Further author information: (Send correspondence to R.Y.)
R.Y: E-mail<EMAIL_ADDRESS>Telephone: 1 347 459 8709

# The Prototype Telescope and Spectrograph System for the AMASE Project

Renbin Yan  Department of Physics and Astronomy, University of Kentucky, 505
Rose St., Lexington, KY, 40506 USA  Matthew A. Bershady  Department of
Astronomy, University of Wisconsin - Madison, 475 N. Charter Street, Madison,
WI 53706-1582, USA.  South African Astronomical Observatory, PO Box 9,
Observatory 7935, Cape Town, South Africa.  Department of Astronomy,
University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa.
Michael P. Smith  Department of Astronomy, University of Wisconsin - Madison,
475 N. Charter Street, Madison, WI 53706-1582, USA.  Nicholas MacDonald  UC
Observatories, UCO/Lick, UC Santa Cruz, 1156 High St., Santa Cruz, CA 95064,
USA  Dmitry Bizyaev  Apache Point Observatory and New Mexico State University,
P.O. Box 59, Sunspot, NM, 88349-0059, USA  Sternberg Astronomical Institute,
Moscow State University, Moscow, Russia  Kevin Bundy  UC Observatories,
UCO/Lick, UC Santa Cruz, 1156 High St., Santa Cruz, CA 95064, USA  Sabyasachi
Chattopadhyay  South African Astronomical Observatory, PO Box 9, Observatory
7935, Cape Town, South Africa.  James E. Gunn  Department of Astrophysical
Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ, 08544  Kyle B.
Westfall  UC Observatories, UCO/Lick, UC Santa Cruz, 1156 High St., Santa
Cruz, CA 95064, USA  Marsha J. Wolf  Department of Astronomy, University of
Wisconsin - Madison, 475 N. Charter Street, Madison, WI 53706-1582, USA.

###### Abstract

We present the design of the prototype telescope and spectrograph system for
the Affordable Multiple Aperture Spectroscopy Explorer (AMASE) project. AMASE
is a planned project that will pair 100 identical multi-fiber spectrographs
with a large array of telephoto lenses to achieve a large area integral field
spectroscopy survey of the sky at the spatial resolution of half an arcminute
and a spectral resolution of R=15,000, covering important emission lines in
the optical for studying the ionized gas in the Milky Way and beyond. The
project will be enabled by a significant reduction in the cost of each
spectrograph unit, which is achieved by reducing the beam width and the use of
small-pixel CMOS detectors, 50${\rm\mu m}$-core optical fibers, and commercial
photographic lenses in the spectrograph. Although constrained by the
challenging high spectral resolution requirement, we realize a 40% reduction
in cost per fiber at constant etendue relative to, e.g., DESI. As the
reduction of cost is much more significant than the reduction in the amount of
light received per fiber, replicating such a system many times is more cost
effective than building a single large spectrograph that achieves the same
survey speed. We present the design of the prototype telescope and instrument
system and the study of its cost effectiveness.

###### keywords:

Integral Field Spectroscopy; Fiber-fed Spectrograph; medium-resolution
spectroscopy, large area sky survey; AMASE.

##  1 Scientific and Technical Motivation

Understanding the rise and fall in the star formation rate within galaxies is
essential for a complete picture of galaxy evolution. Recent advances in
multi-object integral field spectroscopic instrumentation have made IFS
surveys of thousands of nearby galaxies possible, e.g., SAMI[1] and MaNGA[2].
These data have firmly established that many scaling relations seen for
integrated properties of galaxies persist down to $\sim 1$ kpc scales, such as
the relations between stellar mass surface density, star formation rate
surface density, and gas-phase metallicity[3]. These data have also
established that some regions of galaxies stop their star formation in a
relatively rapid manner (referred to as ‘quenching’) to form an underpopulated
“green valley.” To fully understand what gives rise to these scaling relations
as well as to identify quenching mechanisms, we need to connect these ‘macro’
phenomena on kpc scales with the physical processes of star formation and
baryon cycling that occur on even smaller scales. The relevant physical scales
start at the size of molecular clouds (tens of parsecs), and proceed to the
size of star-forming regions at tenths of pc. Comprehensive optical
spectroscopic observations covering from 0.1 pc to kpc scales are challenging
with current instruments.

There is overwhelming observational and theoretical evidence that stellar
feedback is critical for regulating star formation, for driving galactic
winds, and for determining the properties of the ISM. However, the detailed
physical processes involved with this feedback are far from clear. For
example, although we know turbulence plays a major role in keeping star
formation inefficient, we do not know how turbulence is driven and on what
scale(s) turbulent energy is injected. To tackle this problem requires
measuring the power spectra of velocity and emissivity in thin velocity
slices. This in turn requires instruments capable of delivering medium
spectral resolution ($R>15,000$) while covering spatial scales from sub-pc to
100 pc [4]. We also have not confirmed observationally the relative importance
of photoionization and radiation pressure in driving outflows from star-
forming regions. To do so requires 20 km/s separation between different
velocity components [5, 6], again requiring medium spectral resolution optical
spectroscopy over the same spatial scales. Covering the required spatial and
spectral scales is very difficult with current instruments.

Most integral field spectrographs on large telescopes today do not have large
fields of view. For example, VLT/MUSE has a field of view of 1 arcmin2. HII
regions in the Milky Way are too large on the sky for them to cover, while
those in other galaxies are usually too small to resolve at the 0.1-100 pc
scale in gound-based seeing. Local group galaxies, like M31, M33, LMC, and
SMC, are at about the right distance, however, they still spread over degrees
on the sky, making it extremely time-consuming to cover them completely.
Moreover, most integral field spectrographs cannot reach spectral resolutions
of $R\sim 15,000$ (KCWI[7], MEGARA[8] on the GTC, WEAVE[9] on the WHT, and
WIYN IFUs[10, 11] are some exceptions). The instrument we describe here fills
a niche to enable observations over large areas at medium spectral resolution.

We consider the science case here for nebular spectroscopy of Milky Way HII
regions. In this regime 0.1 pc physical resolution corresponds to tens of
arcseconds on the sky. Consequently we do not need the large plate scales
afforded by large telescopes. Compared to the telescope aperture, more
important is the focal ratio of our observing system. Like the aperture
setting for photography, fast focal ratios concentrate more light into each
pixel, allowing fainter features to be detected in less time. It is extremely
challenging to reduce the focal ratio of optics as it leads to degradation of
image quality and mismatch with the numerical aperture of optical fibers.
However, one can significantly reduce the effective focal ratio of the system
by making many identical copies of a telescope and then stacking the data
together, as done by the Dragonfly Telephoto Array [12]. Applying this
strategy to spectroscopy requires the replication of many spectrographs. Most
astronomical spectrographs cost millions of dollars to build, and are
prohibitively expensive to replicate in large numbers. The VIRUS spectrograph
[13] has demonstrated modular designs and massive replication can
significantly reduce cost and improve cost-effectiveness. Our instrument
design takes cost reduction even further via several strategies, while pushing
for higher spectral resolution than typical for survey spectrographs.

The key to achieving this higher-resolution capability at very low cost is to
take advantage of a unique combination of thinner fibers, smaller-pixel CMOS
detectors, and commercial photographic lenses. Thinner optical fibers
($50~{}\mu$m core) require shorter collimator focal length for fixed spectral
resolution. A shorter focal length reduces the beam width of the spectrograph,
allowing the use of off-the-shelf photographic lenses in the spectrograph,
cutting cost significantly. It also reduces the size of the grating, cutting
cost further. At the same time, thinner fibers enable the use of small pixel
commercial CMOS sensors without requiring a large de-magnification factor.

There have been significant technology advances in both CMOS sensors and
photographic lenses, making them more suitable than ever for adoption in
astronomical instruments. Today’s back-illuminated CMOS sensors have
comparable sensitivity to top-level CCDs ($>90\%$ peak efficiency) and have
even lower read noise ($\sim 1$ $e^{-}$), but are much cheaper due to
industry-scale mass production. The advance in CMOS also indirectly made
commercial lenses more suitable for adoption in spectrographs. The fast
readout speed of CMOS has eliminated the need for a flip mirror in commercial
cameras. This reduced the back-focus requirement for commercial lenses,
enabling them to be designed with faster focal ratio (e.g., f/0.95) and have
better image quality. In addition, advanced nano-crystal coating technology
gives these commercial lenses high throughput and low scattered-light,
enabling the use of more elements to correct for aberrations.

In this proceeding, we describe the conceptual design of an innovative, high
spectral resolution, cost-effective integral field spectrograph. When paired
with small telescopes like a commercial Canon telephoto lens, it is ideal for
observing star-forming regions in the Milky Way and Local Group galaxies, and
for conducting a wide area contiguous IFS mapping of the sky.

##  2 Telescope and Spectrograph Design

###  2.1 Design Optimization to Lower Cost

The entire instrument includes a Canon Telephoto lens serving as a telescope,
a double-arm spectrograph, and a fiber-feed that connects them. Because the
telescope is small and much cheaper than the spectrograph, our design
considerations are very different from those guiding traditional astronomical
instruments. When telescopes are large and expensive, it is most cost-
effective to make the instrument accept all the light collected by the
telescope; hence the telescope sets the system aperture stop. In contrast,
since we do not need large telescopes for nebular spectroscopy, the
spectrograph is relatively more expensive. In this case, cost-effective
designs sets the system aperture by the spectrograph.

Figure 1: Optical model of the AMASE-P spectrograph.  Figure 2: Optomechanical
layout of the system. The plates around the grisms are for the blocking of
stray light.

The design of the spectrograph is driven by the need for moderate spectral
resolution ($R=\lambda/\Delta\lambda=15,000$) and large étendue (area $\times$
solid-angle $\times$ efficiency, or $A\Omega\epsilon$) at low cost. Our
approach is to adopt small spectrograph entrance apertures to enable the use
of commercial optics and CMOS detectors. While we sacrifice system étendue per
unit in doing so, significant cost-reduction allows us to more than recoup via
spectrograph multiplex.

The design optimization starts with setting the beam width of the spectrograph
by finding the most optimal commercial optics to use as the spectrograph
camera. The most optimal commercial lens for our application is one that has
the highest étendue and the highest image quality. The recently released
Nikkor 58mm f/0.95 S Noct lens by Nikon in 2019 is the champion in both
aspects. It has an amazingly fast focal ratio of f/0.95 and is designed for
full-frame detectors ($36{\rm mm}\times 24{\rm mm}$), meaning it has the
largest étendue among all commercial photographic lenses. It is also
specifically designed to minimize aberrations for bright point-sources in dark
backgrounds, perfect for astrophotography. ARNEO nano-crystal coatings yield
high throughput and minimize ghosts and flares. We have measured its
throughput and verified the image quality through lab tests and ray-tracing
calculations. As shown in Fig. 1, the lens has 17 elements in 10 groups,
including 4 extreme low dispersion elements and 3 aspherical surfaces. This
level of sophistication would be extremely challenging and expensive to
achieve with custom optics.

This Nikon lens has an entrance pupil of 61 mm, setting the spectrograph beam
width. To match the small-pixel CMOS detectors, we adopt the thinnest
available multi-mode broad-spectrum optical fiber (Polymicro) with a 50
${\rm\mu m}$-diameter core. With these parameters fixed, the setting of the
collimator focal length requires a compromise. On one hand, to maximize
throughput and minimize focal ratio degradation requires the fibers are fed
with a fast focal ratio close to their numerical aperture. At fixed beam
width, this means we want to minimize the focal length of the collimator. On
the other hand, for fixed fiber core diameter, the spectral resolution
requirement drives a longer collimator focal length to ease the angular
dispersion requirement of the grating. We compromise between these two factors
and choose a collimator with 200mm focal length. This compromise means the
collimated beam is larger than the entrance pupil of the Nikon lens, leading
to a significant amount of vignetting. The resulting demagnification factor is
3.45 – the image of a fiber would have a FWHM of 14.5 ${\rm\mu m}$ on the
detector, corresponding to 3.86 pixels for the detector we choose.

To sample HII regions in the Milky Way to sub-parsec scales, we choose the
$f=400$mm f/2.8 Canon telephoto lens as the telescope. While this overfills
the spectrograph pupil, this is not a concern because the cost of the system
is not limited by the telescope but by the spectrograph. The camera optics
(Nikkor lenses) are the limiting factor in the system étendue; because our
beam size is below a break in the grating cost curve, we place the
spectrograph pupil near the front of the camera to maximize system efficiency,
rather than on the grating.

We describe core spectrograph components, telescope, and fiber-feed system
below. Figure 1 shows the optical and optomechanical models with design
parameters summarized in Table 1.

Table 1: Nominal Instrument Design Parameters Item | Value | Notes
---|---|---
Telescope | D=143mm f/2.8 | Canon 400 mm f/2.8 L III IS USM
Spatial resolution | 25.8 arcsec | fiber core:clad:buffer 50:70:85 ${\rm\mu m}$
Fiber IFU fill factor | 31.4% | 85 ${\rm\mu m}$ center-to-center spacing
Number of Fibers | 547 | per spectrograph
Simultaneous field coverage | 79 arcmin2 | per spectrograph
On-sky footprint | 16 arcmin $\times$ 16 arcmin | per spectrograph
Fiber-to-fiber spacing on slit | 100 ${\rm\mu m}$ | 60 mm pseudo-slit length
Collimator focal length | 200 mm | Silver-coated spherical mirror
Camera focal length | 58 mm | Nikkor 58mm f/0.95 S Noct
Detector array |  $9576\times 6388$ pix |  $36~{}{\rm mm}\times 24~{}$mm
PSF FWHM on detector | 3.86 pixel | pixel size: 3.76 $\mu$m
Red grating density | 2343 l/mm | blazed at 6604 Å
Red spectral coverage | 6250-6850 Å ($\epsilon>10\%$ of peak) | 6500-6770 Å ($\epsilon>70\%$ of peak)
Blue grating density | 3156 l/mm | blazed at 4903 Å
Blue spectral coverage | 4640-5092 Å ($\epsilon>10\%$ of peak) | 4820-5020 Å ($\epsilon>70\%$ of peak)
Spectral Resolution ($R$) | 15,000 | red: @6604 Å, blue: @4903Å

###  2.2 Core Spectrograph Components

Collimator–Entrance-slit Assembly: The collimator is a silver-coated spherical
mirror with a focal length of 200mm. We align 547 fibers along a curved slit
with a radius of curvature of 200mm. The fibers are split into a number of
fiber blocks to allow angling fibers to follow the curvature of the slit.
Fibers are spaced 100${\rm\mu m}$ center-to-center to minimize cross-talk.
Three-fiber spacing gaps between adjacent fiber blocks allow for scattered
light measurements. The fibers are fed at f/2.8. The output beam is slightly
faster, estimated to be about f/2.7 due to focal ratio degradation. The
collimator will form a collimated beam of 74mm diameter.

Dichroic: Grism tilt angles of 61 degrees are accommodated via a dichroic beam
splitter designed for 29 deg incidence angle, reflecting below 576 nm. This
optimally positions the blue camera and detector. In order to place the system
pupil closer to the blue camera to reduce vignetting, we place the dichroic as
close to the collimator as possible. This configuration requires a slot in the
dichroic, which is located a few mm in front of the slit, to allow light from
the slit to pass through. The slot width is minimized to be just larger than
the thickness of the slit plate. A representative quote yields reflectance for
both polarizations in the blue above 99%, and transmittance in the red above
98%.

VPH grism: The VPH-grism is a critical design element for achieving the high
spectral resolution necessary for our science goals. Volume-phase holographic
transmission gratings deliver high diffraction efficiency and optimum
geometry. To achieve the desired spectral resolution we adopt a $\sim
35.3\deg$ incidence angle in the grating medium of both channels, optimized to
reach peak diffraction efficiency for both polarizations simultaneously. Line-
densities and super-blaze peaks are given in Table 1. We bracket the gratings
with two small-angle, fused-silica prisms to further increase the angular
dispersion to reach the science requirement. The expected diffraction
efficiency is $\sim$90% at the peak and $>$ 70% over a $>200$Å window in the
blue and $>270$Å window in the red.

Camera Optics: The adopted Nikkor lens has excellent throughput, which we have
measured by comparing it to uncoated fused silica lens. Its absolute
throughput is $>$90% between 550nm and 695nm and $>$ 80% between 462nm and
732nm. It does have significant vignetting for large field angles. The
collimated beam has a 74mm diameter, larger than the 61mm entrance pupil. This
overfilling helps reduce the vignetting of beams at large field angles
relative to the on-axis beam. One can argue that our on-axis beam has
significant vignetting given the f/2.8 input beam at the telescope. Instead of
choosing a shorter focal length for the collimator, we choose to sacrifice
throughput in order to reach the high spectral resolution required by the
science.

Figure 3: Absolute quantum efficiency of the IMX455 measured by QHYCCD. Figure
adapted from
https://www.qhyccd.com/index.php?m=content&c=index&a=show&catid=94&id=55&cut=1
. Also see this page for a discussion:
https://www.qhyccd.com/index.php?m=content&c=index&a=show&catid=23&id=261

Detector Assembly: The detector package is QHY600M monochromatic camera from
QHYCCD. It uses Sony’s state-of-the-art, back-illuminated, 16-bit full-frame
CMOS sensor, IMX455. Quantum efficiency is about 90% at the peak and above 70%
for the whole wavelength range we use (Figure 3). Read-noise is 1.68 e- for a
full well depth of 21.7k e-, or 1.39e- for a full well depth of 3k e-
111https://www.qhyccd.com/index.php?m=content&c=index&a=show&catid=94&id=55&cut=1.
The dark current is 0.0022 e-/pix/s at T=-20C, dropping by a factor of two for
every $\Delta$T = -10C. A water-cooled version at the same price is available
that can bring the detector to -45 degree below ambient temperature. The front
window of the camera is heated to prevent condensation. We will replace the
front window of the detector by a custom plano-concave lens to flatten the
field curvature. No physical shutter is needed for the camera as it has an
electronic rolling shutter.

Figure 4: Calibration system and focusing mechanism conceptual design. The
Canon telephoto lens can be independently mounted or bolted to a small
telescope on an equatorial mount (as shown here). The cone on top of the
telephoto lens is the calibration system. During science observations, the
cone swings off. This allows us to calibrate at any telescope position to
correct for flexure induced throughput and line spread function variation.

##  3 Telescope, Focal Plane and Fiber-Feed System

The telescope is a $f=400$ mm f/2.8 Canon telephoto lens feeding a hexagonal
fiber bundle of 547 fibers. The telescope will be mounted on an equatorial
mount to avoid field rotation. The 50${\rm\mu m}$ fiber core diameter
corresponds to 26 arcsec on sky. Excluding gaps between the fibers, the bundle
covers an area of 80 arcmin2. The bundle will be packaged in a focal-plane
assembly that includes focus and acquisition camera described below. The IFU
fibers route in a cable to the slit assembly (§2.2) in the spectrograph. The
fiber cable is designed and jacketed for the required routing and actuation.
We have identified a suitable vendor who can fabricate the ferrule and fiber
blocks for our smaller fibers using a photolithographic process necessary for
these micro-precision components. Specific design considerations include the
following elements.

IFU feed: Fiber telecentricity is required to maximize throughput. We have
measured the position of the exit pupil for the Canon telephoto lens in the
lab and found it is $183.5\pm 1{\rm mm}$ away from the focal plane. We plan to
cement an AR coated plano-convex lens on the fiber bundle surface to make the
telecentric correction and to minimize Fresnel losses.

Focus and guiding: We defocus the telescope image slightly to smooth radial
light distributions within fibers; this removes the dependence of the line
spread function on the intrinsic light distribution within a fiber footprint
and reduces the impact of differential atmospheric refraction for stars on
fiber edges. Using a rented lens, we found the slightly-defocused PSF for the
Canon 400mm f/2.8 lens is smooth with no sharp boundaries. To keep constant
defocus, we implement a monitoring mechanism via a 45 deg optical/near-IR
dichroic beamsplitter placed before the focal plane (Figure 4). Wavelengths
bluer than 750 nm are reflected to the fiber bundle; redder light is
transmitted to a focus-monitoring CMOS sensor. We adjust the relative
positioning of the fiber bundles and the CMOS sensor so that when a star is in
focus in the focus-monitoring sensor, it has a 26 arcsec FWHM on the fiber
bundle. This adjustment will be done in the lab using a microscope looking
towards the beamsplitter at the bundle and the sensor, which will also provide
the relative registration for astrometry. Then we will test it using an
illuminated pinhole placed several meters away. By scanning the pinhole across
the bundle and the sensor, we can measure the profile of the pin hole image on
the fiber bundle. The relative positions will be locked and checked for
repeatability. During commissioning, by scanning across a bright star, we can
verify the PSF on the focal plane. During observations, we will only need to
keep the star focused in the focus monitoring camera and we will be sure we
have the desired focus offset for the IFU. The transmitted images through the
dichroic beam splitter will have astigmatism. This is useful for
distinguishing the front and back side of focus. The electronic focus control
will be provided by Birger Engineering.

##  4 Calibration System

Our science goals require accurate measurement of line width (to measure gas
turbulence and kinematic temperature) and line ratios (to measure extinction
and abundances). This requires flat/arc calibration at any pointing position
to take out the flexure-induced throughput and line spread function
variations. Figure 4 illustrates the design of the calibration system. It
consists of a cone mounted on top of the telephoto lens. At the top of the
cone is the lamp assembly including a quartz lamp, two arc lamps (Ne, Th/Ar),
and a diaphragm controlling the light output. At the base of the cone, we
place a Fresnel lens and a ground glass diffuser. With the lamps placed at the
focus of the Fresnel lens, the diffuser is illuminated with a collimated beam.
With a coarse grit, the ground glass diffuser provides a wide diffusing beam
with flat intensity over the central few degrees, sufficient to completely
cover the IFU footprint. When not in use, the cone swings out of the beam and
rests on the side of the lens. The Fresnel lens will be slower than f/2 to
ensure sufficiently uniform pupil illumination.

##  5 Preliminary Tests of the Design

###  5.1 Throughput of the Nikkor lens

We have measured the absolute throughput of the Nikkor lens by comparing it to
a reference optical system constructed using uncoated fused silica lenses. The
reference system is made to have the same focal length as the Nikkor lens.
Both systems are used to image the spectrum of a quartz halogen lamp placed a
few meters away, through a 600 l/mm transmission grating from Thorlabs. An
adjustable iris is placed in front of the lenses to minimize intrinsic
vignetting of the systems. A Pen-Ray HgNe lamp is used to provide wavelength
calibration. The reference system is composed of two uncoated fused silica
plano-convex lens whose 4 air-glass surfaces will yield an absolute throughput
of 86.7%. The measured absolute throughput of the Nikkor lens is shown in
Figure 5.

Figure 5: Absolute throughput measurement of the Nikkor 58mm f/0.95 S Noct
lens.

###  5.2 Image Quality

We have simulated the expected image quality using Zemax. The resulting line
widths satisfy our resolution requirements. The spherical aberration
introduced by the spherical collimator can be adequately compensated for by a
very slight adjustment of the Nikkor lens focus. We have verified this in the
lab using a 50$\mu$m core fiber, a spherical collimator, and a rented Nikon
lens. The result is totally consistent with our simulations. With the addition
of the grism, the images at large slit height are diagonally elongated. This
can be accommodated in the analysis pipelines for spectral extraction and
estimating instrumental resolution, using spectral-perfectionism algorithm
[14].

###  5.3 Thin-fiber positioning

The 50$\mu$m core fibers we are using are more challenging to handle and align
accurately on a pseudo-slit or IFU using the technology developed for the
larger MaNGA fiber system. For example, wire-EDM technology is impractical for
cutting V-grooves with radius of curvature smaller than 40$\mu$m required to
precisely seat fibers with 85$\mu$m OD. Thus, we worked with a commercial
fabricator (FEMTOprint) to develop a process combinning femto-pulse laser
drilling and photolithographic etching. This effort resulted in fused-silica
blocks with cylindrical apertures arrayed in a pseudo-slit. Each aperture has
1$\mu$m centration and diameter tolerance. We have acquired test blocks, and
populated them with $50{\rm\mu m}$-core fiber. The test was largely
successful: Fibers were smoothly inserted in most cases. However, the sharp
edges at the aperture entry locations strip buffer, causing fibers to become
excessively brittle. We have modified the block design to include chamfers at
the aperture edges, and a staggered pseudo-slit to accommodate the chamfers
while delivering the same efficient packing density as in our initial design.
We expect the next version to work without problem. Figure 6 shows a picture
under microscope for such a fiber block with $33\times 50{\rm\mu m}$ fibers
inserted.

Figure 6: Fused silica fiber blocks populated with fibers of 85${\rm\mu m}$ OD
and 50${\rm\mu m}$ core. The block is clamped by a mounting fixture so the
central part is obscured.

##  6 Expected Sensitivity

Channel | Collimator | Dichroic | Prisms | Grating | Camera | Detector | Vignetting | Overall
---|---|---|---|---|---|---|---|---
Blue | 93% | 99% | 94% | 91% | 86% | 84% | 50% | 28.2%
Red | 97% | 98% | 94% | 91% | 93% | 72% | 51% | 27.4%
Table 2: Expected system throughput for the central wavelength in each
channel. Vignetting is separately accounted for and discussed in §6.

As listed in Table 2, the expected throughput of our instrument is about 28%
and 27% for the central wavelengths in the blue and red channels,
respectively. The vignetting is a significant hit to the overall throughput.
Part of it comes from the slot in the dichroic beamsplitter and the
obstruction by the slit plate. But most of the vignetting is due to a design
choice we made in order to reach the high spectral resolution (see §2.1). If
we reduce the focal length of the collimator, we could significantly improve
the throughput, but we will not reach the required spectral resolution.

Figure 7: Expected S/N in H$\alpha$ per fiber for AMASE-P and DESI with a
single 15-min exposure, in 3 different background conditions. The S/N for
AMASE-P is doubled for ease of comparison with DESI.

It is convenient to compare the etendue of the AMASE-P spectrograph with the
DESI spectrograph [15], which is a highly-optimized, state-of-the-art broad
band-pass system. DESI has an f/3.86 input, and the throughput at these
wavelengths is measured to be about 64% and 66% [15]. Given our smaller fiber
core diameter (50 ${\rm\mu m}$) compared to DESI’s 107 ${\rm\mu m}$ fiber
cores, our overall etendue per fiber (fiber input beam solid angle $\times$
fiber area $\times$ spectrograph throughput) is 18% of that for DESI. However,
our hardware costs also differ dramatically. Each DESI spectrograph has 500
fibers and cost $1.5M [16], while our prototype instrument, AMASE-P, will have
547 fibers and cost $184k for the spectrograph hardware. Our per fiber cost is
11% of that for DESI. AMASE-P is about 1.6 times more cost effective per unit
etendue at 4 to 5 times higher spectral resolution but 18% of the passband
coverage. To be clear, AMASE-P is not a panacea; DESI delivers 5.4 times
higher bandwidth and 23% more spectral resolution elements. However, for the
specific emission-line survey targeted by AMASE-P, our design offers
significant advances in lowering cost while increasing spectral resolution.

AMASE-P’s telescope and fiber system are expected to yield a throughput of 76%
and 81% in the blue and red channels, respectively. Figure 7 shows the
expected S/N as a function of H$\alpha$ surface brightness (assuming an
unresolved line width), under different conditions for DESI and AMASE-P. This
also takes into account the sky background, detector read noise and dark
current. The AMASE-P curves are scaled up by a factor of 2 for ease of
comparison. The higher spectral resolution allows AMASE-P to be relatively
faster in brighter time, further increasing cost-effectiveness for the narrow
emission-line science. We obtain a S/N of 3 in $5\times 15$min exposures for
an H$\alpha$ surface brightness of 1 Rayleigh.

##  7 Summary

Here we have presented the conceptual design of a cost-effective, high
spectral resolution, integral field spectrograph capable of reaching spectral
resolution of up to R = 15,000. In principle, it can be used on both small and
large telescopes to enable wide-area, integral field, spectroscopic surveys.
On small telescopes with high-angle VPH grisms, it is ideal for covering and
resolving key strong-line diagnostics across large HII regions in the Milky
Way and mapping all components of the interstellar medium in nearby galaxies.
The cost-effectiveness of the design is amenable to lower-angle VPH grism
applications that would yield full spectral coverage between 400-800 nm for
study of stellar populations in integrated light as well as intensity mapping.

###### Acknowledgements.

## References

  * [1] Croom, S. M., Lawrence, J. S., Bland-Hawthorn, J., Bryant, J. J., Fogarty, L., Richards, S., Goodwin, M., Farrell, T., Miziarski, S., Heald, R., Jones, D. H., Lee, S., Colless, M., Brough, S., Hopkins, A. M., Bauer, A. E., Birchall, M. N., Ellis, S., Horton, A., Leon-Saval, S., Lewis, G., López-Sánchez, Á. R., Min, S.-S., Trinh, C., and Trowland, H., “The Sydney-AAO Multi-object Integral field spectrograph,” MNRAS, 421, 872–893 (Mar. 2012).
  * [2] Bundy, K., Bershady, M. A., Law, D. R., Yan, R., Drory, N., MacDonald, N., Wake, D. A., Cherinka, B., Sánchez-Gallego, J. R., Weijmans, A.-M., Thomas, D., Tremonti, C., Masters, K., Coccato, L., Diamond-Stanic, A. M., Aragón-Salamanca, A., Avila-Reese, V., Badenes, C., Falcón-Barroso, J., Belfiore, F., Bizyaev, D., Blanc, G. A., Bland-Hawthorn, J., Blanton, M. R., Brownstein, J. R., Byler, N., Cappellari, M., Conroy, C., Dutton, A. A., Emsellem, E., Etherington, J., Frinchaboy, P. M., Fu, H., Gunn, J. E., Harding, P., Johnston, E. J., Kauffmann, G., Kinemuchi, K., Klaene, M. A., Knapen, J. H., Leauthaud, A., Li, C., Lin, L., Maiolino, R., Malanushenko, V., Malanushenko, E., Mao, S., Maraston, C., McDermid, R. M., Merrifield, M. R., Nichol, R. C., Oravetz, D., Pan, K., Parejko, J. K., Sanchez, S. F., Schlegel, D., Simmons, A., Steele, O., Steinmetz, M., Thanjavur, K., Thompson, B. A., Tinker, J. L., van den Bosch, R. C. E., Westfall, K. B., Wilkinson, D., Wright, S., Xiao, T., and Zhang, K., “Overview of the SDSS-IV MaNGA Survey: Mapping nearby Galaxies at Apache Point Observatory,” ApJ, 798, 7 (Jan. 2015).
  * [3] Sánchez, S. F., “Spatially Resolved Spectroscopic Properties of Low-Redshift Star-Forming Galaxies,” ARAA,58, 99–155 (Aug. 2020).
  * [4] Medina, S. N. X., Arthur, S. J., Henney, W. J., Mellema, G., and Gazol, A., “Turbulence in simulated H II regions,” MNRAS, 445, 1797–1819 (Dec. 2014).
  * [5] Raskutti, S., Ostriker, E. C., and Skinner, M. A., “Numerical Simulations of Turbulent Molecular Clouds Regulated by Radiation Feedback Forces. II. Radiation-Gas Interactions and Outflows,” ApJ, 850, 112 (Dec 2017).
  * [6] Kim, J.-G., Kim, W.-T., and Ostriker, E. C., “Modeling UV Radiation Feedback from Massive Stars. II. Dispersal of Star-forming Giant Molecular Clouds by Photoionization and Radiation Pressure,” ApJ, 859, 68 (May 2018).
  * [7] Morrissey, P., Matuszewski, M., Martin, D. C., Neill, J. D., Epps, H., Fucik, J., Weber, B., Darvish, B., Adkins, S., Allen, S., Bartos, R., Belicki, J., Cabak, J., Callahan, S., Cowley, D., Crabill, M., Deich, W., Delecroix, A., Doppman, G., Hilyard, D., James, E., Kaye, S., Kokorowski, M., Kwok, S., Lanclos, K., Milner, S., Moore, A., O’Sullivan, D., Parihar, P., Park, S., Phillips, A., Rizzi, L., Rockosi, C., Rodriguez, H., Salaun, Y., Seaman, K., Sheikh, D., Weiss, J., and Zarzaca, R., “The Keck Cosmic Web Imager Integral Field Spectrograph,” ApJ, 864, 93 (Sept. 2018).
  * [8] García-Vargas, M. L., Carrasco, E., Mollá, M., Gil de Paz, A., Berlanas, S. R., Cardiel, N., Gómez-Alvarez, P., Gallego, J., Iglesias-Páramo, J., Cedazo, R., Pascual, S., Castillo-Morales, A., Pérez-Calpena, A., and Martínez-Delgado, I., “MEGARA-GTC stellar spectral library: I,” MNRAS, 493, 871–898 (Mar. 2020).
  * [9] Dalton, G., Trager, S., Abrams, D. C., Bonifacio, P., Aguerri, J. A. L., Vallenari, A., Middleton, K., Benn, C., Dee, K., Sayède, F., Lewis, I., Pragt, J., Picó, S., Walton, N., Rey, J., Allende Prieto, C., Lhomé, É., Terrett, D., Brock, M., Gilbert, J., Ridings, A., Verheijen, M., Tosh, I., Steele, I., Stuik, R., Kroes, G., Tromp, N., Kragt, J., Lesman, D., Mottram, C., Bates, S., Gribbin, F., Burgal, J. A., Herreros, J. M., Delgado, J. M., Martin, C., Cano, D., Navarro, R., Irwin, M., Lewis, J., Gonzales Solares, E., O’Mahony, N., Bianco, A., Zurita, C., ter Horst, R., Molinari, E., Lodi, M., Guerra, J., Baruffolo, A., Carrasco, E., Farkas, S., Schallig, E., Hill, V., Smith, D., Drew, J., Poggianti, B., Pieri, M., Jin, S., Dominquez Palmero, L., Fariña, C., Martin, A., Worley, C., Murphy, D., Hidalgo, A., Mignot, S., Bishop, G., Guest, S., Elswijk, E., de Haan, M., Hanenburg, H., Salasnich, B., Mayya, D., Izazaga-Pérez, R., and Peralta de Arriba, L., “Construction progress of WEAVE: the next generation wide-field spectroscopy facility for the William Herschel Telescope,” in [Ground-based and Airborne Instrumentation for Astronomy VII ], Evans, C. J., Simard, L., and Takami, H., eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 10702, 107021B (July 2018).
  * [10] Bershady, M. A., Andersen, D. R., Verheijen, M. A. W., Westfall, K. B., Crawford, S. M., and Swaters, R. A., “SparsePak: A Formatted Fiber Field Unit for The WIYN Telescope Bench Spectrograph. II. On-Sky Performance,” ApJS, 156, 311–344 (Feb. 2005).
  * [11] Eigenbrot, A. and Bershady, M. A., “Vertical Population Gradients in NGC 891\. I. $\nabla$Pak Instrumentation and Spectral Data,” ApJ, 853, 114 (Feb. 2018).
  * [12] Abraham, R. G. and van Dokkum, P. G., “Ultra-Low Surface Brightness Imaging with the Dragonfly Telephoto Array,” PASP, 126, 55 (Jan 2014).
  * [13] Hill, G. J., Tuttle, S. E., Drory, N., Lee, H., Vattiat, B. L., DePoy, D. L., Marshall, J. L., Kelz, A., Haynes, D., Fabricius, M. H., Gebhardt, K., Allen, R. D., Anwad, H., Bender, R., Blanc, G., Chonis, T., Cornell, M. E., Dalton, G., Good, J., Jahn, T., Kriel, H., Landriau, M., MacQueen, P. J., Murphy, J. D., Peterson, T. W., Prochaska, T., Nicklas, H., Ramsey, J., Roth, M. M., Savage, R. D., and Snigula, J., [VIRUS: production and deployment of a massively replicated fiber integral field spectrograph for the upgraded Hobby-Eberly Telescope ], vol. 9147 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 91470Q (2014).
  * [14] Bolton, A. S. and Schlegel, D. J., “Spectro-Perfectionism: An Algorithmic Framework for Photon Noise-Limited Extraction of Optical Fiber Spectroscopy,” PASP,122, 248 (Feb. 2010).
  * [15] Perruchot, S., Guy, J., Le Guillou, L., Blanc, P. E., Ronayette, S., Régal, X., Castagnoli, G., Sepulveda, E., Le Van Suu, A., Jullo, E., Cuby, J. G., Karkar, S., Ghislain, P., Repain, P., Carton, P. H., Magneville, C., Ealet, A., Escoffier, S., Secroun, A., Cousinou, M. C., Honscheid, K., Elliot, A., Jelinsky, P., Brooks, D., and Tarlè, G., “Integration and testing of the DESI multi-object spectrograph: performance tests and results for the first unit out of ten,” in [Ground-based and Airborne Instrumentation for Astronomy VII ], Evans, C. J., Simard, L., and Takami, H., eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 10702, 107027K (July 2018).
  * [16] Schlegel, D., Kollmeier, J. A., and Ferraro, S., “The MegaMapper: a z>2 spectroscopic instrument for the study of Inflation and Dark Energy,” in Bulletin of AAS, 51, 229 (Sep 2019).


set for the first additive component with utility of $\epsilon+\sum_{k\in
R}(\beta^{i}-\beta^{k})$, while the maximum utility from the second additive
component is $\sum_{k\in R}\beta^{i}-\beta^{k}$. Therefore, we have
$\textsc{Alloc}_{j}(\mathcal{D},p)=\beta^{-j}/\sigma$ for all $j\in S$ (as
whenever we draw $j$ in the sampling process of $\mathcal{D}$, we know
automatically that the utility-maximizing set is $\\{j,m\\}$). Hence,

$\sum_{j\in
S}p_{j}\cdot\textsc{Alloc}_{j}(\mathcal{D},p)=\sum_{j=1}^{m-1}\beta^{j}\cdot\frac{\beta^{-j}}{\sigma}=\sigma^{-1}\cdot(m-1).$

We also comment that the allocation probabilities are exponentially small in
$m$, and thus they do not satisfy the assumption of 1.

To complete the proof, it suffices to show that under any pricing $q$, we have

$\textsc{Rev}(\mathcal{D}_{|S},q)\leq 4\sigma^{-1}\sqrt{m-1}.$ (9)

Fix any pricing $q$. We define a labelling vector $\ell\in\mathbb{Z}^{m-1}$,
where $\ell_{j}=\lceil\log_{\beta}q_{j}\rceil$ for all $j\in[m-1]$. We now
consider the maximum revenue we can extract from $v_{i,R|S}$ for any $i$ and
$R$. Since $v_{i,R}$ is defined as the maximum over two additive valuations,
we consider the three following ways via which we can extract revenue from it.

##### Revenue is extracted from $v^{1}_{i,R|S}$ and $\ell_{i}\neq i$.

Since revenue is extracted from $v^{1}_{i,R|S}$ and $m\notin S$, the only item
that can be purchased is $i$, valued at $\beta^{i}$. Furthermore, since
$\ell_{i}\neq i$, we can conclude that $q_{i}\leq\beta^{i-1}$ and therefore,
the total revenue extracted from this case is at most

$\operatorname{E}_{i,R}[\beta^{i-1}]=\sum_{i=1}^{m-1}\beta^{i-1}\cdot\frac{\beta^{-i}}{\sigma}=\sigma^{-1}\cdot\frac{m-1}{\beta}=\sigma^{-1}\sqrt{m-1}.$

##### Revenue is extracted from $v^{1}_{i,R|S}$ and $\ell_{i}=i$.

In this case, the revenue extracted is at most $\beta^{i}$. A critical
observation here is that we need $\ell_{j}\geq i$ for all $j\in R$ in order
for this case to happen; note that the utility of $S$ under $v^{1}_{i,R|S}$ is
$\beta^{i}-q_{i}<\beta^{i}-\beta^{i-1}$, since $\ell_{i}=i$, while if any item
$j\in R$ has $\ell_{j}\leq i-1$, then its utility under $v^{2}_{i,R|S}$ is
$\beta^{i}-q_{j}\geq\beta^{i}-\beta^{i-1}$, and thus the buyer would deviate
away from buying item $i$ and towards buying this $j$ under $v^{2}_{i,R|S}$.

We now define the set $U=\\{i\in S:\ell_{i}=i\\}$, and note that revenue in
this case can only be extracted from items in $U$. Furthermore, consider any
item $i\in U$ and any $R$, we can only extract revenue from item $i$ under
$v^{1}_{i,R|S}$ if $R\cap U=\emptyset$; otherwise, some item $j\in R$ has
$\ell_{j}=j<i$, and by the previous observation we cannot extract revenue from
item $i$ under $v^{1}_{i,R|S}$. Therefore, if we enumerate
$U=\\{u_{1},u_{2},\dots,u_{t}\\}$ where $u_{1}<u_{2}<\ldots<u_{t}$, the total
revenue extracted in this case is at most

$\displaystyle\operatorname*{\operatorname{E}}_{i,R}[\beta^{i}\cdot\mathbbm{1}(\text{revenue
from $v^{1}_{i,R|S}$ and $\ell_{i}=i$})]$
$\displaystyle\leq\operatorname*{\operatorname{E}}_{i,R}[\beta^{i}\cdot\mathbbm{1}[i\in
U\wedge R\cap U=\emptyset]]$
$\displaystyle=\sum_{j=1}^{t}\beta^{k}\operatorname*{\operatorname{Pr}}[i=u_{j}]\operatorname*{\operatorname{Pr}}[R\cap
U=\emptyset\mid i=u_{j}]$
$\displaystyle=\sum_{j=1}^{t}\beta^{u_{j}}\cdot\frac{\beta^{-u_{j}}}{\sigma}\cdot\left(1-\frac{1}{\sqrt{m-1}}\right)^{j-1}$
$\displaystyle\leq\sigma^{-1}\sum_{j=0}^{\infty}\left(1-\frac{1}{\sqrt{m-1}}\right)^{j}=\sigma^{-1}\sqrt{m-1}$

where the second line uses the fact that each item $u_{1},u_{2},\dots,u_{j-1}$
does not appear in $R$ with probability $1-\frac{1}{\sqrt{m-1}}$
independently.

##### Revenue is extracted from $v^{2}_{i,R|S}$.

We first fix any item $j$, and for convenience let $l=\ell_{j}$. Observe that
item $j$ can only be sold in this case if $l\leq i$ (otherwise when $l>i$, we
have $q_{j}>\beta^{l-1}\geq\beta^{i}=v^{2}_{i,R|S}(j)$); furthermore, we
obviously have $q_{j}\leq\beta^{l}$. Therefore, the total revenue extracted
from item $j$ in this case is at most

$\displaystyle\operatorname*{\operatorname{E}}_{i,R}[\beta^{l}\cdot\mathbbm{1}(\text{revenue
from $v^{2}_{i,R|S}$ and $j$ is included})]$
$\displaystyle\leq\operatorname*{\operatorname{E}}_{i,R}[\beta^{l}\cdot\mathbbm{1}[j\in
R\wedge i\geq l]]$
$\displaystyle=\beta^{l}\sum_{k=1}^{m-1}\operatorname*{\operatorname{Pr}}[i=k]\operatorname*{\operatorname{Pr}}[j\in
R\mid i=k]$
$\displaystyle\leq\frac{\beta^{l}}{\sqrt{m-1}}\sum_{k=l}^{m-1}\frac{\beta^{-k}}{\sigma}$
$\displaystyle\leq\frac{\beta^{l}}{\sqrt{m-1}}\cdot\frac{2\beta^{-l}}{\sigma}=\frac{2\sigma^{-1}}{\sqrt{m-1}}$

where the second line uses that $\operatorname*{\operatorname{Pr}}[j\in R\mid
i=k]$ is $\frac{1}{\sqrt{m-1}}$ if $i>j$, or $0$ otherwise; the second line
assumes that $\beta=\sqrt{m-1}\geq 2$, which means we can bound
$\sum_{k=l}^{m-1}\beta^{-k}\leq 2\beta^{-l}$.

Thus, the total revenue extracted from this case is at most

$\operatorname*{\operatorname{E}}_{i,R}\left[\sum_{j=1}^{m}\beta^{\ell_{j}}\cdot\mathbbm{1}(\text{revenue
from $v^{2}_{i,R|S}$ and $j$ is included})\right]\leq 2\sigma^{-1}\sqrt{m-1}.$

Combining the three cases gives us an upper bound of $4\sigma^{-1}\sqrt{m-1}$
on $\textsc{Rev}(\mathcal{D}_{|S},q)$ for any $q$.

##  Appendix E Deferred Proofs from Section 8

We start by reproducing the description of our instance for convenience.

##### The buyers.

Let $n$ buyers $v_{i}\sim\mathcal{D}_{i}$ be (independently) identically
distributed, with valuations sampled according to the following process:

  1. 1.

A set of items $A\subseteq[m]$ with $|A|=k$ is sampled uniformly at random.

  2. 2.

An integer $h\in\\{1,2,\dotsc,\frac{1}{2}\log k\\}$ is drawn uniformly at
random. We denote $\ell\coloneq 2^{h}$.

  3. 3.

The valuation of the buyer is then realized as an XOS function, which is given
by the maximum over the following additive valuations:

    1. (a)

For the set $A$, we define an additive valuation $v_{A}$ such that
$v_{A}(j)=1+t$ for items $j\in A$ and $v_{A}(j)=0$ for items $j\notin A$.

    2. (b)

For every set $B$ of cardinality $|B|=t\cdot\ell$, we define an additive
valuation $v_{B}$ such that $v_{B}(j)=1+\frac{k}{\ell}$ for items $j\in B$ and
$v_{B}(i)=0$ for items $j\notin B$.

We will take $n=k=\sqrt{m}$ and $t=\sqrt{\log k}$. Observe that for our
selected parameters $t\cdot\ell\leq\sqrt{m\cdot\log m}\leq m$, and all
parameters are integer-valued, so this is a well-defined instance. Let $A_{i}$
be the $A$-set for buyer $i$.

In order to prove the instance-specific claims in Section 8, we will require
more notation. We use $\textsc{Rev}(v_{i},p)$ to denote the revenue that we
collect from the $i$-th buyer if their valuation is realized to $v_{i}$ and
the pricing is some fixed vector $p\in\mathbb{R}_{\geq 0}^{m}$. Note that this
is a deterministic quantity, as all sources of randomness are fixed.

For our instance we will define the events $\mathcal{E}^{A}(v_{i},p)$ and
$\mathcal{E}^{B}(v_{i},p)$ that subset of items $[m]$ that buyer $i$ with
valuation $v_{i}$ purchases is of type $A$ or $B$ respectively; more formally,
$\mathcal{E}^{A}(v_{i},p)$ holds if for prices $p$ the buyers’ utility for
their preferred bundle $T\subseteq[m]$ is given by

$u_{i}(T)=v_{i}(T)-p(T)=\sum_{j\in T}(v_{A}(j)-p_{j}),$

and otherwise $\mathcal{E}^{B}(v_{i},p)$ holds if their valuation is given by
some $v_{B}$.

See 14

###### Proof of Lemma 14.

We consider a fixed buyer $v=v_{i}$ (and thus also the set $A$ and the integer
$\ell$ in the valuation description) and a fixed pricing $p$. Since
$\mathcal{E}^{A}(v,p)$ holds, we know that the buyer will purchase a set of
items such that their utility is maximized through the linear valuation
$v_{A}$; in particular, including a subset $A_{T}\subseteq A$ given by

$A_{T}\coloneq\operatorname*{arg\,max}_{T\subseteq A}(v_{A}(T)-p(T))=\\{j\in
A:p_{j}\leq t+1\\}.$

We will show that $|A_{T}|\geq\frac{kt}{t+1}$. To that end, let
$B_{AT}\subseteq A_{T}$ be a maximal $B$-utility subset of $A_{T}$. Observe
that since $t\leq\frac{k}{\ell}$ for our chosen parameters, every item worth
buying under an $A$-valuation $v_{A}$ will also be worth buying under some
$B$-valuation $v_{B}$. Thus, $B_{AT}$ is any maximal-cardinality subset of
$A_{T}$ of size at most $t\ell$.

Since $\mathcal{E}^{A}(v_{i},p)$ holds we know that $i$ prefers $A_{T}$ to
$B_{AT}$, meaning $A_{T}$ has higher utility than $B_{AT}$, so

$\sum_{j\in A_{T}}(t+1-p_{j})\geq\sum_{j\in
B_{AT}}\left(1+\frac{k}{\ell}-p_{j}\right).$ (10)

We first argue that $|A_{T}|>t\ell$. Otherwise $B_{AT}=A_{T}$, and thus (10)
can only be satisfied if $t\geq\frac{k}{\ell}$ which is not the case as
$t\ell\leq\sqrt{\log k}\cdot\sqrt{k}<k$. Therefore $|A_{T}|\geq t\ell$. In
this case, $|B_{AT}|=t\ell$ and (10) can be restated as

$\sum_{j\in A_{T}\setminus B_{AT}}(t+1-p_{j})\geq\sum_{j\in
B_{AT}}\left(\frac{k}{\ell}-t\right)=kt-t^{2}\ell,$ (11)

and since $p_{j}\geq 0$ for all $j$ this implies that

$(t+1)\cdot(|A_{T}|-t\ell)\geq kt-t^{2}\ell.$

This immediately implies that $|A_{T}|\geq\frac{kt}{t+1}$, as claimed. ∎

See 15

###### Proof of Lemma 15.

We will prove this claim for every item pricing $p$ and every buyer valuation
$v=v_{i}\sim\mathcal{D}_{i}$, provided that the event
$\mathcal{E}^{A}(v_{i},p)$ holds. The claim will then follow from taking the
conditional expectation over $v_{i}$.

We use the same setup as in the proof of Lemma 14 to define the sets $A_{T}$
and $B_{AT}$. Since $\mathcal{E}^{A}(v_{i},p)$ holds by assumption, we know
that $\textsc{Rev}(v_{i},p)=p(A_{T})$ because the buyer maximizes their
utility via $v_{A}$ and buys this set. Therefore we will bound $p(A_{T})$.

Just as before, (10) implies that $|A_{T}|\geq t\ell$ and that

$\sum_{j\in A_{T}\setminus B_{AT}}(t+1-p_{j})\geq kt-t^{2}\ell$ (12)

holds for _any_ subset $B_{AT}$ of $A_{T}$ with cardinality $t\ell$; in
particular it holds for $B_{AT}$ the $t\ell$ items in $A_{T}$ of minimum
price. For this choice of $B_{AT}$ the average item price in $A_{T}\setminus
B_{AT}$ exceeds the average item price in $A_{T}$, i.e.

$\frac{p(A_{T}\setminus B_{AT})}{|A_{T}\setminus
B_{AT}|}\geq\frac{p(A_{T})}{|A_{T}|}.$

Plugging this into (12) and observing that $|A_{T}\setminus
B_{AT}|=|A_{T}|-t\ell$ yields

$kt-t^{2}\ell\leq\sum_{j\in A_{T}\setminus
B_{AT}}(1+t-p_{j})=(1+t)(|A_{T}|-t\ell)-p(A_{T}\setminus
B_{AT})\leq(|A_{T}|-t\ell)\cdot\left(1+t-\frac{p(A_{T})}{|A_{T}|}\right).$

Solving for $p(A_{T})$ we obtain

$p(A_{T})\leq|A_{T}|\cdot\frac{|A_{T}|(t+1)-t(k+\ell)}{|A_{T}|-t\ell},$

and using that $|A_{T}|\leq|A|=k$ (by definition) and that
$|A_{T}|\geq\frac{kt}{t+1}$ by Lemma 14 since $\mathcal{E}_{i}^{A}(v,p,S)$
holds, we finally obtain that

$p(A_{T})\leq k\cdot\frac{k(t+1)-t(k+\ell)}{\frac{kt}{t+1}-t\ell}.$

The proof is completed by showing

$\frac{k(t+1)-t(k+\ell)}{\frac{kt}{t+1}-t\ell}\leq c_{1}$

for some constant $c_{1}$ and all sufficiently large $k$, which easily follows
from observing that $k\geq 2t\ell$ for our choice of parameters $t=\sqrt{\log
k}$ and $\ell\in[2,\sqrt{k}]$. ∎

See 16

###### Proof.

Consider a fixed item pricing $p\in\mathbb{R}_{\geq 0}^{m}$. We can relabel
the items so that their prices are non-decreasing, i.e. we assume without loss
of generality that $p_{1}\leq p_{2}\leq\dotsc\leq p_{m}$.

Since we are conditioning on event $\mathcal{E}_{i}^{B}(v,p,S)$, we know that
the buyer always buys a set of items whose utility is maximized through a
$B$-valuation, thus they buy a set of at most $t\ell$ items where $\ell$ is a
buyer-dependent parameter that is drawn uniformly at random from
$L=\\{2,4,8,\dotsc,\sqrt{k}\\}$. Furthermore, the buyer will clearly
prioritize the cheaper items, as all items have the same (additive) valuation
of $1+k/\ell$ under $B$ and cheaper items contribute more to the buyer’s
utility. Thus, for a given $\ell$, the buyer will purchase items $T$ which are
a prefix of the items in ascending-price-order:

$T=\\{1,2,3,\dotsc,x\\}\subseteq[m],$

where $x$ is the maximum index for which $x\leq t\ell$ and $p_{x}\leq
1+\frac{k}{\ell}$. We will bound the conditional expected revenue, which is
precisely the conditional expectation of $p(T)$, by observing that

$\displaystyle\operatorname*{\operatorname{E}}_{v_{i}}\left[\textsc{Rev}(v_{i},p)\mid\mathcal{E}^{B}(v_{i},p)\right]$
$\displaystyle=\sum_{\ell^{\prime}\in
L}\operatorname*{\operatorname{E}}_{v_{i}}\left[\textsc{Rev}(v_{i},p)\mid\mathcal{E}^{B}(v_{i},p)\wedge\ell(v_{i})=\ell^{\prime}\right]\cdot\operatorname*{\operatorname{Pr}}_{v_{i}}\left[\ell(v_{i})=\ell^{\prime}\right]$
$\displaystyle=\frac{1}{|L|}\cdot\sum_{\ell^{\prime}\in
L}\operatorname*{\operatorname{E}}_{v_{i}}\left[p(T)\mid\mathcal{E}^{B}(v_{i},p)\wedge\ell(v_{i})=\ell^{\prime}\right].$
(13)

Here we use $\ell(v_{i})$ to denote the value of $\ell$ that
$v_{i}\sim\mathcal{D}_{i}$ realizes, and we used that
$\operatorname*{\operatorname{Pr}}_{v_{i}}\left[\ell(v_{i})=\ell^{\prime}\right]=\frac{1}{|L|}=\frac{2}{\log
k}$ for all $\ell^{\prime}\in L$. We will bound this sum (13) by
$c_{2}\cdot\frac{tk}{\log k}$ for some constant $c_{2}$ and sufficiently large
$k$. We will do this by a combination of case analysis and grouping the
possible values of $\ell$ as a function of the non-uniform prices. Let

$\ell^{*}\coloneq\max\left(\left\\{\ell\in
L:\>p_{t\ell}\leq\frac{k}{\ell}+1\right\\}\right).$

This set is nonempty and the threshold parameter value $\ell^{*}$ exists
provided that $p_{2t}\leq\frac{k}{2}+1$. This is because the thresholds
$\frac{k}{\ell}+1$ are decreasing in $\ell$, while the prices $p_{t\ell}$ are
increasing in $\ell$. Therefore we have two cases to consider.

##### Case $1$.

First, we consider the case that $p_{2t}>\frac{k}{2}+1$. In this case, since
$t+1\leq\frac{k}{2}+1$ for $t=\sqrt{\log k}$ and also
$\frac{k}{\ell}+1\leq\frac{k}{2}+1$ for $\ell\geq 2$, we know the buyer will
not buy any item $j$ for $j\geq 2t$; since $p_{j}$ is too high it would
necessarily decrease their utility. Furthermore, the maximum revenue
attainable from each of the $2t$ remaining items is at most equal to their
value to the buyer, which is $(\frac{k}{\ell}+1)$ for $\ell\in L$ uniformly at
random. Thus, from (13) we obtain

$\operatorname*{\operatorname{E}}_{v_{i}}\left[\textsc{Rev}(v_{i},p)\mid\mathcal{E}^{B}(v_{i},p)\right]\leq
2t\sum_{\ell\in
L}\frac{1}{|L|}\cdot\left(1+\frac{k}{\ell}\right)=\frac{1}{|L|}\cdot\left(2t|L|+2tk\right)\leq\frac{6tk}{\log
k},$

where the last statement follows from $|L|=\frac{1}{2}\log k$ and the fact
that $L=\\{2,4,8,\dotsc,\sqrt{k}\\}$ follows a geometric progression and so
the $\ell^{-1}$ terms sum to at most $1$.

##### Case $2$.

We now address the case where $\ell^{*}$ is well-defined. We consider a buyer
$v=v_{i}$ and again use $\ell(v)$ to denote the $\ell$ value that their
valuation realized. We analyze their expected revenue based on whether
$\ell(v)<\ell^{*}$, $\ell(v)=\ell^{*}$, or $\ell(v)>\ell^{*}$.

If $\ell(v)=\ell^{*}$, then we know that the buyer will purchase at most
$t\ell^{*}$ items, and also that each item contributes at most the buyer’s
per-item value to the revenue, which is $\frac{k}{\ell^{*}}+1$. Thus we have

$\operatorname*{\operatorname{E}}_{v}\left[\textsc{Rev}(v,p)\mid\mathcal{E}^{B}(v,p)\wedge\ell(v)=\ell^{*}\right]\leq
t\ell^{*}\left(1+\frac{k}{\ell^{*}}\right)=t(k+\ell^{*}).$ (14)

If $\ell(v)<\ell^{*}$, then each item sold can contribute at most
$(\frac{k}{\ell^{*}}+1)$ to the revenue, and for each $\ell$ at most $t\ell$
items can be sold. Thus, we have

$\sum_{\ell\in
L:\ell<\ell^{*}}\operatorname*{\operatorname{E}}_{v}\left[\textsc{Rev}(v,p)\mid\mathcal{E}^{B}(v,p)\wedge\ell(v)=\ell\right]\leq\sum_{\ell\in
L:\ell<\ell^{*}}t\ell\cdot\left(1+\frac{k}{\ell^{*}}\right)\leq
t(k+\ell^{*}),$ (15)

where the last inequality follows from $\sum_{\ell\in
L:\ell<\ell^{*}}\ell\leq\ell^{*}$ again because the terms of $L$ follow a
geometric progression.

Finally, if $\ell(v)>\ell^{*}$ then each individual item can contribute at
most $\frac{k}{\ell(v)}+1$ to the revenue, because this is the buyer’s per-
item value. And by the definition of $\ell^{*}$ there can be at most
$t\ell^{*}$ items such that
$p_{j}\leq\frac{k}{\ell(v)}+1\leq\frac{k}{\ell^{*}}+1$; only these items may
generate revenue. Thus, we have

$\sum_{\ell\in
L:\ell>\ell^{*}}\operatorname*{\operatorname{E}}_{v}\left[\textsc{Rev}_{i}(v,p,S)\mid\mathcal{E}_{i}^{B}(v,p,S)\wedge\ell(v)=\ell\right]\leq
t\ell^{*}\cdot\sum_{\ell\in L:\ell>\ell^{*}}\left(1+\frac{k}{\ell}\right)\leq
t\ell^{*}\cdot|L|+tk,$ (16)

where the last inequality once again follows because the terms of $L$ follow a
geometric progression, so the sum of the $\ell^{-1}$ terms is at most $1$.

Combining everything, applying (14), (15), and (16) to (13) we have

$\displaystyle\operatorname*{\operatorname{E}}_{v_{i}}\left[\textsc{Rev}(v_{i},p)\mid\mathcal{E}^{B}(v_{i},p)\right]$
$\displaystyle=\frac{1}{|L|}\cdot\sum_{\ell\in
L}\operatorname*{\operatorname{E}}_{v_{i}}\left[\textsc{Rev}(v_{i},p)\mid\mathcal{E}^{B}(v_{i},p)\wedge\ell(v)=\ell\right]$
$\displaystyle\leq\frac{1}{|L|}\cdot(2t(k+\ell^{*})+t\ell^{*}|L|+tk),$

and since $\ell^{*}\leq\max(L)=\sqrt{k}$ and $|L|=\frac{1}{2}\log k$, we have
that

$\displaystyle\operatorname*{\operatorname{E}}_{v_{i}}\left[\textsc{Rev}(v_{i},p)\mid\mathcal{E}^{B}(v_{i},p)\right]\leq
c_{2}\cdot\frac{tk}{\log k}$

for some constant $c_{2}$ and all suitably large $k$, as desired.

This holds in both cases, completing the proof. ∎

Before proving the lemma 17, recall the definitions of an _assignment_
$\sigma$ of items to buyers and the definition for $I\subseteq[n]$ to be
$(1-c)$-_feasible_ under $\sigma$ (Definitions 4 and 5). We will use the
following more general claim:

######  Lemma 24.

Let $\mathcal{E}(c,C,n)$ denote the event that there exist some subset of
buyers $I\subseteq[n]$ and assignment $\sigma:[m]\rightarrow[n]\cup\\{\bot\\}$
such that $|I|\geq Cn$ and $I$ is $(1-c)$-feasible under $\sigma$. For any
fixed constant $\gamma>0$, if $c=\gamma\cdot\log^{-1/2}n$ and
$C=8c=8\gamma\log^{-1/2}n$, then

$\operatorname*{\operatorname{Pr}}_{v\sim\mathcal{D}}\left[\mathcal{E}(c,C,n)\right]=O\left(e^{-n^{2}/\log^{2}n}\right).$

###### Proof of Lemma 24.

To begin, consider a fixed $I\subseteq[n]$ let $\alpha\coloneq|I|/n$. We will
be interested in the $I$ for which $\alpha\geq C$. Next let
$N(I)\coloneq\cup_{i\in I}A_{i}$ be the union of the $A$-sets for $i\in I$,
and let $A_{i}^{\sigma}\coloneq A_{i}\cap\sigma^{-1}(i)$, so that $\sigma$ is
$(1-c)$-feasible for $i$ precisely when
$|A_{i}^{\sigma}|\geq(1-c)|A_{i}|=(1-c)n$.

Next observe that for all assignments $\sigma$,

$\min_{i\in I}|A_{i}^{\sigma}|\leq\frac{1}{|I|}\cdot\sum_{i\in
I}|A_{i}^{\sigma}|\leq\frac{1}{|I|}\cdot|N(I)|,$

since $A_{i}^{\sigma}\subseteq A_{i}$ by definition and the $A_{i}^{\sigma}$
are disjoint because $\sigma$ is a well-defined mapping. Therefore if $\sigma$
is $(1-c)$-feasible for $I$, then

$|I|n(1-c)\leq|I|\min_{i\in I}|A_{i}^{\sigma}|\leq|N(I)|.$ (17)

We will argue that for fixed $I$ a $(1-c)$-feasible assignment for $I$ is
unlikely to exist, because this would imply that $|N(I)|\geq|I|(1-c)n$ by
(17), which we will show is unlikely.

To this end, let $X_{ij}\coloneq\mathbbm{1}\\{j\in A_{i}\\}$ indicate the
event that $j\in A_{i}$; then let $Y_{j}\coloneq\max_{i\in I}X_{ij}$ indicate
that $j$ is in at least one set $A_{i}$ for $i\in I$. Then observe that
$Y\coloneq\sum_{j}Y_{j}=|N(I)|$.

######  Claim 25.

$\\{Y_{j}\\}$ is negatively associated (NA).

###### Proof.

This follows from results about NA variables by Joag-Dev and Proschan [1983];
in particular closure and composition properties for collections of NA
variables.

First, consider each collection of indicator random variables
$\mathcal{X}_{i}\coloneq\\{X_{i^{\prime}j}:i^{\prime}=i\\}$. Exactly $n$ of
these are 1, indicating the subset $A$ chosen uniformly at random from $[m]$;
therefore this is a permutation distribution, and so by Joag-Dev and Proschan
[1983] Theorem 2.11 this collection $\mathcal{X}_{i}$ is NA. Next, consider
the larger collection $\mathcal{X}\coloneq\\{{}X_{ij}\\}$; as the union of
independent collections of NA variables, $\mathcal{X}$ is also NA (Joag-Dev
and Proschan [1983], Property $P_{7}$). Finally, consider the collection
$\mathcal{Y}\coloneq\\{Y_{j}\\}$. Since $Y_{j}\coloneq\max_{i\in I}X_{ij}$,
each $Y_{j}$ is defined on a unique subset of the NA variables $\mathcal{X}$;
for any $Y_{j}$ and $Y_{j^{\prime}}$, the sets of $X_{ij}$ upon which they
depend are disjiont. Since $\max(\cdot)$ is an increasing function, therefore
$\mathcal{Y}$ is NA also (Joag-Dev and Proschan [1983], Property $P_{6}$). ∎

Bounding $|N(I)|$: Since 25 demonstrates that $\mathcal{Y}$ is negatively
associated, we can use concentration. First note that for all $j$,

$\operatorname*{\operatorname{E}}_{v\sim\mathcal{D}}\left[Y_{j}\right]=1-\operatorname*{\operatorname{Pr}}_{v\sim\mathcal{D}}\left[\bigwedge_{i\in
I}(j\not\in A_{i})\right]=1-(1-1/n)^{|I|}=1-(1-1/n)^{\alpha n}.$

Then we may bound the probability that $|N(I)|\geq|I|(1-c)n$ by applying a
Hoeffding bound to $\sum_{j}Y_{j}$ for any $I$ with $|I|\geq Cn$ as follows:

$\displaystyle\operatorname*{\operatorname{Pr}}_{v\sim\mathcal{D}}\left[|N(I)|\geq\alpha
n^{2}(1-c)\right]$
$\displaystyle=\operatorname*{\operatorname{Pr}}_{v\sim\mathcal{D}}\left[Y-\operatorname*{\operatorname{E}}_{v\sim\mathcal{D}}\left[Y\right]\geq\alpha
n^{2}(1-c)-\operatorname*{\operatorname{E}}_{v\sim\mathcal{D}}\left[Y\right]\right]$
$\displaystyle\leq\exp\left(\frac{-2\left(\alpha
n^{2}(1-c)-\operatorname*{\operatorname{E}}_{v\sim\mathcal{D}}\left[Y\right]\right)^{2}}{n^{2}}\right)$
$\displaystyle=\exp\left(\frac{-2\left(\alpha
n^{2}(1-c)-n^{2}(1-(1-1/n)^{\alpha n})\right)^{2}}{n^{2}}\right).$ Since
$1-(1-1/n)^{\alpha n}\leq 1-e^{-\alpha}(1-O(1/n))^{\alpha}\leq
1-e^{-\alpha}+e^{-\alpha}\cdot
O(1/n)\leq\alpha-\alpha^{2}/2+\alpha^{3}/6+e^{-\alpha}\cdot O(1/n)$, therefore
$\displaystyle\operatorname*{\operatorname{Pr}}_{v\sim\mathcal{D}}\left[|N(I)|\geq\alpha
n^{2}(1-c)\right]$
$\displaystyle\leq\exp\left(-2\alpha^{2}n^{2}\left((1-c)-(1-\frac{\alpha}{2}+\frac{\alpha^{2}}{6}+\frac{e^{-\alpha}}{\alpha}O(n^{-1})\right)^{2}\right)$
$\displaystyle=\exp\left(-2\alpha^{2}n^{2}\left(\frac{\alpha}{2}-\frac{\alpha^{2}}{6}-\frac{e^{-\alpha}}{\alpha}O(n^{-1})-c\right)^{2}\right)$
$\displaystyle\leq\exp\left(-\frac{1}{8}n^{2}\alpha^{4}\left(1-\frac{2\alpha}{3}-4\frac{e^{-\alpha}}{\alpha^{2}}O(n^{-1})\right)^{2}\right),$
since $\alpha\geq C>4c$, and since $\alpha\geq C\gg n^{-1/3}$ for sufficiently
large $n$ this last term is $o(\alpha)$, so
$\displaystyle\leq\exp\left(-\frac{1}{8}n^{2}\alpha^{4}\left(1-\frac{3\alpha}{4}\right)^{2}\right)$
$\displaystyle\leq\exp\left(-\frac{9}{128}n^{2}\alpha^{4}\right),$

where we used that $1-3\alpha/4\geq 3/4$. Since $I$ is a set for which
$\alpha\geq C$, by (17), we therefore know that

$\operatorname*{\operatorname{Pr}}_{v\sim\mathcal{D}}\left[\exists\sigma:\sigma\text{
is $(1-c)$-feasible for
}I\right]\leq\exp\left(-\frac{9}{128}n^{2}C^{4}\right).$ (18)

Concluding: We finish our argument with a union bound over the sets
$I\subseteq[n]$ for which $|I|\geq Cn$. There are at most $2^{n}<e^{n}$ such
sets; therefore by (18),

$\displaystyle\operatorname*{\operatorname{Pr}}_{v\sim\mathcal{D}}\left[\bigvee_{I\subseteq[n]:|I|\geq
Cn}\left(\exists\sigma:\sigma\text{ is $(1-c)$-feasible for }I\right)\right]$
$\displaystyle\leq\sum_{I\subseteq[n]:|I|\geq
Cn}\operatorname*{\operatorname{Pr}}_{v\sim\mathcal{D}}\left[\exists\sigma:\sigma\text{
is $(1-c)$-feasible for }I\right]$
$\displaystyle\leq\exp\left(n-\frac{9}{128}\cdot n^{2}C^{4}\right)$
$\displaystyle=O(e^{-n^{2}/\log^{2}n})$

for our choice of $C=O(\log^{-1/2}(n))$. The left-hand side is precisely the
probability of the event $\mathcal{E}(c,C,n)$, and so this demonstrates the
stated claim. ∎

With this lemma in hand, Lemma 17 follows as a direct corollary.

See 17

###### Proof.

Since $t=\sqrt{\log n}$ for our choice of parameters, observe that any $I$
which is $(1-\frac{1}{t+1})$-feasible under some assignment $\sigma$ is also
$(1-\frac{2}{t})$-feasible under $\sigma$.

We may take therefore take $\gamma=2$ and $c=\frac{2}{\log^{1/2}n}$ and
$C=\frac{16}{\log^{1/2}n}$ and apply Lemma 24, which implies that

$\operatorname*{\operatorname{Pr}}_{v\sim\mathcal{D}}\left[\mathcal{E}(c,C,n)\right]=O\left(e^{-n^{2}/\log^{2}n}\right).$

Therefore, turning to $|I_{max}(v)|$, we have

$\displaystyle\operatorname*{\operatorname{E}}_{v}\left[|I_{max}(v)|\right]$
$\displaystyle=\operatorname*{\operatorname{E}}_{v}\left[|I_{max}(v)|\mid\neg\mathcal{E}(c,C,n)\right]\operatorname*{\operatorname{Pr}}_{v}\left[\neg\mathcal{E}(c,C,n)\right]+\operatorname*{\operatorname{E}}_{v}\left[|I_{max}(v)|\mid\mathcal{E}(c,C,n)\right]\operatorname*{\operatorname{Pr}}_{v}\left[\mathcal{E}(c,C,n)\right]$
$\displaystyle\leq
Cn+n\cdot\operatorname*{\operatorname{Pr}}_{v}\left[\mathcal{E}(c,C,n)\right]$
$\displaystyle\leq n\cdot\frac{16}{t}+n\cdot c_{1}\cdot
e^{-n^{2}/\log^{2}n}\leq 17\cdot\frac{n}{t}$

for some constant $c_{1}$ and for sufficiently large $k=n$, as claimed. ∎