CCDs ($13.5\mu m$
square $2048\times 2048$ pix). One additional light curve was obtained on the
PWI SDK 17" (43.2 cm corrected Dall-Kirkham) telescope ($FOV=43^{\prime}\times
43^{\prime}$) at AndromedA Observatory - equipped with Alta Apogee U16M CCD
($9\mu m$ square $4096\times 4096$ pix). A total of 70 hours of simultaneous
and quasi-simultaneous observations in two to five bands were acquired. The
majority of observing runs were longer than two hours. A journal of the
observations is presented in Table 1.

Table 1: Journal of observtions Date | Length (hr) | Filters | Telescope
---|---|---|---
2020/08/11 | 2.34 | U B V R I | 25 cm Newton
2020/08/17 | 3.72 | B V | 25 cm Newton
2020/08/22 | 1.56 | U B V R I | 50/70 cm Schmidt
2020/09/22 | 4.99 | U B V R I | 50/70 cm Schmidt
2020/09/23 | 5.88 | U B V R I | 50/70 cm Schmidt
2020/09/24 | 3.14 | U B V R I | 50/70 cm Schmidt
2020/09/25 | 8.34 | B V | 50/70 cm Schmidt
2020/10/13 | 1.84 | U V | 2.0 m RCC
2020/10/14 | 4.48 | U V | 2.0 m RCC
2020/10/14 | 0.74 | B V R I | 50/70 cm Schmidt
2020/12/21 | 0.53 | U V | 2.0 m RCC
2021/01/18 | 3.61 | U V | 2.0 m RCC
2021/01/18 | 2.41 | B R I | 50/70 cm Schmidt
2021/01/20 | 4.25 | U B V R I | 50/70 cm Schmidt
2021/10/04 | 1.80 | U B V R I | 50/70 cm Schmidt
2021/10/06 | 1.50 | U B V R I | 50/70 cm Schmidt
2021/10/30 | 4.11 | V | 43.2 cm Dall-Kirkham
2021/11/06 | 8.21 | V | 2.0 m RCC
2021/11/07 | 2.45 | V | 2.0 m RCC
2021/11/07 | 2.82 | U B V R I | 50/70 cm Schmidt
2021/11/08 | 2.82 | U B V R I | 50/70 cm Schmidt
Figure 1: All light curves in V band listed in Table 1. The calendar date and
Julian date of each observation are in the right side of the panels. The light
curves are shown in relative magnitude (average subtracted) for clarity.

All photometric data were dark or bias subtracted and flat-fielded. To extract
the stellar magnitudes, standard aperture photometry was applied using two to
five comparison stars depending on the angular size of the frame. The gathered
light curves are shown in Figure 1.

In addition to our photometry, data from several sky surveys was also analysed
- The All-Sky Automated Survey for Supernovae (ASAS-SN) (Shappee et al., 2014;
Kochanek et al., 2017), Catalina Real-Time Transient Survey (CRTS) (Drake et
al., 2009), Wide Angle Search for Planets (WASP) (Butters et al., 2010), and
the Northern Sky Variability Survey (NSVS) (Woźniak et al., 2004). A combined
light curve is shown in Figure 2.

Figure 2: The long term behaviour of BG Tri in 1999-2021. This light curve is
built with data from several sky surveys, but the magnitudes are not
transformed to a single band. The color key is as follows - NSVS magnitudes
(orange), WASP magnitudes (purple), CRTS magnitudes (red), ASAS-SN $V$ band
magnitudes (green) and ASAS-SN $g$ band magnitudes (blue).

Photometry of BG Tri with 1426s cadence from sector 17 of the Transiting
Exoplanet Survey Satellite TESS (Ricker et al., 2015) was acquired using the
Python package "Lightkurve" (Lightkurve Collaboration et al., 2018) and its
dependencies (Brasseur et al., 2019; Ginsburg et al., 2019; Astropy
Collaboration et al., 2013, 2018). Custom aperture photometry of the TESS data
was performed using this package. The resulting light curve is shown in the
top left panel of Figure 4.

##  3 Data analysis, results, and discussion

Sky survey data span 22 years. The mean magnitude of the star in the high
state is $\sim$ 11.9 mag in $V$ band and varies with an amplitude of $\sim$
0.5 mag. Only one observed low state was detected in 2018-2019. The decline
and rise rates are described in detail in Hernández et al. (2021). The drop in
brightness with 2.5 mag is typical for VY Scl type CVs, but such amplitude
shows that the star has not reached a deep minimum of brightness as the ones
observed in MV Lyr, TT Ari, and KR Aur (e.g. Leach & Hessman, 1999). The data
from ASAS-SN and CRTS was converted to V band, joined, and then sent to
periodogram analysis. No significant intrinsic large-scale periodicity was
found in this data set.

###  3.1 Search for superhumps

Our first two observations of BG Tri in August 2020 showed light curves with
large amplitude variations. The subsequent multi-band monitoring of the star
reveals variations in brightness within four hours. The expected ellipsoidal
variations would produce double orbital frequency modulation and the secondary
shows no significant contribution to the total flux (Hernández et al., 2021).
Another argument against this interpretation of the variations is that due to
the low system inclination, $i=25(5)^{\circ}$, the system is visible face-on,
and these would be insufficient to modulate the light curves. Thus, the
obvious 4-hour variations may be evidence for the existence of superhumps. To
further study this, several long consequent observational runs were carried
out at NAO Rozhen. The obtained data are presented in Figure 1. All data
points were analysed using the Lomb-Scargle (Lomb, 1976; Scargle, 1982) and
CLEAN algorithms (Roberts et al., 1987). A strictly periodic signal with
$P=0.1516(2)$ days was detected in 2020 light curves. This period is slightly
shorter than the orbital period of the system and represents a negative
superhump. The next step was to analyse the available sky survey data. They
were separated in seasons of visibility - BG Tri is visible in the sky from
June to February. The observing season is labeled by the year of the beginning
of observations, even if some measurements were made in the first months of
the next year. The seasons with enough data points were analysed with the same
period search algorithms cited above. Strong periodic signals are found in
seasons 2006, 2019 and 2020. The resulting power spectra are shown in Figure
3. Season 2006 is entirely made of photometry from the WASP sky survey,
modulated with $P=0.1727(14)$ days - 9$\%$ longer than the orbital period - a
positive superhump. The ASAS-SN photometric data from 2019 and 2020 are in $g$
band. Both seasons contain a periodic variation with $P=0.1515(2)$ days, a
negative superhump 5$\%$ shorter than Porb. All measured periods and their
uncertainties are shown in Table 2. We did not find significant evidence for
the presence of superhumps during the observations in 2017 and 2018 of
Hernández et al. (2021).

Figure 3: Normalised power spectra of the observing seasons containing significant periodic signals. Light curves and power spectra are shown on the right and left respectively. The top two rows contain data form ASAS-SN. The bottom row shows data from the WASP survey. The CLEAN periodograms are shown in blue and Lomb-Scargle in black. The red dashed vertical line is the orbital frequency of BG Tri. Table 2: Significant periods from the power spectra. Data set | Period [days]
---|---
NAO Rozhen season 2020 | 0.1516(2)
ASAS SN season 2020 | 0.1515(2)
ASAS SN season 2019 | 0.1514(2)
TESS sector 17 2019 | 0.1522(8)
TESS sector 17 2019 | 3.94(53)
WASP season 2006 | 0.1727(14)

TESS observations provide continuous photometry of BG Tri from 07.10.2019 to
02.11.2019, with two gaps - one due to data transfer during perigee and one
due to low-quality data. The periodogram analysis shown in Figure 4 reveals
two significant periodicities:

  1. 1.

Negative superhump with $P=0.1522(8)$ days. The superhump period from ASAS-SN
data of the same season has a matching period within the uncertainty range.

  2. 2.

Superorbital variation with $P=3.94(53)$ days. This variation is not present
in ASAS-SN data, likely due to the limited amount of data and low time
resolution.

Figure 4: Top: The extracted TESS light curve of BG Tri and its Lomb-Scargle
power spectrum. The red vertical line represents the orbital frequency of the
binary. Bottom: Data phased with $P=0.152$ days on the left and with $P=3.94$
days on the right.

###  3.2 The negative superhumps of BG Tri

Negative superhumps are interpreted as a result of the gas stream sweeping
across the face of a tilted disc. In a non-tilted disc, the disc overflow is
the same on both sides, but in a slightly tilted disc, the overflow would be
bigger on one side. If the disc is precessing, the overflow would then be
modulated with the beat period of the disc precession and the orbital period
of the binary, thus, creating the negative superhump. If this is the case, we
would expect the precession period of the disc $P_{prec}$ to be:

$\dfrac{1}{P_{prec}}=\dfrac{1}{P_{-sh}}-\dfrac{1}{P_{orb}}.$ (1)

Using the negative superhump period from TESS sector 17 ($P_{-sh}=0.1522(8)$
d) and the orbital period $P_{orb}=0.15845(10)$ d from Hernández et al.
(2021), the expected value for $P_{prec}$ is $3.8(5)$ days. The other major
periodicity found in the TESS data is $P=3.94(53)$ days, co-existing with the
negative superhump period. This we interpret as the precession period of the
tilted accretion disc. The precession phase curve is on the bottom right panel
of Figure 4. The simultaneous presence of negative superhumps and superorbital
period in TESS the data is good evidence that both result from a tilted
precessing disc.

Our multi-color observations show the full amplitudes of the negative
superhump in the five bands to be - $\Delta U=0.30$ mag, $\Delta B=0.28$,
$\Delta V=0.24$, $\Delta R=0.22$, and $\Delta I=0.20$ mag. The $B$ and $V$
magnitudes of the used comparison stars are taken from Henden et al. (2015).
To account for the interstellar extinction, we used $E(B-V)=0.03$ from Green
et al. (2015). The $(B-V)_{0}$ color of BG Tri phased with the negative
superhump period is shown in Figure 5. This $(B-V)_{0}$ phase curve is created
using phase-binned photometry in the two bands from multiple nights with an
equal cadence. Its bluest peak corresponds to the maximum brightness of the
superhump. This result is similar to the color index behaviour of ER UMa
(Imada et al., 2018) and differs from the case of MASTER J1727, where the
bluest peak coincides with the ascending branch of the phase curve (Pavlenko
et al., 2019).

Figure 5: The upper panel shows the binned phase curve (25 bins) of the
negative superhump in the dereddened $B$ with blue dots and in dereddened $V$
with green dots. The error bars represent the standard error of the data in
each bin. The bottom panel is the resulting $(B-V)_{0}$ color for each bin.

###  3.3 Disc tilt

The tilt angle of the disc is a function of the phase curve amplitude of the
nodal precession variability. As the disc precesses, its projected area
increases and decreases periodically, thus creating the superorbital
variability. Most systems that exhibit this type of disc precession have tilts
of a few degrees (e.g., Kimura et al., 2020; Kimura & Osaki, 2021). BG Tri’s
disc tilt can be estimated using the amplitude-tilt relation derived in Smak
(2009):

$\delta=A*52.770\dfrac{(1-u+u\cos{i})\cos{i}}{(1-u+2u\cos{i})\sin{i}},$ (2)

where $u$ is the limb darkening coefficient, $i$ is the system inclination,
and $A$ is the semi-amplitude in magnitudes. Using $u=0.6$, $i=25^{\circ}$
from Hernández et al. (2021), and $A=0.05$ mag, measured by fitting a sine
curve to the phase plot of the precession variability (bottom right panel on
Fig. 4), a value of $\delta\approx 3^{\circ}$ is obtained. This is the typical
value for disc tilts in CVs that display negative superhumps (Smak, 2009).

###  3.4 The positive superhumps of BG Tri

Positive superhumps are interpreted as a result of the apsidal precession of
an elliptical accretion disc, driven by a 3:1 resonance (Whitehurst, 1988;
Hirose & Osaki, 1990; Lubow, 1991). The resonance induces tidal deformations
in the disc, which manifest as brightness variations due to periodic heating.
Positive superhumps in BG Tri are discovered in archival WASP data from 2006.
This is the only dataset with detected positive superhumps. Phase curve of the
data folded with the most significant period - $P_{+sh}=0.1727(14)$ days is
shown in Figure 6.

Figure 6: Phase curve of the positive superhump of BG Tri - the WASP data from
2006 folded with $P=0.1727$ days.

###  3.5 System parameters

Superhumps can be used to reveal the mass ratio $q=M_{2}/M_{1}$ using the
excess/deficit of the positive/negative superhump:

$\varepsilon_{+/-}=\dfrac{P_{+/-sh}-P_{orb}}{P_{orb}}.$ (3)

For BG Tri the values of $\varepsilon_{-}$ and $\varepsilon_{+}$ are
$0.044(1)$ and $0.090(9)$ respectively. The ratio
$\varepsilon_{+}/\varepsilon_{-}=2.0(2)$ is in agreement with the value of
$\sim 2$ for systems that show both types of superhumps (Patterson et al.,
1997; Wood et al., 2009). The excess and deficit are shown to correlate with
$q$. In Wood et al. (2009), relation between $\varepsilon_{-}$ and $q$ is
derived from particle simulations:

$q=-0.192|\varepsilon_{-}|^{1/2}+10.37|\varepsilon_{-}|-99.83|\varepsilon_{-}|^{3/2}+451.1|\varepsilon_{-}|^{2}.$
(4)

This relation gives a value of $q=0.37(2)$. For the positive superhump, we
used the experimentally derived relation $q(\varepsilon_{+})$ in Kato (2022),
assuming that the measured $\varepsilon_{+}$ is a stage B superhump excess.
This gives us again $q=0.40(5)$. The uncertainty estimation for $q$ is done
using only errors from the periodogram analysis. These relations provide
values for $q$ that are in agreement with Hernández et al. (2021), where the
WD mass was assumed to be $0.8M_{\sun}$ since this is the average WD mass for
binaries above the period gap (Zorotovic et al., 2011). With the average value
of the system’s mass ratio $q=0.39(3)$ (averaged by the positive and the
negative superhump estimations) and mass of the secondary
$0.35M_{\odot}$111The same result can be obtained with the mass-radius
relation in Demircan & Kahraman (1991), using the size of the secondary
$R_{2}=0.40R_{\odot}$ by Eq. (2.100) in Warner (2003). from the mass-period
relationship in Warner (2003), we calculate the mass of the white dwarf
$M_{1}=0.91(7)M_{\odot}$. Using these parameters, the binary separation is
$a=1.33(2)R_{\odot}$ and the truncation radius of the accretion disc222 Given
by Eq. (2.61) $r_{d}=0.60a/(1+q)$ in Warner (2003). is
$r_{d}=0.58(2)R_{\odot}$. We note that the primary Roche lobe radius derived
by Eggleton (1983) is $r_{L}=0.62(1)R_{\odot}$, which means that the accretion
disc in BG Tri fills about 93% of the primary Roche lobe. This is in good
agreement with the maximum size of the discs in non-magnetic nova-like CVs
(Harrop-Allin & Warner, 1996).

###  3.6 Quasi-periodic oscillations and flickering

Fast variations in brightness in timescales of seconds to minutes are typical
for cataclysmic variables. Their amplitudes are the largest in $U$ band and
can vary from a few hundredths to a few tenths of a magnitude. They are
erratic and highly unstable and can change significantly from night to night.
QPOs are generally observed in systems with high transfer rates and are
believed to be originating from the inner accretion disc. To study these
phenomena and compare BG Tri with other similar systems, the approach
presented by Bruch (2021) was followed. A Savitzky-Golay filter (Savitzky &
Golay, 1964) with a cutoff time scale $\Delta t=60$ min and a $4^{th}$ degree
smoothing polynomial was applied and subtracted from all light curves in the
$V$ band longer than 2 hours. Then, the weighted wavelet Z-transform (WWZ)
method by Foster (1996) was used to detect and display the evolution of the
QPOs in BG Tri. A sample of the acquired two-dimensional spectra are shown in
Figure 8. Typical QPO periods for BG Tri are in the range of 5-25 min, similar
to other CVs - e.g., TT Ari and V729 Sgr (Bruch, 2019; Sun et al., 2022). A
summary of the WWZ analysis for all selected nights is shown in Table 4.
Distribution of the detected QPO periods is shown in Figure 9. Overlapping of
multiple oscillations with different periods is also present in some of the
data. Most of them remain coherent for 3 to more than 10 cycles. Their period
and stability vary from night to night - there were no QPOs lasting for more
than 3 cycles seen on 14 Oct 2020, but the 14 minute QPO seen in the light
curve from 25 Sep 2020 lasts for 11 cycles. Figure 7 shows this behaviour with
fitted sines to the QPOs in the data. Sudden changes in period similar to the
ones observed in TT Ari by Kraicheva et al. (1999) are also displayed by BG
Tri.

If we suppose that the QPOs in BG Tri occur in dynamical time scale
$(t_{dyn})$ related to the Keplerian rotation around the primary, we could
determine the possible locations of the observed variations. For $t_{dyn}$ we
use the scaled relation in Sokoloski (2003):

$t_{dyn}=4s\left(\dfrac{r}{10^{9}cm}\right)^{3/2}\left(\dfrac{M_{WD}}{0.6M_{\odot}}\right)^{-1/2}.$
(5)

Here $r$ is the distance from the primary (white dwarf) and $M_{WD}$ is the
mass of the primary. Replacing the values of $M_{WD}$ and $r_{d}$ from Section
3.5 into Eq. 5 and taking into account their uncertainties, for $t_{dyn}$ we
obtain: $t_{dyn}=14(1)$ min. This result makes it possible to assume that the
observed QPOs in the range of $12-14$ minutes are connected with the outer
disc rim, or the overflow of matter from the back side of the disc in BG Tri
proposed by Hernández et al. (2021). The full range of the detected QPOs
($5-25$min.) corresponds to the range of distances
$R=0.28(1)-0.82(3)R_{\odot}$ from the white dwarf. The origin of the fastest
ones may be related to the formation of ’clumps’ or turbulent eddies in the
process of transporting matter through the disc.

Figure 7: Sinusoidal fits using periods from the spectral analysis of some of the light curves. Figure 8: Two-dimensional power spectra of three nights using WWZ. Table 3: Average values of the flickering amplitude $A_{60}$ measured in the $UBVRI$ bands for BG Tri. The numbers in brackets are the standard deviation of $A_{60}$ derived from different light curves and the number of light curves upon which the average values are based. Data set | $A_{60}(U)$ | $A_{60}(B)$ | $A_{60}(V)$ | $A_{60}(R)$ | $A_{60}(I)$
---|---|---|---|---|---
Season 2020 | 0.13 (0.02) (7) | 0.10 (0.02) (8) | 0.09 (0.02) (8) | 0.09 (0.01) (6) | 0.08 (0.01) (6)
Season 2021 | 0.06 (0.01) (3) | 0.041 (0.004) (3) | 0.04 (0.01) (6) | 0.052 (0.008) (3) | 0.050 (0.008) (3)
Table 4: Summary of the quasi-periodic oscillations detected in BG Tri. Date | P [minutes] | Date | P [minutes]
---|---|---|---
2020/09/22 | 17.3($\pm$7) | 2021/01/18 | 6.5 ($\pm$1.5)
| 15.8($\pm$7) |  | 12 ($\pm$1)
| 14.5($\pm$7) |  | 9.5 ($\pm$1.1)
| 12.8($\pm$9) |  | 14.3($\pm$1.3)
| 10.7($\pm$2) |  | 4.4 ($\pm$0.6)
2020/09/23 | 18 ($\pm$1) | 2021/10/30 | 20.4($\pm$4.6)
| 16 ($\pm$1) |  | 13.3($\pm$1.1)
| 23.5($\pm$2.5) |  | 3.7 ($\pm$0.6)
2020/09/24 | 21.5($\pm$3.5) | 2021/11/06 | 19.4($\pm$1.3)
| 10.1($\pm$1.6) |  | 22.5($\pm$1.2)
| 6.8 ($\pm$1.3) |  | 11.9($\pm$2.2)
2020/09/25 | 13.5($\pm$0.9) |  | 7.2 ($\pm$0.7)
| 15.6($\pm$1.6) |  | 6.2 ($\pm$0.6)
| 26.6($\pm$2.6) |  | 5.2 ($\pm$0.3)
| 19.2($\pm$1.4) | 2021/11/07 | 16.2($\pm$2.7)
| 6.1 ($\pm$0.8) |  | 23.9($\pm$3.6)
2020/10/14 | 19.3($\pm$3.4) |  | 10.2($\pm$2)
| 10 ($\pm$1) | 2021/11/08 | 18.4($\pm$1.1)
| 12 ($\pm$1) |  | 5.1 ($\pm$0.9)
Figure 9: Distribution of the QPO periods found with the WWZ method.

In his study of flickering in CVs, Bruch (2021) defines the $A_{60}$ amplitude
of flickering. After applying the Savitsky-Golay filter as discussed above,
the magnitude distribution of each light curve is fitted with a normal
distribution, and its FWHM is the $A_{60}$ amplitude. All light curve data
were binned to the same cadence before $A_{60}$ amplitudes were measured. The
contribution of the secondary is insignificant (Hernández et al., 2021) and no
correction was applied. A minimum of 25 hours of high cadence photometric data
in all five bands were acquired during season 2020 on site observations. BG
Tri was in a negative superhump regime on all of these nights. Our data from
this season shows that the $A_{60}$ amplitudes are highest in $U$ band - 0.13
mag. The amplitudes in the other filters are similar but decrease to the red
bands with $\Delta R$ and $\Delta I$ being 0.09 and 0.08 mag, respectively. In
the following season 2021, our observations consist of minimum 8 hours of high
cadence data in the five bands. During this season the negative superhump is
not present in the light curves, and the $A_{60}$ amplitudes in all bands are
half as low those measured in 2020. The amplitudes during season 2020 are
similar to other VY Scl sub-type stars and during season 2021 resemble those
of UX UMa sub-type (Bruch, 2021). The measured amplitudes from both seasons
are shown in Table 3.

##  4 Conclusions

In this work, we present the first photometric study of the bright cataclysmic
variable BG Triangulum. Our light curve analysis yields the following results:

  * •

Positive superhumps are discovered in data from the WASP sky survey. They
appear only in 2006 data with period $P_{+sh}=0.1727(14)$ days. This gives a
value of the excess $\varepsilon_{+}=0.090(9)$. Using this excess, the mass
ratio $q$ is estimated to be $q_{+}=0.40(5)$.

  * •

Negative superhumps are discovered in our observations from season 2020 and in
data from the ASAS-SN sky survey. They appear in season 2019 after an year-
long low VY-Scl state and disappear in season 2021. During these two years, we
did not find a significant change in the superhump period -
$P_{-sh}=0.1515(2)$ days. This gives a superhump deficit
$\varepsilon_{-}=0.044(1)$ and $q_{-}=0.37(2)$.

  * •

A superorbital variation is present in photometry from the TESS mission. It
has a period of $3.94(53)$ days and an amplitude of 0.05 mag. Using this
amplitude, we estimate the tilt of the disc of BG Tri to be $\approx
3^{\circ}$.

  * •

Our study of the quasi-periodic oscillations of BG Tri shows that the most
common quasi-periods are in the range of $5-25$ min. During our observations
in season 2020, BG Tri was in a negative superhump regime, and we find the
amplitude to be the highest in $U$ band $(\Delta U=0.13$ mag) and decrease to
the red bands $(\Delta I=0.08$ mag). In season 2021 when the superhump is
gone, the amplitudes of the QPOs are systematically lower $(\Delta U=0.06$
mag, and in $I$ band is $0.05$ mag).

BG Tri is a good candidate for the study of CV accretion discs. Its relative
brightness in high state allows it to be a great target even for amateur
astronomers. The system should be monitored in the future to study the nature
of superhump appearance and disappearance.

## Acknowledgements

We are grateful to the organizers of the "Beli Brezi" summer school of
astronomy and the team working with the 25 cm Newton telescope. We also wish
to thank the team of AndromedA Observatory for the data they kindly provided
us. This study is using publicly available data from the TESS mission, ASAS-
SN, WASP, CRTS, and NSVS. We express our gratitude to the teams of these sky
surveys for making their data public.

This study is supported by the grant K$\Pi$-6-H28/2 - Binary stars with
compact object (Bulgarian national science fund).

We appreciate the anonymous reviewer’s valuable comments and suggestions,
which helped us to improve the quality of the manuscript.

## Data availability

The non-public data underlying this article will be shared on reasonable
request to the corresponding authors.

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#  $p$-Adic Scaling Set and Generalized Scaling Set

Debasis Haldar  Department of Mathematics
NIT Rourkela
Rourkela 769008
India<EMAIL_ADDRESS>
###### Abstract.

The main goal of this paper is to develop the MRA theory along with wavelet
theory in $L^{2}(\mathbb{Q}_{p})$. Generalized scaling sets are important in
wavelet theory because it determine multiwavelet sets. Although the theory of
scaling set and generalized scaling set on $\mathbb{R}$ and many other local
field of positive characteristic are available but not on $\mathbb{Q}_{p}$.
This article contains discussion of some necessary conditions of scaling set
and characterize generalized scaling set with examples.

###### Key words and phrases:

$p$-adic, scaling set, multiwavelet set, generalized scaling set

###### 2010 Mathematics Subject Classification:

42C40, 11F85

##  1\. Introduction

Wavelet analysis is consider as one of the pillar of modern harmonic analysis
since last three decades. It is integral part of digital signal processing and
image processing. Moreover, it also has wide applications in statistical data
analysis [1], nonlinear dynamics [4, 10], spectral analysis [19], quantum
field theory [5], pseudo-differential operator theory [14, 16, 18], to name a
few. As wavelets are localised in time and frequency, it is useful where
classical Fourier analysis is not effective. Mathematically, multiwavelets are
some functions whose dilates and translates give some kind of basis,
preferably an orthonormal basis.

First example of one dimensional $p$-adic multiwavelet was given by Kozyrev
[19]. It is haar type. In that paper, he also established a relationship
between wavelet analysis and spectral analysis by showing those multiwavelets
are eigen functions of the Vladimirov operator. Later, Khrennikov and
Shelkovich [15, 16] extended the example of $p$-adic multiwavelet given by
Kozyrev and studied its connection with spectral analysis and cauchy problem
in PDE. Their multiwavelet is non-haar type.

Albeverio et al. [3] gave an example of $n$-dimensional $p$-adic multiwavelet
by direct multiplying Kozyrev’s multiwavelet. Moreover, Kozyrev’s multiwavelet
was extended to the ultrametric spaces [13, 14, 18].

J. J. Benedetto and R. L. Benedetto [6, 7] suggested a method for multiwavelet
based on multiwavelet set on the locally compact abelian groups having compact
open subgroup. As $\mathbb{Q}_{p}$ does not have any discrete subgroup, they
expressed their doubt on the development of $p$-adic multiwavelet theory
through MRA. However, Kozyrev’s multiwavelet can be constructed in the
Benedettos’ framework.

Shelkovich and Skopina [21] initiated development of $p$-adic MRA and wavelet
theory. They have given example of scaling set and multiwavelet on
$L^{2}(\mathbb{Q}_{2})$. Khrennikov et al. [17] developed $p$-adic MRA theory
by describing a wide class of $p$-adic refinement equations generating MRA.
Also, Albeverio et al. [2] parallelly developed $p$-adic MRA theory. They
showed 1-periodic test functions only be considered as scaling functions and
those generate haar MRA. Evdokimov and Skopina [9] showed that any band-
limited (periodic) multiwavelet is nothing but a modification of the Kozyrev’s
multiwavelet.

The organization of the present paper is as follows. In Section 2, we discuss
preliminary results related to $p$-adic number field $\mathbb{Q}_{p}$ and
$\mathbb{C}$-valued function defined on it. Section 3 deals with scaling set.
Finally, characterization and examples of generalized scaling set are given in
Section 4.

##  2\. Preliminaries

Let us make here a brief review of $p$-adic analysis. We refer [20, 22, 23]
for details and proof of the results in this section. $\mathbb{Q}$ and
$\mathbb{C}$ denote the set of rational numbers and the set of complex numbers
respectively. For a prime $p$, the $p$-adic norm $|\cdot|_{p}$ on $\mathbb{Q}$
is defined as follows

$|x|_{p}=\begin{cases}0&\text{ if }x=0,\\\ p^{-\gamma}&\text{ if
}x=p^{\gamma}\frac{m}{n}\neq 0,~{}~{}p\not|~{}mn.\end{cases}$

This norm is non-Archimedean i.e. satisfies strong triangle inequality $\lvert
x+y\rvert_{p}\leq\text{max}\\{\lvert x\rvert_{p},\lvert y\rvert_{p}\\}$. Here
equality holds if and only if $|x|_{p}\neq|y|_{p}$. The field $\mathbb{Q}_{p}$
of $p$-adic numbers is defined as the completion of $\mathbb{Q}$ with respect
to the norm $|\cdot|_{p}$. Every non-zero $p$-adic number has following formal
Laurent series expansion in $p$ as follows

$x=\sum\limits_{j=\gamma}^{\infty}x_{j}p^{j},$

where $x_{j}\in\\{0,1,\ldots,p-1\\}$ with $x_{\gamma}\neq 0$ and
$\gamma\in\mathbb{Z}$. The fractional part $\\{x\\}_{p}$ of $x$ is defined by
$\sum\limits_{j=\gamma}^{-1}x_{j}p^{j}$. Note that $\\{x\\}_{p}=0$ if and only
if $\gamma\geq 0$. Moreover, $\\{0\\}_{p}:=0$. Define
$J_{p,m}:=\\{\frac{s_{-m}}{p^{m}}+\ldots+\frac{s_{-1}}{p}:s_{-j}=0,1,\ldots,p-1;~{}j=1,2,\ldots,m;~{}s_{-m}\neq
0\\}$ for $m\in\mathbb{N}$. The set of $p$-adic integers $\mathbb{Z}_{p}$ and
the set of $p$-adic fractional numbers $I_{p}$ are given by

$\mathbb{Z}_{p}=\\{x\in\mathbb{Q}_{p}:\\{x\\}_{p}=0\\}$ and
$I_{p}=\\{x\in\mathbb{Q}_{p}:\\{x\\}_{p}=x\\}$.

We can also write $\mathbb{Z}_{p}=\\{x\in\mathbb{Q}_{p}:|x|_{p}\leq 1\\}$. The
translates of $\mathbb{Z}_{p}$ by the elements of $I_{p}$ are mutually
disjoint and their union equals $\mathbb{Q}_{p}$. Therefore $I_{p}$ can be
considered as a set of translations for $\mathbb{Q}_{p}$. Note that although
$I_{p}$ is a discrete subset of $\mathbb{Q}_{p}$ but does not form a group.
Moreover, any choice of representatives in $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ is
discrete but not a group as there is only trivial discrete subgroup of
$\mathbb{Q}_{p}$. It is become the biggest constrain for development of
$p$-adic MRA theory as well as wavelet theory compaired to $\mathbb{R}$,
Vilenkin group, local field of positive characteristic etc. The additive
character $\chi_{p}$ on the field $\mathbb{Q}_{p}$ is defined by

$\chi_{p}(x)=e^{2\pi i\\{x\\}_{p}},~{}~{}~{}x\in\mathbb{Q}_{p}.$

A ball centered at $c\in\mathbb{Q}_{p}$ with radius $p^{\gamma}$, where
$\gamma\in\mathbb{Z}$, denoted as $B_{\gamma}(c)$ and defined by

$B_{\gamma}(c)=\\{x\in\mathbb{Q}_{p}:\lvert x-c\rvert_{p}\leq p^{\gamma}\\}.$

The ball $B_{\gamma}(c)$ is compact, open and every point of the ball is also
a center of the ball. Further any two balls are either disjoint or one is
contained in the other. The field $\mathbb{Q}_{p}$ is locally compact, totally
disconnected and has no isolated points. There exists Haar measure $dx$ on
$\mathbb{Q}_{p}$ which is positive and invariant under translations, i.e.,
$d(x+a)=dx$, for all $a\in\mathbb{Q}_{p}$. It is normalized by the equality
$\int\limits_{\mathbb{Z}_{p}}dx=1$. Moreover,

$d(ax)=|a|_{p}~{}dx,~{}~{}~{}a\in\mathbb{Q}_{p}\setminus\\{0\\}.$

The Hilbert space of all complex-valued functions on $\mathbb{Q}_{p}$, square
integrable with respect to the measure $dx$, is denoted by
$L^{2}(\mathbb{Q}_{p})$. The inner product in this space is given by

$\langle f,g\rangle=\int\limits_{\mathbb{Q}_{p}}f(x)\overline{g(x)}dx,$ where
$f,g\in L^{2}(\mathbb{Q}_{p}).$

A complex valued function defined on $\mathbb{Q}_{p}$ is said to be locally
constant if for any $x\in\mathbb{Q}_{p}$ there exists an $l(x)\in\mathbb{Z}$
such that $f(x+y)=f(x),~{}y\in B_{l(x)}(0)$. The linear space of locally-
constant and compactly supported functions (also called the test functions) is
denoted by $\mathcal{D}$. The set of all complex valued functions defined on
$\mathbb{Q}_{p}$ which are locally constant, $p^{M}$-periodic and supported on
$B_{N}(0)$ is denoted by $\mathcal{D}_{N}^{M}$. Note that
$f\in\mathcal{D}_{N}^{M}$ is equivalent to say that $f$ is $p^{M}$-periodic
and Fourier transform of $f$ is supported on $B_{M}(0)$. The Fourier transform
of $f\in L^{1}(\mathbb{Q}_{p})\cap L^{2}(\mathbb{Q}_{p})$ is defined as

$\displaystyle\hat{f}(\xi):=\mathcal{F}[f](\xi)=\int\limits_{\mathbb{Q}_{p}}\chi_{p}(\xi
x)f(x)~{}dx$

where $\xi\in\mathbb{Q}_{p}$. This Fourier transform can be extended to
$L^{2}(\mathbb{Q}_{p})$ and we have the Parseval theorem [22] as follows

$\langle f,g\rangle=\langle\hat{f},\hat{g}\rangle,\hskip
28.45274pt\forall~{}f,g\in L^{2}(\mathbb{Q}_{p}).$

The inverse Fourier transform of $f\in L^{2}(\mathbb{Q}_{p})$ is defined as

$\check{f}(\xi):=\int\limits_{\mathbb{Q}_{p}}\chi_{p}(-\xi x)f(x)~{}dx.$

It is easy to see that $\hat{\check{f}}=f$. If $f\in L^{2}(\mathbb{Q}_{p})$
and $a,b\in\mathbb{Q}_{p}$ with $a\neq 0$, then Fourier transform of dilate
along with translate of $f$ is

$\mathcal{F}[f(a\cdot+b)](\xi)=|a|^{-1}_{p}\chi_{p}\bigg{(}-\frac{b}{a}\xi\bigg{)}\mathcal{F}[f]\bigg{(}\frac{\xi}{a}\bigg{)}.$

For example, fourier tansform of the characteristic function of the set of
$p$-adic integers is itself, i.e.,
$\hat{\mathds{1}}_{\mathbb{Z}_{p}}=\mathds{1}_{\mathbb{Z}_{p}}$ and moreover,
$\mathds{1}_{\mathbb{Z}_{p}}$ is 1-periodic.

######  Lemma 2.1.

If $f\in L^{2}(\mathbb{Q}_{p})$, then $\lim\limits_{n\to\infty}|f(p^{-n}x)|=0$
for $x\in\mathbb{Q}_{p}$ a.e.

###### Proof.

Let $f\in L^{2}(\mathbb{Q}_{p})$. By applying monotone convergence theorem we
have,

$\displaystyle\int\limits_{\mathbb{Q}_{p}}\sum_{n\in\mathbb{N}}|f(p^{-n}x)|^{2}dx$
$\displaystyle=$
$\displaystyle\sum_{n\in\mathbb{N}}\int\limits_{\mathbb{Q}_{p}}|f(p^{-n}x)|^{2}dx$
$\displaystyle=$
$\displaystyle\sum_{n\in\mathbb{N}}p^{-n}\int\limits_{\mathbb{Q}_{p}}|f(x)|^{2}dx$
$\displaystyle=$ $\displaystyle\frac{1}{p-1}\|f\|^{2}<\infty$

It follows that, $\sum_{n\in\mathbb{N}}|f(p^{-n}x)|^{2}dx<\infty$ for
$x\in\mathbb{Q}_{p}$ a.e.
Thus, $\lim\limits_{n\to\infty}|f(p^{-n}x)|=0$ for $x\in\mathbb{Q}_{p}$ a.e. ∎

##  3\. Scaling Set

Shelkovich and Skopina [21] gave following real analogous definition of
$p$-adic multiresolution analysis (MRA in sequel).

######  Definition 3.1.

(Multiresolution Analysis) A collection of closed subspaces $V_{j}\subset
L^{2}({\mathbb{Q}}_{p})$, where $j\in\mathbb{Z}$, is called a multiresolution
analysis if following conditions hold

  1. (i)

$V_{j}\subset V_{j+1}$ for all $j\in\mathbb{Z}$,

  2. (ii)

$\overline{\bigcup\limits_{j\in\mathbb{Z}}V_{j}}=L^{2}({\mathbb{Q}}_{p})$,

  3. (iii)

$\bigcap\limits_{j\in\mathbb{Z}}V_{j}=\\{0\\}$,

  4. (iv)

$f(\cdot)\in V_{j}\iff f(p^{-1}\cdot)\in V_{j+1}$ for all $j\in\mathbb{Z}$,

  5. (v)

there exists $\phi\in V_{0}$ such that the system $\\{\phi(\cdot-a):a\in
I_{p}\\}$ is an orthonormal basis for $V_{0}$.

This $\phi$ is called a scaling function of the given MRA. We will also say
that $\phi$ generates the MRA. Axioms $(iv)$ and $(v)$ of Definition 3.1
immediately refer that $\\{p^{\frac{j}{2}}\phi(p^{-j}\cdot-a):a\in I_{p}\\}$
forms an orthonormal basis for $V_{j}$. As $\phi\in V_{0}\subset V_{1}$ and
$\\{p^{\frac{1}{2}}\phi(p^{-1}\cdot-a):a\in I_{p}\\}$ is an orthonormal basis
for $V_{1}$, $\phi$ can be written as

(1) $\phi(x)=\sum\limits_{a\in I_{p}}\alpha_{a}\phi(p^{-1}x-a),\hskip
14.22636pt\text{where
}\alpha_{a}=p\int\limits_{\mathbb{Q}_{p}}\phi(x)\overline{\phi(p^{-1}x-a)}dx.$

In order to construct an MRA, we start with a function $\phi$ whose
$I_{p}$-translates be an orthonormal system and set

(2) $V_{j}=\overline{span}\\{\phi(p^{-j}\cdot-a):a\in
I_{p}\\},~{}~{}~{}~{}j\in\mathbb{Z}.$

Now axioms $(iv)$ and $(v)$ clearly hold here. In order to hold $(i)$, we also
need $\phi$ such that $\phi(\cdot-b)\in V_{1},~{}~{}\forall~{}b\in I_{p}$,
i.e.,

(3) $\phi(\cdot-b)=\sum_{a\in
I_{p}}\alpha_{a,b}\phi(p^{-1}\cdot-a),~{}~{}~{}~{}~{}~{}~{}~{}\forall~{}b\in
I_{p}.$

But, in general, $\frac{b}{p}+a\notin I_{p}$ for $a,b\in I_{p}$, therefore
$\phi(\cdot-b)\notin V_{1},~{}~{}\forall~{}b\in I_{p}$. Nevertheless, there
are $\phi$’s which satisfy equation (3) and we will discuss here.

######  Definition 3.2.

(Scaling Set) A measurable set $S\subset\mathbb{Q}_{p}$ is said to be a
scaling set if $\check{\mathds{1}}_{S}$ is a scaling function of an MRA in
$L^{2}({\mathbb{Q}}_{p})$.

######  Example 3.3.

$\mathbb{Z}_{p}$ is a scaling set.

######  Lemma 3.4.

Measure of a scaling set is 1.

###### Proof.

Let $S\subset\mathbb{Q}_{p}$ be a scaling set. As dilation is an isomorphism,
$\\{p^{\frac{j}{2}}\check{\mathds{1}}_{S}(p^{-j}\cdot-a):a\in I_{p}\\}$ is an
orthonormal basis for the space $V_{j}$. Now

$1=\langle
p^{\frac{j}{2}}\check{\mathds{1}}_{S}(p^{-j}\cdot),p^{\frac{j}{2}}\check{\mathds{1}}_{S}(p^{-j}\cdot)\rangle=p^{-j}\langle\mathds{1}_{p^{-j}S},\mathds{1}_{p^{-j}S}\rangle=p^{-j}\mu(p^{-j}S).$

Therefore $\mu(p^{-j}S)=p^{j}$. ∎

######  Proposition 3.5.

For a scaling set $S$, $\lim\limits_{n\to\infty}\mathds{1}_{p^{n}S}(\xi)=0$
for $\xi\in\mathbb{Q}_{p}$ a.e.

###### Proof.

By Plancherel Theorem, we know that, $\mathds{1}_{S}\in
L^{2}({\mathbb{Q}}_{p})$. Now by Lemma 2.1, for $\xi\in\mathbb{Q}_{p}$ a.e.

$\lim\limits_{n\to\infty}\mathds{1}_{S}(p^{-n}\xi)=0\Rightarrow\lim\limits_{n\to\infty}\mathds{1}_{p^{n}S}(\xi)=0$

∎

Haldar and Singh [11] defined following multiwavelet set analogous to
$\mathbb{R}$.

######  Definition 3.6.

(Multiwavelet Set) A measurable set $W\subset\mathbb{Q}_{p}$ is said to be a
multiwavelet set of order $L$ if $W=\bigcup\limits^{L}_{n=1}W_{n}$ for
measurable sets $W_{n}$; $n=1,\ldots,L$ and $\\{\psi_{1},\ldots,\psi_{L}\\}$
is a multiwavelet for $L^{2}(\mathbb{Q}_{p})$ where
$\hat{\psi}_{n}=\mathds{1}_{W_{n}}$.
When $L=1$, $W$ is simply said to be a wavelet set.

######  Lemma 3.7.

For a MRA, scaling set and wavelet set are disjoint a.e.

###### Proof.

Let $W$ and $S$ be wavelet set and scaling set. As
$\check{\mathds{1}}_{W}\perp V_{0}$

$0=\langle\check{\mathds{1}}_{W},\check{\mathds{1}}_{S}(\cdot-a)\rangle=\int\limits_{S\cap
W}\chi_{p}(a\xi)d\xi$

If $a=0$, then $\mu(S\cap W)=0$. ∎

Note that instead of wavelet set, if it is a multiwavelet set then
$\mu((\bigcup\limits_{n=1}^{L}W_{n})\cap S)=0$ i.e.
$\mu(\bigcup\limits_{n=1}^{L}(W_{n}\cap S))=0.$

######  Definition 3.8.

(Refinable Function) A function $f\in L^{2}(\mathbb{Q}_{p})$ is said to be a
refinable function if it satisfy following refinement equation

$f(x)=\sum\limits_{a\in I_{p}}r_{a}f\left(\frac{x}{p}-a\right),\hskip
28.45274ptr_{a}\in\mathbb{C}.$

Note that every scaling function is refinable function but converse may not be
true.

######  Lemma 3.9.

If $S\subset\mathbb{Q}_{p}$ is a scaling set, then $S\subset p^{-1}S$.

###### Proof.

Let $\phi$ be the scaling function corresponding to $S$. We know scaling
function satisfy following refinement equation

$\phi(x)=\sum\limits_{a\in I_{p}}\alpha_{a}\phi(p^{-1}x-a),\hskip
28.45274pt\text{where }\alpha_{a}\in\mathbb{C}.$

Taking Fourier transform in both side, we have

(4) $\hat{\phi}(\xi)=m_{0}(p\xi)\hat{\phi}(p\xi),\text{ where
}m_{0}(\xi):=\sum\limits_{a\in I_{p}}\frac{\alpha_{a}}{p}\chi_{p}{(a\xi)}$

$\Rightarrow\mathds{1}_{S}(\xi)=\mathds{1}_{p^{-1}S}(\xi)m_{0}(p\xi)$

This completes the proof. ∎

$m_{0}$ is said to be scaling filter. It is to be noted that $m_{0}$ is
trigonometric function satisfying $m_{0}(0)=1$ whenever $\hat{\phi}(0)\neq 0.$

######  Proposition 3.10.

Let $S$ be a scaling set and $m_{0}$ be corresponding filter defined by
equation (4). Then

$m_{0}(p\xi)=\begin{cases}1&\text{ if }\xi\in S\\\ 0&\text{ if }\xi\in
p^{-1}S\setminus S.\end{cases}$

###### Proof.

As $S$ is the scaling set,

$\mathds{1}_{S}(\xi)=\mathds{1}_{p^{-1}S}(\xi)m_{0}(p\xi).$

If $\xi\in S$ then, by Lemma 3.9, $m_{0}(p\xi)=1$ .
If $\xi\notin S$ but $\xi\in p^{-1}S$ then $m_{0}(p\xi)=0$. ∎

######  Proposition 3.11.

Let $S\subset B_{M}(0)$ be scaling set correspond to $\phi$ defined by
$\hat{\phi}(\xi)=R\prod\limits_{j=0}^{\infty}m\left(\frac{\xi}{p^{N-j}}\right)$,
where $m$ is a trigonometric polynomial satisfy $m(0)=1$ and $R$ is a real no.
Then at most $\frac{\text{deg}~{}m}{p-1}$ many integers are in
$p^{M}S\cap[0,p^{M+N}-1]$.

###### Proof.

It easily follows from Lemma 1 of [2]. ∎

######  Proposition 3.12.

Let $S$ be scaling set containg $0$ corresponding
$\phi\in\mathfrak{D}_{N}^{M}$, where $M,N\geq 0$. If $\forall b~{}\in\\{x\in
I_{p}:\left|x\right|_{p}\leq p^{N}\\}$, $\phi(\cdot-b)\in V_{1}$ as defined by
equation (2), then there are at most $p^{N}$ many imterger $n$ such that $n\in
p^{M}S\cap[0,p^{M+N}-1]$.

###### Proof.

It directly follows from Theorem 4 of [2]. ∎

######  Theorem 3.13.

Let $S$ be a scaling set corresponding to scaling function
$\phi\in\mathfrak{D}_{N}^{M}$ where $M,N\geq 0$. If there are maximum $p^{N}$
many integers in $p^{M}S\cap[0,p^{M+N}-1]$, then,
$\forall~{}b\in\mathbb{Q}_{p}$,

$\sum_{a\in I_{p}}\alpha_{a,b}\chi_{p}((a-b)\xi)=1,\hskip
14.22636pt\forall~{}\xi\in S.$

###### Proof.

As there are atmost $p^{N}$ many integers in $p^{M}S\cap[0,p^{M+N}-1]$, using
Theorem 5 of [2], $\phi(x-b)$, where $b\in\mathbb{Q}_{p}$, can be written as

$\phi(x-b)=\sum_{a\in I_{p}}\alpha_{a,b}~{}\phi(x-a)$

Taking Fourier transform in both side, we have

(5) $\chi_{p}(b\xi)\hat{\phi}(\xi)=\sum_{a\in
I_{p}}\alpha_{a,b}\chi_{p}(a\xi)\hat{\phi}(\xi)$

When $\xi\in S$, equation (5) reduces to $\chi_{p}(b\xi)=\sum\limits_{a\in
I_{p}}\alpha_{a,b}\chi_{p}(a\xi)$. This completes the proof. ∎

Scaling set satisfy following similar results to wavelet set.

######  Theorem 3.14.

Let $S\subset\mathbb{Q}_{p}$ be a set such that
$\check{\mathds{1}}_{S}\in\mathfrak{D}_{N}^{M}$, where $M,N\geq 0$. If $S$ be
a scaling set then $p^{N}$ many integers are in $p^{M}S\cap[0,p^{M+N}-1]$.

###### Proof.

As $S$ is a scaling set, $\\{\check{\mathds{1}}_{S}(\cdot-a):a\in I_{p}\\}$ is
a orthonormal set. Now using orthonormality and Parseval theorem

$\displaystyle\delta_{a,0}$ $\displaystyle=$
$\displaystyle\langle\check{\mathds{1}}_{S}(\cdot-a),\check{\mathds{1}}_{S}\rangle$
$\displaystyle=$
$\displaystyle\langle\mathds{1}_{S}\chi_{p}(a\cdot),\mathds{1}_{S}\rangle$
$\displaystyle=$
$\displaystyle\int\limits_{B_{M}(0)}\mathds{1}_{S}(\xi)\chi_{p}(a\xi)d\xi$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{p^{M+N}-1}\int\limits_{B_{-N}\left(\frac{l}{p^{M}}\right)}\mathds{1}_{S}(\xi)\chi_{p}(a\xi)d\xi$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{p^{M+N}-1}\mathds{1}_{S}\left(\frac{l}{p^{M}}\right)\chi_{p}\left(\frac{al}{p^{M}}\right)\int\limits_{B_{-N}\left(0\right)}\chi_{p}(a\xi)d\xi$
$\displaystyle=$
$\displaystyle\frac{1}{p^{M}}\mathds{1}_{\mathbb{Z}_{p}}(ap^{N})\sum_{l=0}^{p^{M+N}-1}\mathds{1}_{S}\left(\frac{l}{p^{M}}\right)\chi_{p}\left(\frac{al}{p^{M}}\right)$

Now $a=0$ give
$\sum\limits_{l=0}^{p^{M+N}-1}\mathds{1}_{S}\left(\frac{l}{p^{M}}\right)=p^{N}$.
This completes the proof. ∎

######  Theorem 3.15.

Let $S$ be a scaling set containing $0$. If $\check{\mathds{1}}_{S}$ is a test
function then $S\subset\mathbb{Z}_{p}$.

###### Proof.

This directly follows from Theorem 8 of [2]. ∎

######  Lemma 3.16.

Let $S$ be a scaling set corresponding $\phi$ and let
$\phi(\cdot-b)\in\bigcup\limits_{j\in\mathbb{Z}}V_{j}$ for any
$b\in\mathbb{Q}_{p}$, where $V_{j}$ is defined by equation (2). Then
$\bigcup\limits_{j\in\mathbb{Z}}p^{j}S=\mathbb{Q}_{p}.$

###### Proof.

From [2], we have $\bigcup\limits_{j\in\mathbb{Z}}\text{supp
}\hat{\phi}(p^{j}\cdot)=\mathbb{Q}_{p}$.
Therefore, $\bigcup\limits_{j\in\mathbb{Z}}\text{supp
}\mathds{1}_{p^{-j}S}=\bigcup\limits_{j\in\mathbb{Z}}p^{-j}S=\mathbb{Q}_{p}$.

This completes the proof. ∎

The concept of generalized scaling set in $L^{2}(\mathbb{R}^{n})$ was given by
Brownik et al. [8]. We are giving here analogous definition of that.

##  4\. Generalized Scaling Set

In this section our intension is to discuss generalized scaling set and give
its characterization.

######  Definition 4.1.

For a fixed $L\in\mathbb{N}$, a set $G\subset\mathbb{Q}_{p}$ is said to be a
generalized scaling set of order $L$ if $\mu(G)=\frac{L}{p-1}$ and
$p^{-1}G\setminus G$ is a multiwavelet set of order $L$.

######  Example 4.2.

$\mathbb{Z}_{p}$ is a generalized scaling set of order $p-1$ corresponding to
Kozyrev’s multiwavelet [19].

######  Example 4.3.

$p^{-m+1}\mathbb{Z}_{p}$ be a generalized scaling set of order $(p-1)p^{m-1}$
for $m\in\mathbb{N}$ correspond to the multiwavelet given by Khrennikov and
Shelkovich [16].

In the following lemma we give a necessary and sufficient condition for a
measurable set to be a generalized scaling set. One can observe similar result
for $\mathbb{R}^{n}$ in [8].

######  Lemma 4.4.

A measurable set $G\subset\mathbb{Q}_{p}$ is a generalized scaling set of
order $L$ if and only if $G=\bigcup\limits_{j=1}^{\infty}p^{j}W$ a.e. for some
multiwavelet set $W$ of order $L$.

###### Proof.

For some multiwavelet set $W$ of order $L$, let
$G=\bigcup\limits_{j=1}^{\infty}p^{j}W$.
Clearly, $p^{-1}G\setminus G=W$. From [11], we know that $\mu(W)=L$. Now

$\mu(p^{-1}G\setminus G)=\mu(W)\Rightarrow\mu(G)=\frac{L}{p-1}.$

So $G$ is a generalized scaling set of order $L$.

Conversely, suppose $G$ is a generalized scaling set of order $L$. Then
$W:=p^{-1}G\setminus G$ is a multiwavelet set of order $L$.
As $\mu(W)=L$, $G\subset p^{-1}G$. Clearly,
$\bigcup\limits_{j=1}^{\infty}p^{j}W\subset G$. Since both have same measure,
$G=\bigcup\limits_{j=1}^{\infty}p^{j}W$ a.e.

∎

######  Example 4.5.

Consider $G=\mathbb{Z}_{p}$ and
$W=\bigcup\limits_{n=1}^{p-1}(-\frac{n}{p}+\mathbb{Z}_{p})$.

$\bigcup\limits_{j=1}^{\infty}p^{j}W=pW\bigcup\bigcup\limits_{j=2}^{\infty}p^{j}W=\bigcup\limits_{n=1}^{p-1}(-n+p\mathbb{Z}_{p})\bigcup\bigcup\limits_{j=2}^{\infty}p^{j}W.$

Here $0\notin pW$ and
$\bigcup\limits_{j=2}^{\infty}p^{j}W\subset\mathbb{Z}_{p}$. Therefore,
$G=\bigcup\limits_{j=1}^{\infty}p^{j}W$ a.e.

######  Example 4.6.

Consider $G=p^{-m+1}\mathbb{Z}_{p}$ and $W=\bigcup\limits_{s\in
J_{p,m}}(-s+\mathbb{Z}_{p})$.

$\bigcup\limits_{j=1}^{\infty}p^{j}W=\bigcup\limits_{s\in
J_{p,m}}(-sp+p\mathbb{Z}_{p})\bigcup\bigcup\limits_{j=2}^{\infty}p^{j}W.$

Here also $0\notin pW$ and
$\bigcup\limits_{j=2}^{\infty}p^{j}W\subset\mathbb{Z}_{p}$. Therefore,
$G=\bigcup\limits_{j=1}^{\infty}p^{j}W$ a.e.

Now we will give a necessary condition for a generalized scaling set.

######  Theorem 4.7.

Let $G\subset\mathbb{Q}_{p}$ be a generalized scaling set of order $L$. Then

  1. (i)

$\mu(G)=\frac{L}{p-1}$.

  2. (ii)

$G\subset p^{-1}G$.

  3. (iii)

$\lim\limits_{n\rightarrow\infty}\mathds{1}_{G}(p^{n}\xi)=1$ for
$\xi\in\mathbb{Q}_{p}$ a.e.

###### Proof.

Let $G$ is a generalized scaling set of order $L$.
Then $(i)$ is guaranteed by Definition 4.1.
$(ii)$ easily follows from Lemma 4.4.
Also from the last lemma we know that $G=\bigcup\limits_{j=1}^{\infty}p^{j}W$
for some multiwavelet set $W$ of order $L$. Furthermore, we know that,
$\mu(p^{j}W\cap p^{l}W)=p^{-j}\delta_{j,l}L$. Therefore

$\mathds{1}_{G}(p^{n}\xi)=\mathds{1}_{\bigcup\limits_{j=1}^{\infty}p^{j}W}(p^{n}\xi)=\sum\limits_{j=1}^{\infty}\mathds{1}_{p^{j}W}(p^{n}\xi)=\sum_{j=1}^{\infty}\mathds{1}_{W}(p^{n-j}\xi).$

As, $\sum\limits_{j\in\mathbb{Z}}\mathds{1}_{W}(p^{j}\xi)=1$ for $\xi$ a.e.,
we get condition $(iii)$. ∎

######  Remark 4.8.

We will conclude by raising following question. Like other field, is every
scaling set is a generalized scaling set in this field?

## Acknowledgment

The author is highly indebted to the fiscal support of Ministry of Human
Resource Development (M.H.R.D.), Government of India.

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