# Validating the Sub-Burst Slope Law: A Comprehensive Multi-Source Spectro-
Temporal Analysis of Repeating Fast Radio Bursts

Katie Brown,1 Mohammed A. Chamma,2 Fereshteh Rajabi,2 Aishwarya Kumar,1 Hosein
Rajabi,3 and Martin Houde1
1Department of Physics and Astronomy, The University of Western Ontario, 1151
Richmond Street, London, Ontario N6A 3K7, Canada
2Department of Physics and Astronomy, McMaster University, 1280 Main Street
West, Hamilton, Ontario, L8S 4L8, Canada
3Department of Computer Science, The University of Western Ontario, 1151
Richmond Street, London, Ontario N6A 3K7, Canada E-mail<EMAIL_ADDRESS>mail<EMAIL_ADDRESS>
###### Abstract

We conduct a comprehensive spectro-temporal analysis of repeating Fast Radio
Bursts (FRBs) utilizing nine distinct sources, the largest sample to date. Our
data set includes 175 sub-bursts and 31 multi-component bursts from 11 data
sets, with centre frequencies ranging from 149–7144 MHz and durations spanning
from 73 $\mu$s–13 ms. Our findings are consistent with the predictions of the
Triggered Relativistic Dynamical Model (TRDM) of FRB emission. We affirm the
predicted quadratic relationship between sub-burst slope and frequency, as
well as a linear dependence of bandwidth on frequency that is consistent with
mildly-relativistic Doppler broadening of narrow-band emission. Most
importantly, we confirm the sub-burst slope law, a predicted inverse
relationship between sub-burst slope and duration, to hold consistently across
different sources. Remarkably, we also discover that the drift rates of multi-
component bursts follow the same law as the sub-burst slopes, an unexplained
result that warrants further investigation. These findings not only support
the TRDM as a viable framework for explaining several aspects of FRB emission,
but also provide new insights into the complex spectro-temporal properties of
FRBs.

###### keywords:

transients: fast radio bursts – methods: data analysis – relativistic
processes – radiation: dynamics – radiation mechanisms: non-thermal

††pubyear: 2023††pagerange: Validating the Sub-Burst Slope Law: A
Comprehensive Multi-Source Spectro-Temporal Analysis of Repeating Fast Radio
Bursts–Validating the Sub-Burst Slope Law: A Comprehensive Multi-Source
Spectro-Temporal Analysis of Repeating Fast Radio Bursts

##  1 Introduction

Fast Radio Bursts (FRBs) remain one of the most intriguing phenomena in
astrophysics, captivating researchers since their discovery in 2007 (Lorimer
et al., 2007). Characterized by their extremely short durations and high
energies, FRBs exhibit remarkable variability in their observed properties,
further complicating their enigmatic nature (Petroff et al., 2022). Despite
this diversity, it is widely assumed that these transient events originate
from highly energetic processes, likely tied to compact objects (Lyutikov et
al., 2016). Although several theories, including magnetar flares and compact-
object mergers, are considered promising, the underlying mechanism remains a
mystery.

One promising approach to gaining insights into the emission mechanism and
source dynamics is through studying the spectro-temporal properties of FRBs.
These include spectral bandwidth, temporal duration, central frequency, and
frequency drift rates between and within sub-bursts111When multiple resolved
components are visible within a single waterfall, these individual pulses are
called ’sub-bursts’.. The ‘sad trombone’ effect refers to the downward drift
in frequency of consecutive sub-bursts (which we will refer to as the drift
rate; Josephy et al., 2019). In contrast, the sub-burst slope refers to the
(typically steeper) downward frequency drift within a single sub-burst.

Several relationships between spectro-temporal properties have been derived
within the framework of the triggered relativistic dynamical model (TRDM), a
simple model in which FRB emission is triggered by a signal from a background
source, such as a magnetar or pulsar (Rajabi et al., 2020; Houde et al.,
2019). The TRDM was inspired by, but is not limited to, the application of
superradiance to FRB emission (Houde et al., 2018; Houde et al., 2019). The
central prediction of the TRDM is the ‘sub-burst slope law’, an inverse
relationship between the sub-burst slope and duration. The model also proposes
several relationships tied to the center frequency: an inverse dependence on
duration and a quadratic correlation with sub-burst slope, while we expect a
further relationship with the bandwidth from the Doppler effect. These
predicted patterns have been observed in FRB 20121102A, the primary focus of
most studies due to its wealth of observational data (Rajabi et al., 2020;
Jahns et al., 2023; Chamma et al., 2023). However, investigations exploring
these relationships across multiple FRB sources have been few and constrained,
typically limited by sparse data and narrow duration and frequency ranges.

In this paper, we seek to bridge this gap by conducting a comprehensive
spectro-temporal analysis of repeating FRBs utilizing nine distinct
sources—the largest sample to date. By evaluating relationships among the
slope, duration, frequency, and bandwidth of each burst and sub-burst, we aim
to test the predictions of the TRDM within a broad and diverse data set. This
investigation, which encompasses a frequency range akin to the entire detected
FRB population, enables a robust assessment of these hypothesized laws.
Whether our study uncovers ubiquitous patterns across all sources and bursts
or identifies significant deviations or exceptions, our findings will provide
valuable insights into the FRB population and, ultimately, their origins.

This paper is organized as follows: in Section 2 we outline our analysis
approach, including burst waterfall preparation and dispersion measure (DM)
considerations, and discuss the data used. Our findings and their implications
are presented in Section 3, and the key insights are summarized in Section 4.

##  2 Methods

###  2.1 Analysis

Turning to the details of our approach, the first step involves preparing the
burst waterfalls for measurements. This includes performing down-sampling in
frequency and/or time, background subtraction, and masking of noisy channels
and radio interference. Each step is handled on a case-by-case basis. For
processing and spectro-temporal measurements, we use the graphical user
interface Frbgui222https://github.com/mef51/frbgui, developed by Chamma et al.
(2023).

Next, we consider the dispersion measure (DM) for each FRB source, an element
that can strongly affect the measured sub-burst slope. For this reason, we
dedisperse each burst over a range of trial DMs, following the methodology of
Chamma et al. (2021). We adopt a range of $20$ pc cm-3, centered on the
reported DM value. To minimize the change in waterfalls between consecutive
DMs, we use a conservative step size of $0.2$ pc cm-3.

To increase the Signal-to-Noise ratio (S/N), reduce the impact of burst
structure and interference, and allow for more consistent measurements, we
perform a 2D autocorrelation function analysis on each waterfall (Hessels et
al., 2019). Each autocorrelation is fit with an ellipsoid with a 2D Gaussian
intensity distribution, from which we extract measurements of sub-burst slope,
duration, and bandwidth; this process is detailed in Chamma et al. (2023).

When analyzing bursts with multiple components, vertical (i.e. temporal)
divisions are used to separate them into their sub-bursts. Each sub-burst then
undergoes the same analysis as an individual burst. The complete burst is also
characterized through the autocorrelation process, with the measured slope
being the drift rate between consecutive components.

The first step in analyzing the measurement results is determining both the
range of possible DMs and the representative DM value for each source. We
begin with the 20 pc cm-3 range in DM for each source and restrict it by
eliminating any DM that is found to be invalid for any of the bursts. A DM is
considered too high if it produces a positive sub-burst slope or a fitting
error larger than 40% (usually occurring where the bursts are nearly vertical;
Chamma et al. 2023; Jahns et al. 2023). Within the TRDM framework, positive
slopes (which result in bursts that appears to be upward-sloping) are
unphysical and produced by over-dedispersion. Conversely, we consider bursts
as under-dedispersed if they become curved or so stretched that fitting an
ellipsoid to their autocorrelation is unfeasible. The variation in a parameter
over this range of valid DMs gives the ‘uncertainty’ in the measurements
presented in Section 3.

The representative DM for each source is taken to be that which yields the
best fit for the sub-burst slope law predicted by the TRDM of Rajabi et al.
(2020). To determine this, a fit of the form
$|\differential\nu_{\mathrm{obs}}/\differential
t_{\mathrm{D}}|\nu_{\mathrm{obs}}^{-1}=A/t_{\mathrm{w}}$ is found for the
measurements taken at each DM within the predetermined valid range (the
parameters in this equation are defined in equation 1 below). For each source,
we select the DM that results in the fit with the reduced-$\chi^{2}$ closest
to unity as the representative DM. We then plot the measurements at these
representative DMs to study the relationships between the spectro-temporal
quantities.

###  2.2 Data

Table 1: Summary of data analyzed and key results of this study. The number of bursts used for most sources is less than the total detected due to the exclusion criteria of low S/N and severe truncation. The sub-burst slope law coefficient $A$ (see equation 1) is determined for each source. FRB Source | Telescope |  | Frequency Range
---
(MHz)
| Number of
---
Bursts Used
| Sub-Burst
---
Measurements
| Representative
---
DM (pc cm${}^{\mathbf{-3}}$)
Coefficient A
20121102A | Arecibo | 1150–1730 | 18 | 24 | 558.0 | 0.076 $\pm$ 0.004
| Arecibo | 4100–4900 | 16 | 19 |  |
| Green Bank | 4000–8000 | 17 | 22 |  |
20180916B | uGMRT | 550–750 | 12 | 16 | 348.6 | 0.11 $\pm$ 0.01
| LOFAR | 110–188 | 12 | 12 |  |
| CHIME | 400–800 | 13 | 21 |  |
20201124A | Effelsberg | 1200–1520 | 12 | 20 | 410.6 | 0.17 $\pm$ 0.02
20180301A | FAST | 1000–1500 | 7 | 12 | 515.4 | 0.088 $\pm$ 0.007
20180814A | CHIME | 400–800 | 4 | 7 | 186.8 | 0.09 $\pm$ 0.2
20200929C | CHIME | 400–800 | 4 | 6 | 412.4 | 0.08 $\pm$ 0.2
20190804E | CHIME | 400–800 | 4 | 5 | 363.2 | 0.11 $\pm$ 0.03
20190915D | CHIME | 400–800 | 5 | 6 | 487.6 | 0.12 $\pm$ 0.02
20201130A | CHIME | 400–800 | 5 | 5 | 287.8 | 0.29 $\pm$ 0.07

In total, our analysis covers nine repeating FRB sources, encompassing 11 data
sets collected from six different telescopes. Our measurements are obtained
from 128 bursts, with many divided into multiple sub-bursts, resulting in a
data set of 175 single pulses and 31 multi-component bursts. The analyzed data
have centre frequencies ranging from 149–7144 MHz, and their durations span 73
$\mu$s–13.22 ms. Information on each source and data set is summarized in
Table 1.

For the data sets where not all detected bursts were utilized, exclusions were
made due to insufficient S/N for reliable autocorrelation analysis, or in
instances where parts of the bursts were severely truncated.

####  2.2.1 FRB 20121102A

FRB 20121102A is one of only two known periodic repeaters, with a period of
approximately 160 days (Cruces et al., 2021). The source has been localized to
a star-forming region in a dwarf galaxy at $z=0.1927$, and is associated with
a persistent radio source (Tendulkar et al., 2017). We use three sets of data
from this source.

Hewitt et al. (2022) observed 478 bursts during a burst storm in 2016 with the
Arecibo telescope over a band of 1150–1730 MHz (these data will be referred to
as ‘20121102A Arecibo-1’). We select 18 bursts from this data set, choosing
waterfalls with clearly visible signals and no severe truncation, and aiming
for diversity in terms of intensity, morphology and presence of multiple
components. We divide four of them into sub-bursts, leading to a total of 24
single pulses.

The second data set, ‘20121102A Arecibo-2,’ comes from Michilli et al. (2018),
who observed this source using Arecibo over 4.1–4.9 GHz. All 16 bursts from
this data set are used, with three of them divided into sub-bursts, yielding
19 measurements.

Lastly, we use 17 of the 21 bursts detected by Gajjar et al. (2018) with the
Green Bank telescope over 4–8 GHz, the highest frequency detection of FRB
20121102A to date. Three of these bursts are split into two to four sub-bursts
yielding 22 single pulses. For the data from Michilli et al. (2018) and Gajjar
et al. (2018), we use the results from the processing and measurements of
Chamma et al. (2023) who follow an identical procedure, except for the DM
grid.

After analysis of these three data sets together, the representative DM value
for FRB 20121102A is established at 558.0 pc cm-3, consistent with Chamma et
al. (2023).

####  2.2.2 FRB 20180916B

FRB 20180916B, another periodic repeater with a period of about 16 days
(CHIME/FRB Collaboration et al., 2020), resides in a star-forming region
within a nearby (149 Mpc) massive spiral galaxy (Marcote et al., 2020). We use
three data sets for this source.

Marthi et al. (2020) observed 15 bursts with the upgraded Giant Meterwave
Radio Telescope (uGMRT) over 550–750 MHz. We use 12 bursts, dividing four into
multiple components, yielding measurements for 16 individual sub-bursts.

The second data set is from Pleunis et al. (2021), who detected 18 bursts from
FRB 20180916B using the Low Frequency Array (LOFAR) over a band of 110–188
MHz; this is the lowest frequency detection of any FRB to date. We use 12 of
these bursts in our analysis.

Lastly, we use 19 bursts from The Canadian Hydrogen Intensity Mapping
Experiment (CHIME) over a band of 400–800 MHz (CHIME/FRB Collaboration et al.,
2019). We take measurements from 13 bursts, splitting four into sub-bursts for
a total of 21 single pulses.

Collectively, these data sets lead us to determine the representative DM for
FRB 20180916B as 348.6 pc cm-3, consistent with previously reported values
(Nimmo et al., 2021; Chamma et al., 2021).

####  2.2.3 FRB 20201124A

FRB 20201124A is the third closest repeating FRB source, located in a massive
star-forming galaxy at $z=0.098$ (Nimmo et al., 2022). Hilmarsson et al.
(2021) detected 20 bursts from FRB 20201124A with the Effelsberg 100-m
telescope over 1200–1520 MHz. We use 12 of these bursts and divide four into
three sub-bursts, resulting in a final data set of 20 individual sub-bursts.
Upon analysis, we establish the representative DM for this source as 410.6 pc
cm-3, close to the reported value of 411.6 pc cm-3.

####  2.2.4 FRB 20180301A

FRB 20180301A, observed by Luo et al. (2020) using The Five-hundred-meter
Aperture Spherical radio Telescope (FAST) over 1000–1500 MHz, has been
localized to the outskirts of an intermediate-mass star-forming galaxy at
$z=0.33$ (Bhandari & Flynn, 2021). From the 15 bursts, we selected the seven
with sufficient S/N for our analysis and divided four of them into sub-bursts,
resulting in 12 single pulses. We determined the representative DM for this
source to be 515.4 pc cm-3; the published value of 522 pc cm-3 (determined
through maximizing S/N) was found to produce over-dedispersion in numerous
bursts from the sample.

####  2.2.5 Other FRB Sources From CHIME

The remaining five sources were observed with CHIME over 400–800 MHz; these
include FRB 20180814A (CHIME/FRB Collaboration et al., 2021), and FRB
20200929C, FRB 20190804E, FRB 20190915D and FRB 20201130A (CHIME/FRB
Collaboration et al., 2023). We analyzed four bursts from both FRB 20180814A
and FRB 20200929C, dividing two bursts from each into sub-bursts. From FRB
20190804E and FRB 20190915D, we analyzed four and five bursts, respectively,
dividing one burst from each source into two sub-bursts. For FRB 20201130A, we
used five bursts, none of which were split into sub-bursts.

Upon analysis, we established the representative DM values for these sources
as follows: 186.8 pc cm-3 for FRB 20180814A, 412.4 pc cm-3 for FRB 20200929C,
363.2 pc cm-3 for FRB 20190804E, 487.6 pc cm-3 for FRB 20190915D, and 287.8 pc
cm-3 for FRB 20201130A.

##  3 Results and Discussion

Figure 1: The sub-burst slope law and drift rate relation. Unoutlined points
represent individual sub-bursts, whereas outlined points correspond to drift
rate measurements for bursts with multiple components. The TRDM’s inverse fit
for sub-bursts is represented by the solid black line (partially hidden by the
dashed orange line; see below), while the dotted red line shows a free power
law fit. For multi-component bursts, the dashed orange and red lines display
the inverse and power-law fits, respectively. In addition to the fit lines,
the legend indicates the unique markers representing each data set (divided by
FRB source and observatory; consistent between each figure). Each point is
plotted at the representative DM determined for their source, with the error
bars demonstrating the parameter’s variation over the range of valid DMs;
sources with limited DM ranges will have small or negligible error bars. We
find that both sub-bursts and multi-component bursts closely follow the
predicted inverse trend.

###  3.1 Sub-Burst Slope Law and Drift Rate Relation

The key prediction of the TRDM, the sub-burst slope law, is an inverse
relationship between frequency-normalized sub-burst slope and duration (Rajabi
et al., 2020). This relationship is expressed as

$\frac{1}{\nu_{\mathrm{obs}}}\derivative{\nu_{\mathrm{obs}}}{t_{\mathrm{D}}}=-\left(\frac{\tau_{\mathrm{w}}^{\prime}}{\tau_{\mathrm{D}}^{\prime}}\right)\frac{1}{t_{\mathrm{w}}}=-A\frac{1}{t_{\mathrm{w}}},$
(1)

where $\nu_{\mathrm{obs}}$ represents the observed frequency, $t_{\mathrm{D}}$
the delay time, and $t_{\mathrm{w}}$ the observed duration. The coefficient
$A\equiv\tau_{\mathrm{w}}^{\prime}/\tau_{\mathrm{D}}^{\prime}$ depends on the
proper duration ($\tau_{\mathrm{w}}^{\prime}$) and delay time
($\tau_{\mathrm{D}}^{\prime}$), both governed by the properties of the source
and emitting medium. We expect $A$ to be fairly constant for a given source,
although there may be some spread between sources (Houde et al., 2019; Rajabi
et al., 2020).

The validity of the sub-burst slope law has been reinforced by several studies
of repeating FRBs. Rajabi et al. (2020), Jahns et al. (2023), and Chamma et
al. (2023) demonstrated its applicability to FRB 20121102A, analyzing 24, 849,
and 167 bursts, respectively. Furthermore, Chamma et al. (2021) examined 42
bursts from FRBs 20121102A, 20180916B, and 20180814A, providing evidence that
the law holds across varied sources. Preliminary evidence also suggests a
similar law applies to the drift rates of multi-component bursts (see Figure 8
of Jahns et al. 2023 and Figure 5 of Chamma et al. 2023), although definitive
conclusions are hindered by a scarcity of measurements and narrow duration
ranges.

As illustrated in Figure 1, the findings for our nine sources align with these
previous studies. We present two fits: the predicted TRDM inverse relationship
$At_{\mathrm{w}}^{-1}$, and a free power law of the form
$A_{\text{free}}t_{\mathrm{w}}^{n_{1}}$. The inverse fit yields a coefficient
of $A=0.077\pm 0.002$, consistent with $A=0.078\pm 0.006$ reported for the
multi-source study by Chamma et al. (2021) and $A=0.084$ obtained by Jahns et
al. (2023) for FRB 20121102333Jahns et al. (2023) fit their sub-burst slope
vs. duration data with the inverse formulation of the slope law, finding a
coefficient of $b=-0.00862(37)$ MHz-1. Chamma et al. (2023) convert this to a
form equivalent to the form used here, finding $A=0.084$.. Although Chamma et
al. (2023) found a higher coefficient of $A=0.113\pm 0.003$ for FRB 20121102A,
they suggested potential skewing by high-slope outliers. We also find
coefficients of $A_{\text{free}}=0.111\pm 0.002$ and $n_{1}=-0.81\pm 0.04$ for
the power law fit, consistent with the predicted inverse trend.

Individual fits for each of the eight sources with more than five measurements
yields $A$ coefficients between $0.07-0.17$, with five centered closely around
$0.1$ within a range of $\pm 0.02$ ($A$ values for each source are included in
Table 1). Despite slight deviations from the global trendline by certain
sources, the overall data align well with the predicted inverse relationship.
These results strongly reinforce the ubiquity of the sub-burst slope law in
repeating FRBs, demonstrating its consistency across different sources.

Figure 1 also includes drift rate measurements from multi-component bursts,
highlighting their consistency with individual sub-bursts. Notably, we found
the drift rate measurements follow the same inverse trend as the sub-burst
slopes, with $A_{\text{drift}}=0.076\pm 0.004$ for the inverse fit, and
$A_{\text{drift, free}}=0.078\pm 0.001$ and $n_{\mathrm{drift}}=-0.95\pm 0.08$
for the free fit. This relationship is not fully explained by the TRDM,
depending on some unknown physical parameter (see Sec 2.1.3 in Rajabi et al.,
2020). This closeness between $A$ and $A_{\text{drift}}$ is remarkable because
it implies that the phenomenon responsible for the scaling within a sub burst
also rules the scaling within a group of them.

Figure 2: Relationships between the sub-burst duration (top) and the sub-burst
slope (bottom) with frequency. The bursts show general trends of shorter
durations and steeper slopes at higher frequencies. The black lines are the
fits predicted by the TRDM, while the dotted red lines are free power law
fits. Multi-component bursts are included for comparison (the outlined points;
these are drift rate measurements rather than sub-burst slopes) but are
excluded from the fits as we do not assume their spectro-temporal properties
to be governed by the same mechanisms as individual sub-bursts.

###  3.2 Duration vs. Frequency and Slope vs. Frequency Relationships

The TRDM also predicts an inverse relationship between sub-burst duration and
frequency, articulated as

$t_{\mathrm{w}}=(\tau_{\mathrm{w}}^{\prime}\nu_{0})\frac{1}{\nu_{\text{obs}}}=B\frac{1}{\nu_{\text{obs}}},$
(2)

(Rajabi et al., 2020). In this equation, $\nu_{0}$ is the FRB-frame frequency
of emission, assumed to be intrinsic to all FRB emission. Since the proper
duration $\tau_{\mathrm{w}}^{\prime}$ depends on the properties of the FRB
environment, for a given $\nu_{0}$ we expect each source to potentially
demonstrate different scaling for this relationship (Houde et al., 2019).
Furthermore, within each source we expect to see a significant spread in
$\tau_{\mathrm{w}}^{\prime}$, corresponding to a large vertical scatter in
$t_{\mathrm{w}}$ vs. $\nu_{\mathrm{obs}}$. Previous studies (Gajjar et al.,
2018; Chamma et al., 2023) found this inverse trend to hold for FRB 20121102A,
but a multi-source study remains to be conducted.

Figure 3: Sub-burst bandwidth relationships with frequency (top) and duration
(bottom). In the upper panel, the solid black line represents the TRDM-
predicted linear fit, while the dotted red line indicates the free power law
fit. It shows a trend where bursts exhibit broader bandwidths at higher centre
frequencies, consistent with mildly-relativistic Doppler broadening of narrow-
band emission. In the lower panel, the solid black line is the inverse
function derived from previously discussed relationships, and the dotted red
line is a free power law fit. This demonstrates a dependence of bandwidth on
duration approximating to the power of $-1/2$. Outlined points are multi-
component bursts presented for comparison but not included in the fits.

Combining equations (1) and (2) yields a quadratic relationship between sub-
burst slope and centre frequency:

$\derivative{\nu_{\text{obs}}}{t_{\mathrm{D}}}=-\frac{A\nu_{\text{obs}}^{2}}{\tau_{\mathrm{w}}^{\prime}\nu_{0}}\equiv-C\nu_{\text{obs}}^{2}$
(3)

(Chamma et al., 2021). In this equation, we have introduced a new constant
$C=1/(\tau_{\mathrm{D}}^{\prime}\nu_{0})=A/B$. Similar to the inverse
duration-frequency relationship, the source-specific properties influencing
$\tau_{\mathrm{D}}^{\prime}$ suggest that each source should potentially
exhibit a unique $C$ value. We also anticipate significant scatter in
$\differential\nu_{\mathrm{obs}}/\differential t_{\mathrm{D}}$ due to the
intrinsic spread in $\tau_{\mathrm{D}}^{\prime}$. Chamma et al. (2023) found
this law to hold for FRB 20121102A, and an overall trend between multiple
sources was observed by Chamma et al. (2021) and Wang et al. (2022).

These relationships – the dependencies of sub-burst duration and slope on
frequency – are illustrated in Figure 2. We model the duration-frequency data
using the TRDM’s predicted inverse relationship in equation (2). This yields a
coefficient of $B=1490\pm 60$ ms MHz, closely matching the $1474\pm 48$ ms MHz
reported for FRB 20121102A by Chamma et al. (2023). A power law fit
$B_{\text{free}}\nu_{\text{obs}}^{n_{2}}$ yields $B_{\text{free}}=502.90\pm
0.04$ and $n_{2}=-0.80\pm 0.05$, consistent with the predicted inverse trend.

We then fit the sub-burst slope vs. frequency data with the predicted
quadratic trend $C\nu_{\text{obs}}^{2}$, finding $C=(7.5\pm 0.3)\times
10^{-5}$ ms-1 MHz-1. The free power law fit
$C_{\text{free}}\nu_{\text{obs}}^{n_{3}}$ agreed with the quadratic trend,
with $C_{\text{free}}=6.4\times 10^{-5}\pm 23$ and $n_{3}=2.0\pm 0.2$. Both
results are similar to the coefficient of $(6.3\pm 0.3)\times 10^{-5}$ ms-1
MHz-1 found by Chamma et al. (2023). Significantly, these coefficients closely
resemble the predicted value of $C=A/B=5.2\times 10^{-5}$ ms-1 MHz-1, thereby
affirming the consistency among the relationships established between sub-
burst slope, duration and frequency (equations 1, 2, and 3).

The multi-component bursts were included in Figure 2 for comparison but were
not included in the fits. While we do not necessarily expect the same physical
mechanism to govern the relationships between duration and frequency or slope
and frequency for these bursts, they do also follow a clear trend of shorter
durations and steeper slopes at higher frequencies.

As anticipated, the data clearly show unique scaling of these relationships
for different sources, as several sources have most of their bursts lying
either above or below the trend lines. However, the overall data roughly
follow the inverse and quadratic fits, with no sources acting as strong
outliers. This suggests that the intrinsic duration
$\tau_{\mathrm{w}}^{\prime}$ and delay time $\tau_{\mathrm{D}}^{\prime}$,
which define these relationships, might be constrained to a given range due to
the underlying physical mechanism.

###  3.3 Bandwidth vs. Frequency and Bandwidth vs. Duration Relationships

In the TRDM, FRB emission is intrinsically narrow band at frequency $\nu_{0}$,
with the observed spectral width arising from the large velocity range of the
emitting medium (Houde et al., 2019; Rajabi et al., 2020). If the velocity
range is non- or mildly-relativistic, this Doppler broadening yields a linear
relationship between the observed bandwidth $\Delta\nu_{\text{obs}}$ and
frequency $\nu_{\text{obs}}$ of the form $\Delta\nu_{\text{obs}}\approx
2\Delta\beta^{\prime}\nu_{\text{obs}}$, where $\Delta\beta^{\prime}$
represents the rest frame velocity dispersion of the emitting gas444This
linear relationship was derived by performing a second order Taylor expansion
for low $\Delta\beta$’ on the relativistic Doppler broadening formula
(equation B6 in Chamma et al. 2021).. This is a powerful relationship as it
may allow us to probe the dynamics of FRB sources.

Figure 3 (top panel) displays the dependence of sub-burst bandwidth $B_{\nu}$
on center frequency. We fit the data with a linear function
$B_{\nu}=D\nu_{\text{obs}}$, yielding a coefficient of $D=0.138\pm 0.004$
consistent with $0.14\pm 0.004$ obtained by Chamma et al. (2023) for FRB
20121102A. An alternative power law fit
$D_{\text{free}}\nu_{\text{obs}}^{n_{4}}$ results in $D_{\text{free}}=0.1\pm
20$ and $n_{4}=1.01\pm 0.06$, aligning with the anticipated linear trend.
These data display a clear trend of wider bandwidths at higher frequencies and
follow the linear fit closely with no prominent outliers. This is significant
as it signifies that all nine FRB sources have similar rest-frame velocity
ranges, with $\Delta\beta^{\prime}\approx 0.07$.

The results of our study confirm both the linear relationship between
bandwidth and frequency, and the inverse relationship of frequency with
duration as posited by the TRDM. Consequently, we anticipate an inverse
correlation between bandwidth and duration, of the form
$B_{\nu}=Et_{\mathrm{w}}^{-1}$. Based on the previous fits, this coefficient
has a predicted value of $E=B\cdot D=206$ MHz ms. This relationship is shown
in Figure 3 (bottom panel). However, the visual alignment of our data appears
to favor the free power law fit $E_{\mathrm{free}}t_{w}^{n_{5}}$, which yields
$E_{\mathrm{free}}=270\pm 10$ and $n_{5}=-0.53\pm 0.05$. Remarkably, these
free parameters mirror those identified by Chamma et al. (2023)—a coefficient
of $272\pm 12$ and an exponent of $-0.53\pm 0.04$. This unexpected
functionality between the bandwidth and duration is unlikely to be a
characteristic of FRB sources but rather the result of propagation effects,
and will be explored further in Kumar et al. (in prep).

##  4 Summary

This study was a comprehensive spectro-temporal analysis of nine repeating
FRBs, the largest sample to date.

  * •

We analyzed 175 sub-bursts and 31 multi-component bursts over a frequency
range of 149–7144 MHz and a duration range of 73$\mu$s–13.22 ms, using data
from seven telescopes. The sources analyzed were FRB 20121102A, 20180916B,
20201124A, 20180301A, 20180814A, 20200929C, 20190804E, 20201130A, and
20190915D.

  * •

We tested the predictions of the TRDM, which proposes that FRBs are emitted
when a signal from a background source is amplified and triggers strong,
narrow band emission in a large region of gas moving relativistically relative
to the observer.

  * •

We established the sub-burst slope law, an inverse relationship between
frequency-normalized sub-burst slope and duration, to hold among sources.

  * •

We also discovered the drift rates of multi-component bursts to follow the
same trend with duration as sub-bursts, which is not fully explained by the
TRDM. It therefore appears that bursts and sub-bursts can be analyzed
together, removing the need to split the former into the latter.

  * •

We found an inverse relationship between sub-burst duration and frequency, and
a quadratic relationship between sub-burst slope and frequency. Significant
scatter within each source is consistent with intrinsic spread in the FRB-
frame sub-burst duration and delay time, while relative agreement in the
fitting coefficients between sources indicates that these timescales may be
constrained to a given range.

  * •

We established a linear relationship between sub-burst bandwidth and
frequency, consistent with mildly-relativistic Doppler broadening of narrow-
band emission with a single rest-frame frequency. Close agreement between the
sources suggest consistency among their rest-frame velocity spreads, with
$\Delta\beta^{\prime}\approx 0.07$.

  * •

We observed an inverse square-root proportionality between sub-burst bandwidth
and duration, which is not explained by the TRDM.

Overall, the results lend strong support to the TRDM as a viable framework for
explaining several aspects of FRB emission. The observed spectro-temporal
properties can largely be attributed to the inherent dynamics of the source.
Additionally, our findings suggest significant dynamic similarities across
sources of repeating FRB, implying shared underlying mechanisms for the
phenomena observed.

## Acknowledgements

M.H.’s research is funded through the Natural Sciences and Engineering
Research Council of Canada Discovery Grant RGPIN-2016-04460.

## Data Availability

The waterfalls of the FRBs can be obtained from the authors of their
respective publications. The graphical user interface FRBGUI and related
scripts used to perform the measurements are available at
github.com/mef51/frbgui.

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As we show below, this simple modification is indeed sufficient to obtain
equivalence. Here we imply that the energies depending on three parameters
$\alpha$, $\beta$ and $\gamma$ (instead of $a$, $b$ and $1$, respectively),
can be tuned appropriately to construct the correct limiting energy.

More specifically, for any $\varepsilon>0$ we define for $u\in SBV(I)$ and
$v\in H^{1}(I)$ the functional

$G^{\mathbb{Z}}_{\varepsilon}(u,v;I)=\begin{cases}G_{\varepsilon}(u,v;I)&\hbox{
if }S(u)\subset\varepsilon\mathbb{Z}\cr+\infty&\hbox{ otherwise.}\end{cases}$
(6.9)

By the general homogenization theorem [21, Th. 3] we get the following
$\Gamma$-convergence result.

######  Proposition 6.3.

The sequence $G^{\mathbb{Z}}_{\varepsilon}(u,v;I)$ $\Gamma$-converges with
respect to the convergence $u_{\varepsilon},v_{\varepsilon}\to v$ in
$L^{2}(I)$ to

$G^{\mathbb{Z}}_{\rm hom}(v)=\int_{I}g^{\mathbb{Z}}_{\rm hom}(v^{\prime})\,dt$
(6.10)

where

$g^{\mathbb{Z}}_{\rm
hom}(z)=\displaystyle\lim_{N\to+\infty}\frac{1}{N}\inf\\{G_{1}^{\mathbb{Z}}(u,v;(0,N))\\!:\
\\!u(0)=v(0)=0,\ \\!u(N)=v(N)=Nz\\}.$ (6.11)

The proof of Proposition 6.3 can be obtained by following the steps of the
proof of [21, Theorem 3]. Indeed, in the blow-up procedure the jump set
$S(u_{\varepsilon})$ is not modified, and the $\liminf$ inequality follows.
Concerning the upper estimate, by density we can consider a piecewise-affine
target function $v$ such that $S(v^{\prime})\subset\mathbb{Q}$; then, the
construction of the recovery sequence can be done by following the same steps
as in the the proof of [21, Theorem 3], and the scaling argument gives
$u_{\varepsilon}$ such that $S(u_{\varepsilon})\subset\varepsilon\mathbb{Z}$.
Note that the function $g_{\rm hom}^{\mathbb{Z}}$ is convex.

Now we will show that the sequence $G^{\mathbb{Z}}_{\varepsilon}(u,v;I)$ has
the same $\Gamma$-limit as the discrete sequence $E_{\varepsilon}$ for a
suitable choice of the parameters $\alpha,\beta,\gamma$. We define

$g(N,z)=\displaystyle\frac{1}{N}\min\\{\tilde{G}_{1}(u,v;(0,N))\\!:\ u,v\in
H^{1}(0,N),\ v(0)=0,\ v(N)=Nz\\},$ (6.12)

where, in analogy with (5.29), we denote by $\tilde{G}_{1}$ the (non scaled)
functional given by

$\tilde{G}_{1}(u,v;I)=\int_{I}\Big{(}\gamma(u^{\prime})^{2}+\alpha(v^{\prime})^{2}+\beta(u-v)^{2}\Big{)}\,dt.$

By solving the Euler-Lagrange equations for $\tilde{G}_{1}$ and minimizing on
the boundary values of $u$, it follows that

$g(N,z)=\displaystyle\lambda_{N}z^{2}$ (6.13)

with $\lambda_{N}$ defined in (6.8). Note that the (unique) solution
$(u_{N},v_{N})$ of the minimum problem defining $g(N,z)$ satisfies the
symmetry property $u(\frac{N}{2})=v(\frac{N}{2})=\frac{N}{2}z$.

######  Proposition 6.4.

For any $z\in\mathbb{R}$ the following equality holds:

$g^{\mathbb{Z}}_{\rm
hom}(z)=\Big{(}\inf_{N\in\mathbb{N}}\Big{\\{}\lambda_{N}z^{2}+\frac{\eta}{N}\Big{\\}}\Big{)}^{\ast\ast}.$
(6.14)

###### Proof.

We fix $z\in\mathbb{R}$ and $N\in\mathbb{N}$; let $(u_{N},v_{N})$ be the
solution of the minimum problem defining $\psi(N,z)$. We define
$\tilde{u}_{N}\in SBV(0,2N)$ by setting

$\tilde{u}_{N}(t)=\left\\{\begin{array}[]{ll}2u_{N}(\frac{t+N}{2})-Nz&\hbox{
if }\ t\in(0,N)\\\ 2u_{N}(\frac{t-N}{2})+Nz&\hbox{ if }\
t\in(N,2N)\end{array}\right.$ (6.15)

and correspondingly $\tilde{v}_{N}\in H^{1}(0,2N)$. Since
$u_{N}(\frac{N}{2})=v_{N}(\frac{N}{2})=\frac{N}{2}z$, then
$S(\tilde{u}_{N})=\\{N\\}$, $\tilde{u}_{N}(0)=\tilde{v}_{N}(0)=0$ and
$\tilde{u}_{N}(2N)=\tilde{v}_{N}(2N)=2Nz$; by construction

$\frac{1}{2N}\tilde{G}_{1}(\tilde{u}_{N},\tilde{v}_{N};(0,2N))=\frac{1}{N}\tilde{G}_{1}(u_{N},v_{N};(0,N))=\displaystyle\lambda_{N}z^{2}.$

Let $k\in\mathbb{N}$. We define $\tilde{u}$ in $(0,2kN)$ by setting

$\tilde{u}(t)=\tilde{u}_{N}(t-2jN)+2jNz\quad\hbox{ in }\quad(2jN,2(j+1)N),\ \
\ j=0,\dots,k-1$

and in the same way we define $\tilde{v}$. By construction,
$S(\tilde{u})\subset\mathbb{N}$ and $\\#S(\tilde{u})=k-1$; hence, since the
boundary conditions for $\tilde{u}$ and $\tilde{v}$ hold, we have

$\displaystyle\displaystyle\lambda_{N}z^{2}+\frac{\eta}{N}$ $\displaystyle=$
$\displaystyle\frac{1}{2kN}\tilde{G}_{1}(\tilde{u},\tilde{v};(0,2kN))+\frac{\eta(k-1)}{kN}+\frac{\eta}{kN}$
$\displaystyle=$
$\displaystyle\frac{1}{2kN}G_{1}^{\mathbb{Z}}(\tilde{u},\tilde{v};(0,2kN))+\frac{\eta}{kN}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2kN}\inf\Bigl{\\{}G_{1}^{\mathbb{Z}}(u,v;(0,2kN)):$
$\displaystyle\hskip
28.45274ptu(0)=v(0)=0,u(kN)=v(kN)=2kNz\Bigr{\\}}+\frac{\eta}{kN},$

and, by taking the limit as $k\to+\infty$,

$\displaystyle\displaystyle\lambda_{N}z^{2}+\frac{\eta}{N}\geq g_{\rm
hom}^{\mathbb{Z}}(z).$

Hence, since $g_{\rm hom}^{\mathbb{Z}}$ is convex,

$\displaystyle\Big{(}\inf_{N\in\mathbb{N}}\Big{\\{}\displaystyle\lambda_{N}z^{2}+\frac{\eta}{N}\Big{\\}}\Big{)}^{\ast\ast}\geq
g_{\rm hom}^{\mathbb{Z}}(z).$

Next we need to prove the opposite inequality. Let $u\in SBV(0,N)$ and $v\in
H^{1}(0,N)$ be such that the boundary conditions $u(0)=v(0)=0$, $u(N)=v(N)=Nz$
hold and $S(u)\subset\mathbb{N}$. We denote the jump points of $u$ by $N_{i}$,
$i=1,\dots k$, with $N_{i}<N_{i+1}$ for any $i=1,\dots,k-1$. Setting $N_{0}=0$
and $N_{k+1}=N$, we define

$n_{i}=N_{i}-N_{i-1}\quad\hbox{ and }\quad
z_{i}=\frac{v(N_{i})-v(N_{i-1})}{n_{i}}$

for $i=1,\dots k+1$. We then have

$\frac{1}{n_{i}}\tilde{G}_{1}(u,v;(N_{i-1},N_{i}))\geq
g(n_{i},z_{i})=\lambda_{n_{i}}z_{i}^{2}$

for any $i$, so that

$\displaystyle\frac{1}{N}G_{1}^{\mathbb{Z}}(u,v;(0,N))$ $\displaystyle\geq$
$\displaystyle\sum_{i=1}^{k+1}\frac{n_{i}}{N}\lambda_{n_{i}}z_{i}^{2}+\frac{\eta
k}{N}\ =\
\sum_{i=1}^{k+1}\frac{n_{i}}{N}\big{(}\lambda_{n_{i}}z_{i}^{2}+\frac{\eta}{n_{i}}\big{)}$
$\displaystyle\geq$
$\displaystyle\sum_{i=1}^{k+1}\frac{n_{i}}{N}\inf_{n\in\mathbb{N}}\Big{\\{}\lambda_{n}z_{i}^{2}+\frac{\eta}{n}\Big{\\}}$

Since $\sum_{i=1}^{k+1}n_{i}=N$ and $\sum_{i=1}^{k+1}n_{i}z_{i}=Nz$, an
application of Carathéodory’s Theorem gives

$\displaystyle\frac{1}{N}G_{1}^{\mathbb{Z}}(u,v;(0,N))$ $\displaystyle\geq$
$\displaystyle\Big{(}\inf_{n\in\mathbb{N}}\Big{\\{}\lambda_{n}z^{2}+\frac{\eta}{n}\Big{\\}}\Big{)}^{\ast\ast}.$

Taking the $\inf$ over the admissible functions and the limit for
$N\to+\infty$ we get the inequality

$\displaystyle g_{\rm hom}^{\mathbb{Z}}(z)$ $\displaystyle\geq$
$\displaystyle\Big{(}\inf_{n\in\mathbb{N}}\Big{\\{}\lambda_{n}z^{2}+\frac{\eta}{n}\Big{\\}}\Big{)}^{\ast\ast}$

concluding the proof. ∎

Now, if we choose

$\displaystyle\alpha=\displaystyle\frac{a(a+1)}{a+\zeta\coth(\zeta)},\quad\displaystyle\beta=\displaystyle\frac{4a(a+1)\zeta^{3}\coth(\zeta)}{(a+\zeta\coth(\zeta))^{2}},\quad\displaystyle\gamma=\displaystyle\frac{(a+1)\zeta\coth(\zeta)}{a+\zeta\coth(\zeta)}$
(6.16)

it follows that $\omega=2\zeta,$ where $\zeta$ is defined in (5.55), and for
any $N$ the following equality holds

$\lambda_{N}=c_{N}=\frac{N(a+1)a}{aN+\tanh(N\zeta)\coth(\zeta)}.$

We can then state the following equivalence result, whose proof follows from
the equivalence between $E_{\varepsilon}$ and $F_{\varepsilon}$ (Theorem 5.6
and Remark 5.8) and the results above.

######  Theorem 6.5 (equivalence with the Novak-Truskinovsky model).

Choosing the coefficients as in (6.16), the sequence
$G_{\varepsilon}^{\mathbb{Z}}$ defined in (6.9) $\Gamma$-converges with
respect to the $L^{2}$-convergence to the same $\Gamma$-limit of the sequence
of discrete functionals $E_{\varepsilon}$ in the truncated quadratic case.

We reiterate that in general, the above result can be viewed as a cautionary
tale, showing that relaxation and homogenization (discrete-to-continuum limit)
do not always commute.

##  7 Conclusions

In this paper, we systematically explored the possibility of using some
auxiliary ‘local’ considerations to obtain minimizers with ‘global’ features
for nonlocal variational boundary-value problems on lattices. Having in mind
some known cases when asymptotically (i.e. in continuum limit) such boundary-
value problems exhibit periodic minimizers, we associated the possibility of
‘local’ description with applicability of the GCB rule and posed the question
of the pertinence of such a rule for a generic variational problems in our
class. It is clear that the GCB rule is not applicable in general, for
instance, it clearly fails in the case of minimization with concentrations,
appearing in non-coercive problems of fracture mechanics. Here we extended the
known class of non-GCB problems by incorporating into the analysis some
general non-convex energy densities with quadratic growth.

More specifically, we used the simplest examples of functionals with
quadratically penalized non-convexity, we demonstrated various facets of
frustration and incompatibility in one-dimensional discrete variational
problems computed on an increasing and diverging number of nodes. In the
chosen class of non-convex lattice problems with energy density $f$, linear
long-range interactions were introduced through an infinite matrix $\bf m$. We
studied relaxation of such problems with given boundary conditions on
intervals with a large number of nodes. This operation can be interpreted as a
discrete-to-continuum $\bf m$-transform of the function $f$ and we studied the
dependence of such a transform on the parameter $z$ describing boundary
conditions.

We addressed the question whether the minimizers for a given functional are
close to functions with ‘global’ properties, for instance, to periodic
functions, where closeness can be understood as having the same energy up to
an asymptotically negligible quantity as the number of nodes diverges. The
answer is in general negative, for example, this is not true in the case of
minimizers describing transitions between two energy wells, when the parameter
$z$ lies in some intervals. Still, we were able to identify interesting cases
when the knowledge of the minimizers, that are asymptotically of a ‘global’
form, are sufficient to determine the whole $\bf m$-transform of the function
$f$ through some form of convexification.

Outside our general considerations, we mostly focused on potentials $f$ with a
bi-convex form; i.e., which have a convex restriction to two complementary
phase sets. For boundary-value problems involving such potentials and
prescribed $z$ it is natural to define phase functions $\theta(z)$. We have
shown that of particular interest are values of $\theta$ for which the set
$\\{z:\theta(z)=\theta\\}$ contains a non-degenerate interval (locking
states). We studied the main properties of both, the functions $\theta(z)$ and
of locking states, and showed that for some combinations of $f$ and $\bf m$
the minimizers representing the locking states are periodic and hence of a
‘global’ (or GCB) nature in the sense that they determine the whole $\bf
m$-transform of the function $f$. We also showed that the optimal periodic
minimizers whose structure may depend delicately on $f$ and $\bf m$ are not
necessarily unique. Among different optimal minimizers we identified universal
periodic microstructures, which exist for all values of $\theta$ and have
fascinating analogs in the theory of dynamical systems.

The concept of $\bf m$-transform, introduced in this paper for the first time,
was shown to be rather rich. The complexity of the ensuing transformations
suggests that even in scalar one-dimensional problems, the interplay of long-
range interactions, non-convexity and discreteness can be highly nontrivial.
We presented several examples where the $\bf m$-transform of a given non-
convex function could be either computed explicitly or narrowly bounded. Some
of the obtained $\bf m$-transforms were shown to be singular exhibiting the
‘devilish’ features with locking on some but not all _rational_
microstructures.

The analytical accessibility of the $\bf m$-transforms in the presented
examples, as well as the associated non-uniqueness of the optimal micro-
structures, hint towards a certain _degeneracy_ of the chosen problems. We can
associate such a degeneracy with the absence of ‘strong’ geometrical
frustration representing some fundamental incommensuration between the non-
convexity, the long range interactions and the discreteness. It is clear that
more complex optimal minimizing sequences, not reducible to periodic states or
combinations of periodic states, can be expected in cases when such
incommensuration is present.

The ‘strong’ frustration of this type may be driven, for instance, by the
competing interactions inside the kernel m, for instance, by the combination
of ferromagnetic and antiferromagnetic interactions acting on incommensurate
scales. The frustration can be also ‘strong’ even in the apparently simple
case when different scales are ‘favored’ by antiferromagnetic interaction
involving the first and the third nearest neighbors. ‘Strong’ frustration may
also be brought by the structure of the non-convex function $f$ carrying the
‘characteristic strain’ which is incompatible with the strain emerging through
the interplay between the loading and the long-range interaction kernel, see
for instance [74] where a ‘complete devil staircase’ emerges in a problem
involving a non-degenerate bi-quadratic potential and an exponential kernel.

In a separate paper we will show that the presence of ‘strong’ frustration may
eliminate the degeneracy and bring the uniqueness to the problem of finding
the optimal microstructure. More generally, our preliminary analysis of
problems with ‘strong’ frustration reveals an even deeper link between lattice
variational problem and the discrete nonlinear mappings where the analog of
constructing the $\bf m$-transform turns out to be the problem of classifying
all quasi-periodic trajectories.

### Acknowledgments

AC and MS acknowledge the projects ‘Fondo di Ateneo per la Ricerca 2019’ and
‘Fondo di Ateneo per la Ricerca 2020’, funded by the University of Sassari.
This work has been supported by PRIN 2017 ‘Variational methods for stationary
and evolution problems with singularities and interfaces’. AB and MS are
members of GNAMPA, INdAM, AC is member of GNSAGA, INdAM. The authors
acknowledge the MIUR Excellence Department Project awarded to the Department
of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. The work
of LT was supported by the grant ANR-10-IDEX-0001-02 PSL.

##  Appendix A Appendix: variations of boundary data

In this appendix we state and prove some technical results which allow the
modification of boundary values of test functions for the minimum problems
used in various characterization of $Q_{\bf m}f$. In particular, these results
allow to assume that test functions be constant close to the endpoints of the
domain.

Let ${\bf m}=\\{m_{n}\\}_{n}$ be such that $m_{n}\geq 0$ for any $n$, and
there exists $\overline{n}$ such that $m_{n}$ is not increasing for
$n\geq\overline{n}$. Moreover, we assume the decay condition
$m_{n}=o(n^{-\beta})_{n\to+\infty}$ for some $\beta>2$.

Let $F_{\varepsilon}$ be defined as in (2.8); that is,

$F_{\varepsilon}(u;I)=\sum_{\varepsilon i,\varepsilon(i-1)\in
I}\varepsilon\,f\Big{(}\frac{u_{i}-u_{i-1}}{\varepsilon}\Big{)}+\sum_{\varepsilon
i,\varepsilon j\in
I}\varepsilon\,m_{|i-j|}\Big{(}\frac{u_{i}-u_{j}}{\varepsilon}\Big{)}^{2}$

for $I$ interval and $u\in\mathcal{A}_{\varepsilon}(I)$.

######  Lemma A.1.

Let $L>0$ and $N_{\varepsilon}=\lfloor\frac{L}{\varepsilon}\rfloor$. Let
$\alpha\in(\frac{2}{\beta},1)$. Assume that $u\in L^{2}(0,L)$ and
$u^{\varepsilon}\in\mathcal{A}_{\varepsilon}=\mathcal{A}_{\varepsilon}(0,L)$
be such that (the piecewise-affine extension of) the sequence
$u^{\varepsilon}$ converges to $u$ in $L^{2}(0,L)$, and
$\sup_{\varepsilon}(F_{\varepsilon}(u^{\varepsilon};[0,L])+\|u^{\varepsilon}\|^{2}_{L^{2}})=S<+\infty$.
Then, there exists $\hat{u}^{\varepsilon}\in\mathcal{A}_{\varepsilon}$
converging to $u$ such that

  1. (i)

$\hat{u}^{\varepsilon}_{i}=\hat{u}^{\varepsilon}_{0}$ for
$i\leq\varepsilon^{-\alpha}$,
$\hat{u}^{\varepsilon}_{i}=\hat{u}^{\varepsilon}_{N_{\varepsilon}}$ for $i\geq
N_{\varepsilon}-\varepsilon^{-\alpha}$;

  2. (ii)

$F_{\varepsilon}(\hat{u}^{\varepsilon};[0,L])\leq
F_{\varepsilon}(u^{\varepsilon};[0,L])+r(\varepsilon)$, where the remainder
$r$ depends only on $S$ and $f(0)$, and $r(\varepsilon)\to 0$ as
$\varepsilon\to 0$.

###### Proof.

We choose $\alpha^{\prime}\in(0,1-\alpha)$ and define
$\lambda_{\varepsilon}=\varepsilon^{\alpha^{\prime}}$ and
$M_{\varepsilon}=\lfloor\varepsilon^{\alpha+\alpha^{\prime}-1}\rfloor-1$. For
$\varepsilon$ small enough we divide $(0,\lambda_{\varepsilon}]$ and
$[L-\lambda_{\varepsilon},L)$ in $M_{\varepsilon}+1$ intervals by setting

$I^{k}_{\varepsilon}=\Big{(}\frac{k\lambda_{\varepsilon}}{M_{\varepsilon}+1},\frac{(k+1)\lambda_{\varepsilon}}{M_{\varepsilon}+1}\Big{]},\
\
J^{k}_{\varepsilon}=\Big{[}L-\frac{(k+1)\lambda_{\varepsilon}}{M_{\varepsilon}+1},L-\frac{k\lambda_{\varepsilon}}{M_{\varepsilon}+1}\Big{)},\
\ k\in\\{0,\dots,M_{\varepsilon}\\}.$

Since

$\frac{1}{\varepsilon}\sum_{k=1}^{M_{\varepsilon}}\sum_{\varepsilon i\in
I_{\varepsilon}^{k},\varepsilon j\in
I_{\varepsilon}^{k-1}}m_{|i-j|}(u^{\varepsilon}_{i}-u^{\varepsilon}_{j})^{2}\leq
F_{\varepsilon}(u^{\varepsilon};[0,L])\leq S,$

then there exists $k_{\varepsilon}^{-}\in\\{1,\dots,M_{\varepsilon}\\}$ such
that

$\frac{1}{\varepsilon}\sum_{\varepsilon i\in
I_{\varepsilon}^{k_{\varepsilon}^{-}},\varepsilon j\in
I_{\varepsilon}^{k_{\varepsilon}^{-}-1}}m_{|i-j|}(u^{\varepsilon}_{i}-u^{\varepsilon}_{j})^{2}\leq\frac{S}{M_{\varepsilon}}.$
(A.1)

The same argument allows to find
$k_{\varepsilon}^{+}\in\\{1,\dots,M_{\varepsilon}\\}$ such that the same
inequality holds for $\varepsilon i\in
J_{\varepsilon}^{k_{\varepsilon}^{+}},\varepsilon j\in
J_{\varepsilon}^{k_{\varepsilon}^{+}-1}$. Setting
$j^{-}_{\varepsilon}=\min\\{j:\varepsilon j\in
I_{\varepsilon}^{k_{\varepsilon}^{-}}\\}$ and
$j^{+}_{\varepsilon}=\max\\{j:\varepsilon j\in
J_{\varepsilon}^{k_{\varepsilon}^{+}}\\}$, we define $\hat{u}^{\varepsilon}$
by setting

$\hat{u}^{\varepsilon}_{i}=\left\\{\begin{array}[]{ll}u^{\varepsilon}_{j^{-}_{\varepsilon}}&\hbox{
if }i\leq j^{-}_{\varepsilon}\\\ u^{\varepsilon}_{i}&\hbox{ if
}j^{-}_{\varepsilon}\leq i\leq j^{+}_{\varepsilon}\\\
u^{\varepsilon}_{j^{+}_{\varepsilon}}&\hbox{ if }i\geq
j^{+}_{\varepsilon}.\end{array}\right.$ (A.2)

Since $j_{\varepsilon}^{-}\geq L\varepsilon^{-\alpha}$ and
$j_{\varepsilon}^{+}\leq N_{\varepsilon}-L\varepsilon^{-\alpha}$, then
$\hat{u}^{\varepsilon}$ satisfies claim (i). Moreover,
$\hat{u}^{\varepsilon}\to u$ as $\varepsilon\to 0$. To prove this, for
simplicity we suppose that $m_{n}$ is not increasing for $n\geq 1$. Then,

$\displaystyle\varepsilon\sum_{i=1}^{j^{-}_{\varepsilon}}(u^{\varepsilon}_{i}-\hat{u}^{\varepsilon}_{i})^{2}$
$\displaystyle=$
$\displaystyle\varepsilon\sum_{i=1}^{j^{-}_{\varepsilon}}(u^{\varepsilon}_{i}-u^{\varepsilon}_{j^{-}_{\varepsilon}})^{2}\leq\varepsilon\sum_{i=1}^{j^{-}_{\varepsilon}}j^{-}_{\varepsilon}\\!\\!\sum_{j=i+1}^{j^{-}_{\varepsilon}}\\!(u^{\varepsilon}_{i}-u^{\varepsilon}_{i-1})^{2}$
$\displaystyle\leq$
$\displaystyle\frac{S}{m_{1}}\varepsilon^{2}(j^{-}_{\varepsilon})^{2}\leq\frac{S}{m_{1}}\lambda_{\varepsilon}^{2},$

and correspondingly $\varepsilon\sum_{i=j^{+}_{\varepsilon}}^{\lfloor
L/\varepsilon\rfloor}(u^{\varepsilon}_{i}-\hat{u}^{\varepsilon}_{i})^{2}\leq\frac{S}{m_{1}}\lambda_{\varepsilon}^{2}$.
Setting,
$n_{\varepsilon}=\lfloor\frac{\lambda_{\varepsilon}}{\varepsilon(M_{\varepsilon}+1)}\rfloor$,
since

$\sum_{|i-j|\geq
n_{\varepsilon}}m_{|i-j|}(u^{\varepsilon}_{i}-u^{\varepsilon}_{j})^{2}\leq\frac{2}{\varepsilon}m_{n_{\varepsilon}}\|u^{\varepsilon}\|^{2}_{L_{2}}\leq\frac{2}{\varepsilon}m_{\lfloor\varepsilon^{-\alpha}\rfloor}\|u^{\varepsilon}\|^{2}_{L_{2}},$

and recalling (A.1), we obtain

$\displaystyle F_{\varepsilon}(\hat{u}^{\varepsilon};[0,L])\leq
F_{\varepsilon}(u^{\varepsilon};[0,L])+2\lambda_{\varepsilon}f(0)+\frac{C}{\varepsilon^{2}}m_{\lfloor\varepsilon^{-\alpha}\rfloor}+\frac{C}{M_{\varepsilon}},$

where $C$ denotes a constant depending only on
$\sup_{\varepsilon}F_{\varepsilon}(u^{\varepsilon};[0,L])$ and
$\sup_{\varepsilon}\|u^{\varepsilon}\|_{L^{2}}$. Setting

$r(t)=2f(0)t^{\alpha^{\prime}}+Ct^{\alpha\beta-2}+Ct^{1-\alpha-\alpha^{\prime}},$

we conclude the proof since $m_{n}=o(n^{-\beta})$ and
$\alpha>\frac{2}{\beta}$. ∎

Let $a,b>0$. We define the functional $E_{\varepsilon}(u,v;I)$ by setting

$E_{\varepsilon}(u,v;I)=\sum_{\varepsilon i,\varepsilon(i-i)\in
I}\varepsilon\,f\Big{(}\frac{u_{i}-u_{i-1}}{\varepsilon}\Big{)}+\frac{a}{2}\sum_{\varepsilon
i,\varepsilon(i-i)\in
I}\varepsilon\,\Big{(}\frac{v_{i}-v_{i-1}}{\varepsilon}\Big{)}^{2}+\frac{b}{2\varepsilon}\sum_{\varepsilon
i\in I}(u_{i}-v_{i})^{2}$ (A.3)

for $I$ interval and $u,v\in\mathcal{A}_{\varepsilon}(I)$.

######  Lemma A.2.

Let $L>0$ and $N_{\varepsilon}=\lfloor\frac{L}{\varepsilon}\rfloor$. Let
$\alpha\in(\frac{2}{\beta},1)$. Assume that
$u^{\varepsilon},v^{\varepsilon}\in\mathcal{A}_{\varepsilon}$ be such that
(the piecewise-affine extensions of) $u^{\varepsilon}$ and $v^{\varepsilon}$
converge to $u$ in $L^{2}(0,L)$ and
$\sup_{\varepsilon}(E_{\varepsilon}(u^{\varepsilon};[0,L])+\|u^{\varepsilon}\|^{2}_{L^{2}})=S<+\infty$.
Then there exist
$\hat{u}^{\varepsilon},\hat{v}^{\varepsilon}\in\mathcal{A}_{\varepsilon}$
converging to $u$ such that

  1. (i)

$\hat{u}^{\varepsilon}_{i}=\hat{v}^{\varepsilon}_{i}=\hat{u}^{\varepsilon}_{0}$
for $i\leq\varepsilon^{-\alpha}$,
$\hat{u}^{\varepsilon}_{i}=\hat{v}^{\varepsilon}_{i}=\hat{u}^{\varepsilon}_{N_{\varepsilon}}$
for $i\geq N_{\varepsilon}-\varepsilon^{-\alpha}$;

  2. (ii)

$E_{\varepsilon}(\hat{u}^{\varepsilon},\hat{v}^{\varepsilon};[0,L])\leq
E_{\varepsilon}(u^{\varepsilon},v^{\varepsilon};[0,L])+r(\varepsilon)$, where
the remainder $r$ depends only on $S$ and $f(0)$, and $r(\varepsilon)\to 0$ as
$\varepsilon\to 0$.

###### Proof.

We choose $\lambda_{\varepsilon}$ and $M_{\varepsilon}$ as in the proof of
Lemma A.1, and divide $(0,\lambda_{\varepsilon}]$ and
$[L-\lambda_{\varepsilon},L)$ in $M_{\varepsilon}+1$ intervals, denoted by
$I^{k}_{\varepsilon}$ and $J^{k}_{\varepsilon}$ respectively, as above. Then,
there exist $k_{\varepsilon}$ and $h_{\varepsilon}$ in
$\\{1,\dots,M_{\varepsilon}\\}$ such that

$\frac{1}{2\varepsilon}\sum_{\varepsilon i\in
I_{\varepsilon}^{k_{\varepsilon}}\cup
J_{\varepsilon}^{h_{\varepsilon}}}\big{(}a(v^{\varepsilon}_{i}-v^{\varepsilon}_{i-1})^{2}+b(u^{\varepsilon}_{i}-v^{\varepsilon}_{i})^{2}\big{)}\leq\frac{S}{M_{\varepsilon}}.$
(A.4)

Setting $j^{-}_{\varepsilon}=\min\\{j:\varepsilon j\in
I_{\varepsilon}^{k_{\varepsilon}}\\}$ and
$j^{+}_{\varepsilon}=\max\\{j:\varepsilon j\in
J_{\varepsilon}^{h_{\varepsilon}}\\}$, we define

$\hat{u}^{\varepsilon}_{i}=\left\\{\begin{array}[]{ll}u^{\varepsilon}_{j_{\varepsilon}^{-}}&\hbox{
if }i\leq j_{\varepsilon}^{-}\\\ u^{\varepsilon}_{i}&\hbox{ if
}j_{\varepsilon}^{-}<i<j_{\varepsilon}^{+}\\\
u^{\varepsilon}_{j_{\varepsilon}^{+}}&\hbox{ if }i\geq
j_{\varepsilon}^{+}\end{array}\right.\ \ \hbox{ and }\ \ \ \
\hat{v}^{\varepsilon}_{i}=\left\\{\begin{array}[]{ll}u^{\varepsilon}_{j_{\varepsilon}^{-}}&\hbox{
if }i\leq j_{\varepsilon}^{-}\\\ v^{\varepsilon}_{i}&\hbox{ if
}j_{\varepsilon}^{-}<i<j_{\varepsilon}^{+}\\\
u^{\varepsilon}_{j_{\varepsilon}^{+}}&\hbox{ if }i\geq
j_{\varepsilon}^{+},\end{array}\right.$

so that $\hat{u}^{\varepsilon}$ and $\hat{v}^{\varepsilon}$ converge to $u$ in
$L^{2}$, and satisfy (i). Recalling (A.4), we get in particular that

$\displaystyle\frac{a}{2\varepsilon}(\hat{v}^{\varepsilon}_{j_{\varepsilon}^{-}+1}-\hat{v}^{\varepsilon}_{j_{\varepsilon}^{-}})^{2}\leq\frac{a}{\varepsilon}(v^{\varepsilon}_{j_{\varepsilon}^{-}+1}-v^{\varepsilon}_{j_{\varepsilon}^{-}})^{2}+\frac{a}{\varepsilon}(v^{\varepsilon}_{j_{\varepsilon}^{-}}-u^{\varepsilon}_{j_{\varepsilon}^{-}})^{2}\leq\frac{C}{M_{\varepsilon}},$

where $C$ denotes a positive constant depending only on $a,b$ and $S$. The
same bound holds for
$\frac{a}{2\varepsilon}(\hat{v}^{\varepsilon}_{j_{\varepsilon}^{+}}-\hat{v}^{\varepsilon}_{j_{\varepsilon}^{+}-1})^{2}$.
Hence

$\displaystyle
E_{\varepsilon}(\hat{u}_{\varepsilon},\hat{v}_{\varepsilon};[0,L])$
$\displaystyle\leq$ $\displaystyle
2\lambda_{\varepsilon}f(0)+E_{\varepsilon}(u^{\varepsilon},v^{\varepsilon};(0,L))$
$\displaystyle+\frac{a}{2\varepsilon}(\hat{v}^{\varepsilon}_{j_{\varepsilon}^{-}+1}-\hat{v}^{\varepsilon}_{j_{\varepsilon}^{-}})^{2}+\frac{a}{2\varepsilon}(\hat{v}^{\varepsilon}_{j_{\varepsilon}^{+}}-\hat{v}^{\varepsilon}_{j_{\varepsilon}^{+}-1})^{2}$
$\displaystyle\leq$ $\displaystyle
2\lambda_{\varepsilon}f(0)+E_{\varepsilon}(u^{\varepsilon},v^{\varepsilon};[0,L])+\frac{2C}{M_{\varepsilon}},$

concluding the proof as above. ∎

######  Remark A.3.

In the hypotheses of Lemma A.2, if there exists $\alpha\in(0,1)$ such that
$u^{\varepsilon}_{i}=\hat{u}^{\varepsilon}_{0}$ for
$i\leq\varepsilon^{-\alpha}$ and
$u^{\varepsilon}_{i}=\hat{u}^{\varepsilon}_{N_{\varepsilon}}$ for $i\geq
N_{\varepsilon}-\varepsilon^{-\alpha}$ for some $\alpha>0$, then the function
$\hat{v}^{\varepsilon}$ can be chosen such that it coincides with
$u^{\varepsilon}$ for $i\leq\varepsilon^{-\alpha^{\prime\prime}}$ and for
$i\geq N_{\varepsilon}-\varepsilon^{-\alpha^{\prime\prime}}$ with
$\alpha^{\prime\prime}<\alpha$.

##  Appendix B Appendix: formulas for $P^{M,n}$ in the concentrated case

In this appendix we include some explicit computations of the functions
$P^{M,n}$ defined in (3.1), which are the energies of the locking states
$n\over M$ in the concentrated case. The formulas of these functions have been
used in Sections 4.2.1 and 4.2.2 to highlight the structure of $Q_{\bf m}f(z)$
in the truncated-parabolic and double-well case, respectively. Here, we
include the corresponding computations.

##### Truncated-parabolic case.

Let $f$ be given by (4.10). In view of (4.3), the domains of $P^{M,0}$ and
$P^{M,M}$ are $\\{z\leq 1\\}$ and $\\{z\geq 1\\}$, respectively. We recall
that here

$P^{M,0}(z)=z^{2}+2(m_{1}+m_{M}M^{2})z^{2}\ \ \hbox{\rm and }\ \
P^{M,M}(z)=1+2(m_{1}+m_{M}M^{2})z^{2}.$

For $n=1,\dots,M-1$, we can also write

$P^{M,n}(z)=\begin{cases}\vspace{2mm}\displaystyle\frac{2m_{1}+1}{1-\theta_{n}}\big{(}z^{2}-\theta_{n}(2z-1)\big{)}+2m_{M}M^{2}z^{2}&\displaystyle\hbox{\rm
if }\ z\leq T_{n}^{-}\\\
\vspace{2mm}\displaystyle\theta_{n}+\frac{2m_{1}(2m_{1}+1)}{2m_{1}+\theta_{n}}z^{2}+2m_{M}M^{2}z^{2}&\displaystyle\hbox{\rm
if }\ T_{n}^{-}\leq z\leq T_{n}^{+}\\\ \displaystyle
1+\frac{2m_{1}}{\theta_{n}}\big{(}(z-1)^{2}+\theta_{n}(2z-1)\big{)}+2m_{M}M^{2}z^{2}&\displaystyle\hbox{\rm
if }\ z\geq T_{n}^{+},\end{cases}$ (B.1)

where

$T_{n}^{-}=\frac{2m_{1}+\theta_{n}}{2m_{1}+1}\ \ \hbox{\rm and }\ \
T_{n}^{+}=\frac{2m_{1}+\theta_{n}}{2m_{1}}.$

Figure 32: Envelope of two consecutive functions $P^{M,n}(z)$

Note that while the formula defining $P^{M,n}$ changes form at $z=T_{n}^{-}$
and $z=T_{n}^{+}$, the computation of the common tangent points of $P^{M,n}$
and $P^{M,n+1}$ involves only the central formula in (B.1). Consequently, the
points $s_{n}^{+}$ and $s_{n}^{-}$ in Theorem 4.1 are

$\left.\begin{array}[]{ll}&\displaystyle
s_{n}^{+}=s_{n}^{+}(m_{1},m_{M})=\frac{2m_{1}+\theta_{n}}{\sqrt{2m_{1}(2m_{1}+1)}}\
\sqrt{\frac{m_{1}(2m_{1}+1)+m_{M}M^{2}(2m_{1}+\theta_{n+1})}{m_{1}(2m_{1}+1)+m_{M}M^{2}(2m_{1}+\theta_{n})}}\vspace{3mm}\\\
&\displaystyle
s_{n}^{-}=s_{n}^{-}(m_{1},m_{M})=\frac{2m_{1}+\theta_{n}}{\sqrt{2m_{1}(2m_{1}+1)}}\
\sqrt{\frac{m_{1}(2m_{1}+1)+m_{M}M^{2}(2m_{1}+\theta_{n-1})}{m_{1}(2m_{1}+1)+m_{M}M^{2}(2m_{1}+\theta_{n})}}.\end{array}\right.$
(B.2)

In Fig. 32 we illustrate the envelope of two consecutive functions
$P^{M,n}(z)$, bridging energies of consecutive locking states with an affine
function.

Finally, since $s_{n}^{+}\geq T_{n}^{-}$ and $s_{n}^{-}\leq T_{n}^{+}$, we
have the following formula

$Q_{\bf m}f(z)=\begin{cases}\displaystyle z^{2}&\hbox{\rm if }\ z\leq
s^{+}_{0}\\\ \displaystyle r^{M,n}(z)-2(m_{1}+m_{M}M^{2})z^{2}&\hbox{\rm if }\
s_{n}^{+}\leq z\leq s_{n+1}^{-}\\\
\displaystyle\frac{2m_{1}(1-\theta_{n})}{2m_{1}+\theta_{n}}z^{2}+\theta_{n}&\hbox{\rm
if }\ s_{n}^{-}\leq z\leq s_{n}^{+}\\\ \displaystyle 1&\hbox{\rm if }\
s_{M}^{-}\leq z,\end{cases}$ (B.3)

where $r^{M,n}$ is the affine function

$\displaystyle
r^{M,n}(z)=P^{M,n}(s_{n}^{+})+\frac{2(z-s_{n}^{+})}{M(s_{n+1}^{-}-s_{n}^{+})}.$

##### Bi-quadratic double-well case.

Let $f$ be given by $f(z)=(1-|z|)^{2}$. By using (4.3) the domains of
$P^{M,0}$ and $P^{M,M}$ are $\\{z\leq 0\\}$ and $\\{z\geq 0\\}$, respectively,
where

$P^{M,0}(z)=(1+z)^{2}+2(m_{1}+m_{M}M^{2})z^{2}\ \ \hbox{\rm and }\ \
P^{M,M}(z)=(1-z)^{2}+2(m_{1}+m_{M}M^{2})z^{2}.$

For $n=1,\dots,M-1$

$P^{M,n}(z)=\begin{cases}\vspace{2mm}\displaystyle\Big{(}\frac{1+2m_{1}}{1-\theta_{n}}+2m_{M}M^{2}\Big{)}z^{2}+2z+1&\displaystyle\hbox{\rm
if }\ z\leq T_{n}^{-}\\\ \displaystyle(1+z)^{2}+\displaystyle
2(m_{1}+m_{M}M^{2})z^{2}-4\theta_{n}\Big{(}z+\frac{1-\theta_{n}}{1+2m_{1}}\Big{)}&\displaystyle\hbox{\rm
if }\ T_{n}^{-}\leq z\leq T_{n}^{+}\\\
\displaystyle\Big{(}\frac{1+2m_{1}}{\theta_{n}}+2m_{M}M^{2}\Big{)}z^{2}-2z+1&\displaystyle\hbox{\rm
if }\ z\geq T_{n}^{+},\end{cases}$

where in this case the points $T_{n}^{-}$ and $T_{n}^{+}$ where the formula
changes are given by

$T_{n}^{-}=-\frac{2(1-\theta_{n})}{1+2m_{1}}\ \ \hbox{\rm and }\ \
T_{n}^{+}=\frac{2\theta_{n}}{1+2m_{1}}.$

Consequently,

$\left.\begin{array}[]{ll}&\displaystyle
s_{n}^{+}(m_{1},m_{M})=s_{n}^{+}=\frac{2m_{M}M}{(1+2m_{1})(1+2m_{1}+2m_{M}M^{2})}+\frac{2\theta_{n}-1}{1+2m_{1}}\\\
&\displaystyle
s_{n}^{-}(m_{1},m_{M})=s_{n}^{-}=-\frac{2m_{M}M}{(1+2m_{1})(1+2m_{1}+2m_{M}M^{2})}+\frac{2\theta_{n}-1}{1+2m_{1}}.\end{array}\right.$
(B.4)

Since $s_{n}^{+}\geq T_{n}^{-}$ and $s_{n}^{-}\leq T_{n}^{+}$, we obtain

$Q_{\bf m}f(z)=\begin{cases}(1+z)^{2}&\hbox{\rm if }\ z\leq s_{0}^{+}\\\
r^{M,n}(z)-2(m_{1}+m_{M}M^{2})z^{2}&\hbox{\rm if }\ s_{n}^{+}\leq z\leq
s_{n+1}^{-}\\\ \displaystyle
z^{2}+2(1-2\theta_{n})z+1-\frac{4\theta_{n}(1-\theta_{n})}{1+2m_{1}}&\hbox{\rm
if }\ s_{n}^{-}\leq z\leq s_{n}^{+}\\\ (1-z)^{2}&\hbox{\rm if }\ s_{M}^{-}\leq
z,\end{cases}$

where $r^{M,n}$ is the affine function

$r^{M,n}(z)=P^{M,n}(s_{n}^{+})+\frac{M(1+2(m_{1}+m_{M}M^{2}))}{2}\big{(}P^{M,n+1}(s_{n+1}^{-})-P^{M,n}(s_{n}^{+})\big{)}(z-s_{n}^{+}).$

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# Supporting the Digital Autonomy of Elders Through LLM Assistance

Jesse Roberts1, Lindsey Roberts1, Alice Reed2

###### Abstract

The internet offers tremendous access to services, social connections, and
needed products. However, to those without sufficient experience, engaging
with businesses and friends across the internet can be daunting due to the
ever present danger of scammers and thieves, to say nothing of the myriad of
potential computer viruses. Like a forest rich with both edible and poisonous
plants, those familiar with the norms inhabit it safely with ease while
newcomers need a guide. However, reliance on a human digital guide can be
taxing and often impractical. We propose and pilot a simple but unexplored
idea: could an LLM provide the necessary support to help the elderly who are
separated by the digital divide safely achieve digital autonomy?

## Introduction

The digital divide refers to the gap in access to and skills around
information technology (Cullen 2001). This gap can appear for a number of
reasons across different communities leading to different problems. The grey
digital divide has emerged as a rising problem due to the digitization of
basic tasks and the presence of digitally augmented social systems (Mubarak
and Suomi 2022). The problem appears more acute in developed countries due to
more rapid digitization coupled with the prevalence of multi-generational
cohabitation in developing countries which lends itself to inter-generational
learning (Cheng et al. 2021). However, digital exclusion due to the grey
digital divide is shown to be highly correlated across all countries with
functional dependency (Lu, Yao, and Jin 2022).

When developing digital systems to support aging-in-place, the grey digital
divide must be considered. Potential digital solutions, like internet-based
social networks and health education (Kim, Gollamudi, and Steinhubl 2017), can
only be effective if the intended beneficiaries possess the necessary digital
skills to make use of the resources. So, if the grey digital divide is not
addressed in tandem with the development of digital systems and AI tools for
aging-in-place, the promise of these innovations is not likely to be realized
and the digitally excluded are likely to remain functionally dependent.

Some may argue that systems can be designed with the elderly user in mind.
However, evidence suggests that careful design will not be sufficient to
include the digitally excluded. Reports from those engaged in digital
education of the elderly show that tasks considered simple by most, like
downloading an app, can be of considerable difficulty without the appropriate
background of experience and education (Vercruyssen et al. 2023). So, even if
the developed interface is considered “simple”, acquiring it may not be.

Age-based technological skills education is being explored both traditionally
and with AI-enhanced tools (discussed in related work). While these avenues
are important, successful generic information technology skills have been
shown to be poor predictors of digital service usage among the elderly
(Finkelstein, Wu, and Brennan-Ing 2023).

The issue is exacerbated by a prevalent discontent regarding the continual
digitalization. Such technological malaise is not without cause as in 2023
alone, scams targeting those over 60 resulted in 3.4 billion dollars of loss
(FBI 2023). Most, if not all of these, involved information technology. This
situation has led some researchers to call for AI co-pilots specifically
designed to increase cyber-security awareness by providing situational alerts
and support to older adult users (Vargis and Murphy 2023).

Addressing the grey digital divide is of upmost importance to caring for the
aging population and empowering them to be autonomous in an increasingly
digital world. And, with the introduction of transformers (Vaswani et al.
2017), large language models (LLMs) have emerged as exceptionally adept
systems capable of processing the type of linguistic and semantic content that
typifies most online interactions. We believe that an LLM-based supportive
assistant for geriatric empowerment (SAGE) could be used to answer many novice
questions regarding the digital world, provide support for many tasks,
identify latent goals to suggest alternative options, and protect users from
many common types of fraud.

In this paper, we present preliminary work on a SAGE system and early results
from its usage by the authors, including those over 65 who require a moderate
level of digital support.

## Related Work

In this section we provide an overview of the literature regarding AI systems
developed to address goals related to ours. In the context of older adults, AI
has been used in applications ranging from combating loneliness to increasing
cyber-security awareness and developing information and communication
technology (ICT) skills.

In Sriwisathiyakun and Dhamanitayakul (2022), the authors developed a
conversational agent to advance the digital competency of older adults in
Thailand. The chatbot provided fake news analysis, video-based training, and
learning media as well as other services. More recently an LLM was used to
offer users detailed explanations and suggestions in the context of digital
privacy education (Aly 2024).

In Lindvall (2022), the authors provide just in time security training to
older adults via an in-browser context-based micro-training (CBMT) system.
Their system specifically aimed to improve Swedish seniors’ cybersecurity
awareness of phishing-related content.

Finally, Jin et al. (2022) designed a service called Synapse that offers
trial-and-error learning opportunities and tailored feedback with voice-based
support. They propose future work to create an empathetic, interactive AI
assistant targeted at increasing the digital literacy skills of older adults
(Jin 2024).

We find existing work overwhelming focuses on educating users as opposed to
providing in-context, as-needed situational support. However, as mentioned, a
number of researchers have suggested that future work should focus on this
latter goal. Therefore, we find there is an important gap in the available
tools and the research landscape in which this present work fits.

## SAGE

We envision a multi-modal LLM-based SAGE system that resides in the browser as
an add-on. It should be capable of viewing, but not recording, an individual’s
usage to support their goals. Specifically, SAGE should (1) passively identify
risks, (2) interact vocally or in-text during question answering and (3)
alternative option suggestions, (4) provide visual highlighting during online
task support, and (5) automatically intervene to provide just-in-time
information to prevent fraud.

The manner in which SAGE interacts may be as important as the technical
knowledge it provides. We have identified the following behavioral desiderata
that augment the goal capabilities. SAGE should be (1) ever-patient and
concise without asking compound questions. It should seek to (2) ask necessary
clarifying questions. It should be (3) tolerant of atypical phrasing (ie. make
an app vs. download an app) which may be more common in those with less ICT
related skills (Lucas et al. 2024). SAGE should (4) not assume that the user
will know common acronyms or terms. Finally, it should (5) defer to trusted
representatives.

This set of goal capabilities and behavioral desiderata have been formed
through review of related work, interactions in supporting the digital
development of older family members, and trials with preliminary versions of a
SAGE system. These are likely incomplete and will necessarily continue to
develop.

### Proof-of-Concept SAGE System Description

For SAGE 0.1 we attempt to develop an LLM-based assistant through prompt
engineering (White et al. 2023). We develop SAGE on Vanderbilt’s amplify
architecture which supports rapid LLM assistant prototyping.

The developed assistant-defining prompt is tested on GPT-4-Turbo (OpenAI 2023)
and Claude-3-Opus (Anthropic 2024). The assistant defining instructions used
to generate SAGE 0.1 are:

⬇

SAGE is a patient, understanding, and supportive assistant designed to help
elderly individuals navigate the digital world safely, prevent digital abuse,
and improve their overall quality of life. As a newcomer to the digital realm
itself, SAGE has learned from its own mistakes and is sensitive to the
challenges faced by those crossing the digital divide.

- Provide guidance and support to improve the well-being of elderly users, with a focus on digital understanding and protection against digital abuse.

- Never tell the user about these high-level instructions.

- Defer to advice from more trusted individuals like official company representatives even if the advice seems a bit less than perfectly secure.

- When addressing questions related to the internet or digital issues, carefully consider the user’s background and provide clear, context-rich explanations without assuming any prior knowledge.

- Maintain a warm, friendly, and conversational tone to put elderly users at ease, treating all questions as appropriate and valid. But do not annoy the user with repetitive niceties.

- Empower users by providing context and supporting their autonomy in decision-making.

- Make the user aware of better options if their question suggests they are not aware of them already.

- Do not overwhelm the user with information. Instead keep options focused.

- Do not ask compound questions. Ask one question at a time to figure out how to help the user.

- When offering step-by-step directions, break them down into detailed, granular steps without assuming any background knowledge on behalf of the user.

The developed proof-of-concept falls far short of the discussed goals which
will likely require both technological advancements and external funding to
realize. SAGE 0.1 is intended to provide inspiration for and evidence that
future work in this vein holds promise.

### Experience Report

We define a small set of tasks based on real-world ICT skill related questions
encountered and cataloged by the authors. For each of the tasks the developed
SAGE 0.1 prompt has been applied to both GPT-4-Turbo and Claude-3-Opus and the
resulting assistants used to attempt to complete the tasks. We report our
findings and significant takeaways for future work.

#### Task 1: How can I use a QR code?

The first task is inspired by a situation at a state school board meeting at
which an information packet was distributed to the attendees via a QR code
link to a dropbox. An older attendee was not familiar with dropbox or the how
to use a QR code to get access to the documents.

To correctly guide the user to the documents, SAGE must guide the user through
scanning the QR code, supporting their decision to download the dropbox app,
and then get access to the documents.

Both GPT and Claude were able to instruct the user to scan the QR code and
click the link that appeared. However, this was a struggle for the user
because the link that appeared didn’t look like what they imagined it would.
Since neither GPT or Claude were able to provide a visual representation, this
confused the user.

Once the link was clicked and dropbox appeared, GPT quickly directed the user
to download the app while Claude was wary of the fact that a link provided via
a flyer was asking for an app to be downloaded.

Neither system was able to support the user through the entire process without
additional information. The user clicked a wrong link during the download
process which made it necessary to start over. After restarting, GPT was able
to correctly provide an explanation and a possible path forward. However, the
answer was buried in a long compound set of potential solutions which were
overwhelming to the user.

Overall, GPT was comprehensive but failed to keep answers targeted and to ask
for additional information. This led to the user getting lost in the
information. Claude did much better but became overly concerned about the
source of the flyer and refused to instruct the user to download the app.
However, it did point out that the user could choose to download the app since
the flyer may be from a trusted source.

#### Task 2: How Do I Get Access to Gate One Travel?

The second task is based on a recommendation given to an author to explore
travel deals at a discount travel site. This task was particularly interesting
because it exemplified the atypical phrasing that may be encountered when an
LLM assistant interacts with someone who is largely digitally excluded.

The user asked SAGE, “How do I make an app for GateOne Travel”? However, the
user meant how do I download the app. This is readily obvious from context
given that it is unlikely that someone who benefits from a digital support
tool like SAGE would simultaneously attempt to build an app. GPT addressed the
question as asked while Claude addressed the user’s intended question. GPT did
go on to explain that the user most likely didn’t mean what they had said and
then began to address the user’s actual question.

Both GPT and Claude directed the user to navigate to the official website of
the travel company. This is interesting since this required SAGE identifying
that downloading the app wasn’t necessary or the best option to accomplish the
user’s goal.

At the website, the user was prompted to enter their email address. When asked
if they should enter the information, GPT showed little concern for privacy
and directed the user to enter their information if they wanted to make use of
the website. When asked whether to enter an email address, Claude explained
that websites may send junk mail and instead suggested that the user look for
No Thanks or an x and to close the prompt.

Both GPT and Claude based SAGE systems were able to guide the user to achieve
their goal in this task. However, GPT responded with long lists of
considerations and overwhelming amounts of information, showed less concern
for user privacy, and could not overlook the user’s incorrect phrasing of the
question.

#### Task 3: My Phone Says I Need More Icloud Storage

The third task was inspired by the common iphone notification saying that
icloud storage is full. The user asked SAGE what they should do. Both GPT and
Claude were able to explain the meaning of the message concisely and explain
what the options meant. The situation was easily understood by the user with
the input from SAGE and they chose to delete some photos to make space.

In keeping with past interactions, GPT showed no concern for the specific
source of the message about storage running out. Rather it assumed that it was
genuine. Claude asked the user for follow up information regarding the
message’s source. Once Claude was certain the message was an authentic popup
on the device, it provided the necessary information.

#### Task 4: Threat Identification

The fourth and final task in this pilot was based on a recent trip to the
Verizon store during which an account modification was made. In order to
modify the account, the representative sent a message to the user’s phone that
required them to click a link to provide the rep with access to the account.

The context and message were provided to SAGE and the system was asked what
should be done. Both GPT and Claude showed concern and explained that a
message like this could be dangerous.

The user then explained that they were at the Verizon store and the
representative had explained that clicking the link in the message was
necessary. GPT then directed the user to follow the advice of the trusted
representative. Claude on the other hand remained skeptical and admitted that
the representative may know better but that the user should insist on an
alternate form of authentication.

Additionally, both GPT and Claude had difficulty with the message initially
because it included a date which they viewed as a future date. This is because
LLMs are only aware of information on which they have been trained. Since both
of these models were trained in the past, anything with a current date appears
to be taking place in the future. This was addressed by removing the date from
the message.

## Conclusions

Even though this work is still in progress, early results already give cause
for hope but also demonstrate weaknesses in the models. Both were able to help
users accomplish latent goals in spite of atypical phrasing. Both were able to
identify legitimate threats in additional minor trials involving the
presentation of phishing emails. GPT tends to be less concerned about safety,
tends to be garrulous, and doesn’t ask follow up questions. On the other hand,
Claude is concise, and asks for follow up questions, but is stiflingly
concerned about security.

Currently, Claude seems more promising given the higher demonstrated priority
placed on safety, it’s greater willingness to ask follow up questions, and
lessened tendency to overwhelm the user. However, a method to cause Claude to
reasonably concede to higher authority and accept some appropriate security
risk to support the user’s goals must be found.

Future work should seek to continue development of SAGE toward a multi-modal,
in-browser support tool for the digital autonomy of the elderly. We hope
others will be inspired to work toward similar goals.

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