[1,2,6]Maria Chekhova

1]Max Planck Institute for the Science of Light, Staudtstr. 2, Erlangen,
91058, Germany

2]Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7/B2,
Erlangen, 91058, Germany

3]Department of Physics, Technion – Israel Institute of Technology, Technion
City, Haifa, 32000, Israel

4]Solid State Institute, Technion – Israel Institute of Technology, Technion
City, Haifa, 32000, Israel

5]Helen Diller Quantum Center, Technion – Israel Institute of Technology,
Technion City, Haifa, 32000, Israel

6]Faculty of Electrical and Computer Engineering, Technion – Israel Institute
of Technology, Technion City, Haifa, 32000, Israel

7]femtoQ Laboratory, Department of Engineering Physics, Polytechnique
Montréal, Montréal, Québec H3T 1J4, Canada

8]Univ. Lille, CNRS, UMR 8523—PhLAM—Physique des Lasers Atomes et Molécules,
Lille, F-59000, France

# High Harmonic Generation by Bright Squeezed Vacuum

Andrei Rasputnyi  Zhaopin Chen  Michael Birk  Oren Cohen  Ido Kaminer  Michael
Krüger  Denis Seletskiy<EMAIL_ADDRESS>Francesco Tani  [  [  [  [
[  [  [  [

###### Abstract

We observe non-perturbative high harmonic generation in solids pumped by a
macroscopic quantum state of light, bright squeezed vacuum (BSV), which we
generate in a single spatiotemporal mode. Due to its broad photon-number
distribution, covering states from $0$ to $2\times 10^{13}$ photons per pulse,
and sub-cycle electric field fluctuations over $\pm 1\hbox{V}/\hbox{\r{A}}$,
BSV provides access to free carrier dynamics within a much broader range of
peak intensities than accessible with coherent light. It is also considerably
more efficient in the generation of high harmonics than coherent light of the
same mean intensity.

###### keywords:

Extreme Quantum Optics, High Harmonic Generation, Bright Squeezed Vacuum,
Photon Statistics

##  1 Introduction

High harmonic generation (HHG) is an extreme nonlinear optical process
resulting in the generation of coherent high-frequency radiation under intense
laser illumination of matter, such as gases [1, 2, 3, 4] and solids [5, 6, 7].
It enables the generation of attosecond light pulses [8, 9], giving access to
a vast range of ultrafast phenomena at the natural time scale of electron
motion. In contrast to conventional nonlinear optics, HHG involves complex
electron dynamics, which is explained by a three-step mechanism [10, 11, 12].
Figure 1(b) depicts the mechanism in bulk solids: creation of an electron-hole
pair through tunneling, its acceleration in the laser field (intraband HHG),
and finally, electron-hole recombination, with the acquired kinetic energy
released in the form of high-frequency photons (interband HHG). The control of
electron dynamics at optical frequencies paves the way to petahertz
electronics [13], outperforming current standards of data processing. HHG
allows all-optical energy band structure retrieval [6], extreme ultraviolet
spectroscopy [14], laser picoscopy of the valence electron structure [15],
investigations of quantum phase transitions [16] and Berry phases [17], to
name a few.

Although HHG essentially relies on the quantum properties of the matter, the
intense light field driving it is usually considered to be classical (Fig.
1(a)). Hitherto, there have been no experiments employing quantum states of
light for driving HHG. Meanwhile, recent works reveal quantum-optical features
in the radiation of both high harmonics [18, 19, 20] and the driver as a
result of generating them [21, 22, 23]. Still, all these works deal with the
HHG from classical (coherent) pulses of light. The use of quantum light for
pumping HHG has been considered theoretically [24, 25] but not realized up to
now since HHG requires intensities on the order of $1\hbox{ TW}\
\hbox{cm}^{-2}$, or photon numbers about $10^{13}$ at a timescale of tens of
femtoseconds, which so far were inaccessible for quantum state engineering.
Indeed, quantum states of light produced in quantum optics typically contain
only a few photons per mode. A macroscopic quantum state can be obtained by
adding a strong coherent component to a squeezed vacuum [26] or to a single
photon [27, 28], but even in this case, most of the energy is contained in the
classical part [29].

Figure 1: (a) The electric field of a coherent pulse exhibits a mean value
oscillating at the optical frequency and a constant variance. (b) The electric
field of a bright squeezed vacuum pulse. Its mean value is zero at every time
moment, while the variance oscillates at twice the carrier frequency. (c) High
harmonic generation by coherent light in a semiconductor in the interband
mechanism. An electron-hole pair emerges via perturbative (multiphoton) or
non-perturbative excitation, is accelerated by the electric field, gaining
energy, and finally recombines, emitting high-energy photons. (d) High
harmonic generation by bright squeezed vacuum in a semiconductor. Due to the
extremely noisy nature of the BSV field, the high harmonic generation is
driven by a quantum state of light comprised of a wide range of photon numbers
and an ambiguity in the sign of the electric field.

An exception is bright squeezed vacuum (BSV), a macroscopic quantum state of
light generated at the output of a strongly pumped unseeded optical parametric
amplifier. BSV is a quantum superposition of even-photon-number states, with
the strongest contribution from the vacuum but with very high photon numbers
contributing considerably. The mean photon number $\langle N\rangle$ of BSV
achieved in recent experiments was as high as $10^{13}$ [30, 31], with the
photon-number probability distribution so broad that photon numbers exceeding
the mean by a factor of 6 or more occurred with 2 % probability. This feature
is in striking contrast with the photon-number distribution of coherent
(classical) light, whose width is as small as $\Delta N=\sqrt{\langle
N\rangle}$ (the shot-noise limit). For coherent light with $\langle
N\rangle=10^{13}$, events happening with 2% probability have photon numbers
that differ from the mean by less than $10^{-4}\%$.

BSV has been shown to manifest non-classical features as sub-shot-noise
photon-number correlations [32] and polarization entanglement [33]. It was
also predicted to enhance sub-cycle sensing of quantum optical fields [34].
The broad photon-number probability distribution of BSV leads to superbunching
[35]: its intensity correlation function of order $n$,
$g^{(n)}\equiv\langle:N^{2}:\rangle/\langle N\rangle^{2}$, is considerably
higher than for thermal light. This feature becomes more pronounced at large
$n$: while for thermal light, the correlation functions are $g_{th}^{(n)}=n!$,
for BSV they are $g_{BSV}^{(n)}=(2n-1)!!$. Superbunching enhances multiphoton
effects like perturbative harmonic generation [30] or nonlinear electron
emission from nanotips [36].

Recent theoretical works propose applications of BSV in strong-field physics,
predicting an extended plateau for HHG [24], modification of electron dynamics
[25, 37], and quadrature squeezing in the extreme ultraviolet spectral range
[38]. Motivated by these proposals, here we demonstrate the experimental
observation of BSV-driven HHG in solid-state targets.

In contrast to a bright coherent pulse, which has a well-defined amplitude of
the electric field with very weak fluctuations (Fig. 1(a)), the mean electric
field of the BSV remains at zero for all times, while its variance oscillates
at twice the light frequency (Figure 1(c)) [39, 40]. This feature indicates
strong quantum fluctuations of the field on a sub-cycle timescale. Due to
these fluctuations, a BSV pulse is capable of driving electrons in a solid not
only with a certain quasi-momentum $k$ at a time, as in the case of a coherent
pulse (Fig. 1(b)), but within a broad range of quasi-momenta simultaneously
(Fig. 1(d)), including oppositely directed. Accordingly, within half of the
optical cycle the electron-hole pairs can be spread all over the energy bands,
emitting a broad spectrum of harmonic radiation. It also means that multi-
photon and tunneling mechanisms of HHG, corresponding to different field
values, can be realized simultaneously.

##  2 Results

Figure 2: (a) The spectrum of harmonics generated in a 6 $\mu$m x-cut doped
lithium niobate crystal by bright squeezed vacuum (solid line) and by coherent
radiation (dashed line). The photon energies of the 6th and the 7th harmonics
exceed the direct energy bandgap of doped lithium niobate. Upper panel:
Harmonic yield enhancement factor BSV vs. coherent pump for the same mean
intensity ($\sim 2\times 10^{13}\ \mathrm{W\,cm}^{-2}$). We show experimental
(solid lines) and simulated (dashed) factors. (b),(c) Joint photon-number
probability distribution of the input pump (b, coherent radiation, c, bright
squeezed vacuum) and the 6th harmonic for two different mean pump photon
numbers (red and black). In the case of coherent pump, the joint photon-number
distribution is Gaussian while the photon statistics in the BSV case show non-
trivial features.

In this work we report the first observation of HHG excited by quantum light.
We compare HHG from solid targets driven by ultrashort pulses of coherent and
quantum light. Both pulses are centred at 1.6 $\mu$m (0.77 eV), with the first
having a duration of 70 fs at full-width half maximum (FWHM) and the latter of
25 fs (FWHM). We use two targets: 6 $\mu$m-thick x-cut Mg:LiNbO3 (LN) and 1
$\mu$m-thick amorphous silicon (a-Si), observing emission of harmonics up to
the 7th order.

The HHG spectrum of LN (Fig. 2(a)) shows both odd and even harmonics, due to
the non-centrosymmetric crystal structure, while there are only odd harmonics
from a-Si. The photon energies of the 2nd, the 3rd and the 4th harmonics from
LN are below the indirect bandgap (3.3 eV [41]), the 5th harmonic photon
energy is between the direct (4.3 eV [41]) and indirect badgaps, and the 6th
and 7th harmonics are above the direct bandgap (Fig. 2(a)). Despite the
shorter duration, the BSV-generated 4th-7h harmonics show at least 5 to 15
times higher yield than the same harmonics driven by coherent pulses with
identical mean intensity ($2\hbox{ TW}\ \hbox{cm}^{-2}$).

Simulations based on the semiconductor-Bloch equations and accounting for the
BSV photon statistics shows a similar behavior (see Fig. 2(a) and Methods).

Besides measuring the mean value of the harmonic yield, we also study the
joint photon-number statistics between the input pump light and the harmonic
radiation. For this purpose, we measure the shot-to-shot input light intensity
and the resulting harmonic intensities (see Methods). Figures 2(b,c) show the
measured two-dimensional photon-number distributions between the pump and the
6th harmonic from LN in the case of the coherent pump (panel b) and BSV pump
(panel c) for two different mean photon numbers (red and black points). The
photon-number probability distribution for coherent light is narrow and
Gaussian, centered at the mean photon number $\langle N_{\omega}\rangle$. The
photon-number probability distribution of the harmonic is also Gaussian with
the mean photon number $\langle N_{6\omega}\rangle$ (Figure 2(b)). The
increase of the pump’s mean photon number shifts the center of the joint
probability distribution without significantly changing its width. The
measurement of the mean photon number is thus sufficient to find the harmonic
yield $\langle N_{6\omega}\rangle/\langle N_{\omega}\rangle$, but obtaining
the power scaling requires a systematic scan of the pump mean photon number.

In contrast to coherent radiation, BSV has a broad photon-number distribution
with the maximum at zero photon number. The photon-number distribution of the
harmonic inherits the large width of the driving pulse; moreover, there is a
correlation between the BSV photon number and the harmonic photon number (Fig.
2(c)). As a result, the large variance of the BSV photon number reveals the
entire power dependence through a single joint probability distribution
measurement. In other words, a scan in the mean photon number of the BSV
excitation is not required, underscored by the fact that both low and high BSV
setting (red vs black in Figure 2(c)) can reveal the entire power dependence.

Figure 3: Power scaling of the 5th and 7th harmonics generated in amorphous
silicon by bright squeezed vacuum (a,b), and by coherent radiation (c,d)
(solids lines: mean photon number of harmonic, shaded area: photon number
variance of harmonic for particular input pump peal intensity). For coherent
light excitation, the range above 2 TW/cm2 is inaccessible because of sample
damage.

Figure 3 shows the (shot-to-shot) power scaling of the 5th and 7th harmonics
from amorphous silicon generated by BSV (a,b) and coherent radiation (c,d).
For coherent pumping, we observe non-perturbative power scaling for both the
5th and the 7th harmonics: $N_{5\omega}\propto
N_{\omega}^{2.8},N_{7\omega}\propto N_{\omega}^{2.7}$. These results point to
the non-perturbative regime of the 5th and the 7th harmonic generation in
a-Si. However, in the case of BSV the scaling is different. For the 5th
harmonic, the scaling with the power is perturbative, $N_{5\omega}\propto
N_{\omega}^{5}$, up to the saturation regime above $\sim 2$ TW cm-2. This can
be explained by the superbunched photon statistics of BSV, which leads to the
enhancement of $n$-photon processes by a factor of $g_{BSV}^{(n)}$, compared
to coherent light [30]. In the case of the 5th harmonic generation by BSV, the
enhancement is $g_{BSV}^{(5)}=945$. Although the 7th harmonic generated by BSV
already shows the non-perturbative scaling as a sign of tunneling scenario,
its exponent is larger than in case of coherent pump, which points to a
possible competition between the 7-photon multiphoton process and the non-
peturbative regime of HHG. Another feature of harmonics generation by BSV is
the saturation of the yield, which we attribute to the strong depletion of the
valence band population. Remarkably, this is not observed in the case of
coherent pump due to the optical damage of the sample.

For the x-cut magnesium-doped LN sample, the dependence of the harmonics power
on the pump power is more complicated (Fig. 4) because this sample possesses
much more complex free-carrier dynamics, non-centrosymmetric crystalline
structure and doping. In the case of coherent pump, the below-bandgap
harmonics show perturbative power scalings, starting to saturate below 1 TW
cm-2 (Fig. 4(e,f)). The 6th and 7th harmonics demonstrate non-perturbative
power scaling within the whole range of intensities (Fig. 4(g,h)).

Figure 4: Power scaling of the 4th-7th harmonics generated in x-cut lithium
niobate by bright squeezed vacuum (a-d), and by coherent radiation (e-h)
(solids lines: mean photon number of harmonic, shaded area: photon number
variance of harmonic for particular input pump peal intensity). Dotted lines:
numerical simulation. Dashed lines: power law fit. For coherent light
excitation, the range above 3 TW cm-2 is inaccessible because of the sample
damage.

Meanwhile, the 4th and 5th harmonics generated by BSV show perturbative power
scaling for peak intensities up to 2 TW cm-2, within a range broader than
under pumping with coherent light. Similar to the a-Si case, this effect can
be attributed to the multiphoton transitions enhanced due to the BSV
superbunching. The 6th and 7th harmonics, which are above the direct bandgap,
exhibit non-perturbative power scaling within the whole range of intensities,
similar to the coherent case (Fig. 4(a-d)), with a higher exponent in the case
of the 7th harmonic. This feature explains the lower statistical enhancement
for these harmonics compared to the 5th harmonic (Fig. 2a). A power scaling
with the exponent around 3 leads to the enhancement by only a factor of
$g_{BSV}^{(3)}=15$, instead of $g_{BSV}^{(5)}=945$ in the case of the 5th
harmonic. Above 1.5 TW cm-2, the scaling exhibits two plateaus, especially
pronounced for the 4th and 7th harmonics. These features are visible only in
the case of the BSV pump, due to its possibility of ‘non-invasive’ testing. We
attribute these features to the complex structure of the sample. Numerical
simulations can reproduce our experimental results fairly well both for
coherent and BSV excitation (see Fig. 4 and Methods).

##  3 Discussion

The power scaling of harmonics from lithium niobate obtained with the coherent
pump (Fig. 4(e-h)) terminates at around 2 TW cm-2, which is close to the
sample damage threshold. However, when pumping with BSV pulses, we can extend
the range of peak intensities for harmonics generation. Figure 4(a-d) shows
the harmonic power scaling reaching a peak intensity of 10 TW cm-2, which is
inaccessible with our coherent pump due to the instant damage of the sample.
Under coherent-light irradiation, each pulse contains a definite amount of
photons, which will cause the optical damage of the sample above a certain
threshold. The BSV pump has a broad photon-number distribution, meaning that
each pulse contains an uncertain number of photons, with the mean value far
below the damage threshold. However, the large-photon-number events in the
tail of the distribution can cause HHG and probe the material without damaging
it in an instant or due to prolonged exposure. Furthermore, continuous
irradiation of the sample within temporal intervals shorter than its
relaxation time can result in optical damage to the sample. However, the BSV
photon-number distribution also implies a longer temporal interval between
intense events, thus effectively providing the sample with sufficient time to
relax to its initial state. Thus, the excitation of a material with BSV is
more gentle than with coherent light. It enables the study of solid-state
samples in extreme regimes, such as the harmonic yield saturation in amorphous
silicon (Fig. 3(a, b)), which can be related to the creation of an electron-
hole plasma.

HHG driven by BSV can be used for quantum state engineering. By adjusting the
input state of light, or selecting samples with proper band structure, free
carriers will recombine into particular quantum states of light. Figure 2(c)
shows that the harmonic radiation has a non-trivial photon-number
distribution.

BSV-driven HHG is also useful for spectroscopy. Within a single optical cycle
of BSV, the superposition of photon-number states of BSV is mapped to the
quasi-momentum of an electron; therefore, its wavefunction is distributed over
the whole energy band. The joint photon-number distribution between the BSV
and its harmonic, which results from the electron-hole recombination, contains
the information about the band structure and electron dynamics inside the
sample.

The harmonics power scaling obtained with BSV for lithium niobate (Fig.
4(a-d)) shows different features compared to silicon (Fig. 3(a,b)) which are
attributed to solid state physics. On par with conventional frequency or time-
domain spectroscopic approaches, photon-number statistics become a new
spectroscopic knob, heralding the emergence of extreme nonlinear quantum
spectroscopy.

##  4 Methods

###  4.1 Experimental setup

Figure 5(a) shows a sketch of our experimental setup for HHG in solids. For
comparison, we use two linearly polarised pump sources delivering (i) coherent
pulses and (ii) BSV pulses both centered at 1600 nm. As a pump laser for both
sources, we use amplified titanium sapphire laser (Coherent Legend) with a
central wavelength of 800 nm, 45 fs FWHM pulse duration, and repetition rate
of 1 kHz. Combining this laser with a femtosecond OPA (Light Conversion, TOPAS
Prime), we obtain coherent pulses with a duration of $70$ fs (FWHM).

Figure 5: (a) Experimental setup for HHG by coherent radiation and BSV. (b)
Photon-number distribution of BSV. The solid line is a theoretical fit, which
corresponds to single-mode BSV, the only fitting parameter being the mean
photon number (dashed line). (c) Measured average FROG trace of BSV.

In a parallel setup, after reducing the pump laser beam size to 3 mm
(1/$e^{2}$), we generate BSV in a $3$-mm-thick BBO crystal cut for type-I
collinear frequency-degenerate phase matching via high-gain parametric down-
conversion. Here, the BBO crystal is tuned out of the exact phase matching to
reduce the frequency bandwidth of the BSV, so that the BSV becomes single-mode
temporally. However, at this stage, BSV pulses still contain around 150
spatial modes. In order to make it single-mode also spatially, we reflect both
the BSV and the pump back into the same crystal with a plane silver mirror
placed at a distance of $75$ cm from the crystal. Back at the BBO, only the
BSV spatial mode with the lowest diffraction (e.g, the fundamental mode)
overlaps significantly with the pump beam and undergoes phase-sensitive
amplification, while all the higher-order modes diffract faster without being
further amplified [42]. After the amplification, the BSV pulses exhibit a
spectral bandwidth of 150 nm centered around 1.6 $\mu$m , and we separate them
from the residual 800 nm light using a shortpass dichroic mirror (Thorlabs
DMSP1180) and two longpass filters (Thorlabs FELH1000).

To characterize the temporal profile of the coherent and BSV pulses, and thus
determine the actual intensity of the light irradiating the sample, we use a
home-build device for second-harmonic frequency-resolved optical gating (SH-
FROG). The retrieved FROG traces reveal that both pulses are almost transform-
limited with a duration of 70 fs FWHM for the coherent pulses and 25 fs FWHM
for the BSV pulses (the measured FROG trace of the latter ones is shown in
5(c)).

Figure 5(b) displays the photon-number distribution of BSV measured with a
fast InGaAs photodiode (Thorlabs PDA20C2) without spatial or spectral
filtering. The data are well fitted with a theoretical single-mode
distribution [30] (black line), with the mean number of photons per pulse
$2\cdot 10^{12}$ (dashed vertical line), confirming that the BSV is single-
mode. The distribution stretches up to photon numbers of $2\cdot 10^{13}$,
corresponding to pulse energies of 9 $\mu$J.

For HHG, either the BSV or the coherent radiation is focused into a 30 $\mu$m
spot (at $1/e^{2}$) on the sample with a silver off-axis parabolic mirror
(Edmund Optics, $f=25.4$ mm) for driving HHG. The spot size has been
determined from a knife-edge measurement for both sources. We use two samples:
an x-cut 6 $\mu$m Mg:LiNbO3 crystal on a $1$ mm fused silica substrate and a 1
$\mu$m amorphous silicon layer on a $1$ mm fused silica substrate. We mount
the samples so that the substrate faces the pump pulses. Because fused silica
possesses a significantly higher bandgap, it does not contribute to the
outcome of the HHG measurement. Both the harmonics and the input light are
collimated by a UV-enhanced aluminum off-axis parabolic mirror (Thorlabs
MPD119-F01, $f=25.4$ mm) and sent to a MgF2 prism. After this, the harmonics
are focused by a UV-enhanced aluminum spherical mirror (Thorlabs
CM254-250-F01) to different positions.

We detect the 4th–7th harmonics by a fast UV-enhanced silicon avalanche
photodiode (Thorlabs APD440A2) separately by rotating the spherical mirror
within a range of 1°. Additionally, we place bandpass filters for the 4th
harmonic (Thorlabs FBH400-40), the 5th harmonic (Edmund Optics 12094,
BP325/50) and the 6th harmonic (Edmund Optics 39313, BP266/10), and also an
iris (1 mm opening) in front of the detector to reduce the contribution of
scattered light. We acquire 200,000 single-shot records of input coherent
light or BSV pulses and each harmonic synchronously.

###  4.2 Numerical simulation

To simulate and elucidate our experimental findings, we utilize the
semiconductor-Bloch equations (SBEs) [43] to describe the interaction between
BSV (or coherent) light and LiNbO3 along $\Gamma$-$Z$:

$\frac{\partial P(k,t)}{\partial
t}=[\epsilon_{c}(k)-\epsilon_{v}(k)+iE(t)\nabla_{k}]P(k,t)-(1-f_{k}^{e}-f_{v}^{h})d_{k}\cdot
E(t)-i\frac{1}{T_{2}}P(k,t),$ (1)

$\frac{\partial f_{k}^{e(h)}(k,t)}{\partial t}=-2\text{Im}[d_{k}\cdot
E(t)\cdot P_{k}^{*}]+eE(t)\nabla_{k}f_{k}^{e(h)}(k,t).$ (2)

Here, $P(k,t)$ represents a dimensionless polarization depending on the time
$t$ and the quasi-momentum $k$. $f_{k}^{e(h)}$ denotes the population of
electrons and holes in the conduction and valence bands, respectively. The
energy-momentum relation for the conduction band $\epsilon_{c}(k)$ and the
valence band $\epsilon_{v}(k)$, as well as the transition dipole moment (TDM)
$d_{k}$ are derived from density functional theory (DFT) [44], with some
modifications. Most importantly, we set the bandgap manually to 4.3 eV,
corresponding to the direct bandgap most relevant for the HHG process.

To eliminate multiple recollision interferences and maintain the pronounced
harmonic structure in our 1D simulation, we set the phenomenological
decoherence time to half of the optical cycle duration of the pump laser,
$T_{2}=0.5T_{\text{pump}}$. Extending the decoherence time to one optical
cycle $T_{\text{pump}}$ does not notably alter our results. In general, a
longer decoherence time exceeding a few cycles allows electrons to return over
longer durations than a laser cycle, leading to aperiodic photon emission,
whereas a shorter decoherence time less than a quarter cycle
$0.25T_{\text{pump}}$ may significantly suppress long trajectories.

Considering LiNbO3’s ferroelectric nature, we account for the spontaneous
polarization effect, which breaks the material’s symmetry and allow for even-
order harmonics generation (see Ref. [44] for more details). The HHG spectrum
is computed as follows:

$S_{\text{HHG}}(\omega)\sim\left|\int_{-\infty}^{\infty}[J_{\text{inter}}+iJ_{\text{intra}}]e^{i\omega
t}dt\right|^{2},$ (3)

where

$J_{\text{inter}}=\frac{d}{dt}\int d_{k}\cdot P(k,t)dk+\text{c.c.},$ (4)
$J_{\text{intra}}=\sum_{n=c,v}\int\nu_{n}(k)\cdot f_{n}(k,t)dk$ (5)

where $J_{\text{inter}}$ and $J_{\text{intra}}$ denote the interband and
intraband contributions to the HHG yield, and
$\nu_{n}(k)=\nabla_{k}\epsilon_{n}(k)$ represents the group velocity of the
electrons .

For simulating the BSV-pumped HHG spectrum underlying the enhancement plot
(Fig. 2(a), we employ the Husimi Q function for BSV light [24]:

$Q(\epsilon_{\alpha})\approx\frac{2}{\sqrt{2\pi|\bar{\epsilon}|^{2}}}\exp\left(-\frac{|\epsilon_{\alpha}|^{2}}{2|\bar{\epsilon}|^{2}}\right).$
(6)

Here, $\epsilon_{\alpha}$ is the electric field amplitude with coherent
parameter $\alpha$, and $\bar{\epsilon}$ is mean value of absolute amplitude
of the BSV field.

The BSV-pumped HHG spectrum is obtained by integrating the coherent HHG
spectra with the Q function distribution [24]:

$S^{\text{BSV}}_{\text{HHG}}(\omega,\bar{\epsilon})=\int
d\epsilon_{\alpha}Q(\epsilon_{\alpha})S^{\text{coh}}_{\text{HHG}}(\omega,\epsilon_{\alpha}).$
(7)

Acknowledgements F.T., M.C. and A.R. thank Philip St.J. Russell for supporting
the project. The authors thank Matan Even Tzur and Misha Ivanov for fruitful
discussions. A.R. thanks Patrick Cusson for the help with the FROG measurement
and Isaac Soward for the help at the first stage of the experiment. A.R.
acknowledges funding from the International Max Planck Research School for
Physics of Light. D.S. acknowledges partial support by the European Union’s
Horizon Europe Research and Innovation Programme under agreement 101070700
(project MIRAQLS). Z.C., M.B., O.C., I.K. and M.K. thank the Helen Diller
Quantum Center for partial financial support.

Author contributions M.C. and F.T. conceived the project, supervised the work
and acquired the funding. A.R., F.T. and M.C. designed the experiment. A.R.
carried out the experiment under supervision of M.C., F.T. and D.S. Z.C. and
M.B. developed the theory and carried out the numerical simulations under the
supervision of M.K., I.K. and O.C. A.R., M.C., D.S., M.K. and F.T. wrote the
article. All authors contributed to discussions and the interpretation of the
results.

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# Most Iterations of Projections Converge

Daylen K<EMAIL_ADDRESS>
The author was supported by the doctoral scholarship of the University of
Innsbruck.

Institut für Mathematik, Universität Innsbruck
Technikerstraße 13, 6020 Innsbruck, Austria

## Abstract

Consider three closed linear subspaces $C_{1},C_{2},$ and $C_{3}$ of a Hilbert
space $H$ and the orthogonal projections $P_{1},P_{2}$ and $P_{3}$ onto them.
Halperin showed that a point in $C_{1}\cap C_{2}\cap C_{3}$ can be found by
iteratively projecting any point $x_{0}\in H$ onto all the sets in a periodic
fashion. The limit point is then the projection of $x_{0}$ onto $C_{1}\cap
C_{2}\cap C_{3}$. Nevertheless, a non-periodic projection order may lead to a
non-convergent projection series, as shown by Kopecká, Müller, and
Paszkiewicz. This raises the question how many projection orders in
$\\{1,2,3\\}^{\mathbb{N}}$ are “well behaved” in the sense that they lead to a
convergent projection series. Melo, da Cruz Neto, and de Brito provided a
necessary and sufficient condition under which the projection series converges
and showed that the “well behaved” projection orders form a large subset in
the sense of having full product measure. We show that also from a topological
viewpoint the set of “well behaved” projection orders is a large subset: it
contains a dense $G_{\delta}$ subset with respect to the product topology.
Furthermore, we analyze why the proof from the measure theoretic case cannot
be directly adapted to the topological setting.

##  1 Introduction

Let $C_{1}$ and $C_{2}$ be two closed and convex subsets of a Hilbert space
with nonempty intersection. We are interested in the problem of finding some
point in the intersection $C_{1}\cap C_{2}$, which is usually is referred to
as the convex feasibility problem. Von Neumann proposed the method of
alternating projections for solving it [8]. As a more general setting, we may
consider a Hilbert space $(X,\langle\cdot,\cdot\rangle)$, an index set
$I\coloneqq\\{1,\dots,N\\}$ for $N\in\mathbb{N}$, and a finite number of
closed convex sets $(C_{n})_{n\in I}$ within $X$. By $P_{n}$ we denote the
distance minimizing projection from $X$ to the set $C_{n}$, $n\in I$, which is
well defined, as $C_{n}$ is convex and closed. Let us choose some starting
point $\xi_{0}\in X$ and a sequence $x=(x_{n})_{n\in\mathbb{N}}\in
I^{\mathbb{N}}$. We denote the space of all such sequences by $K\coloneqq
I^{\mathbb{N}}$. Iteratively we may now define a projection sequence
$(\xi_{n})_{n\in\mathbb{N}}$ by

$\xi_{n}=P_{x_{n}}(\xi_{n-1}),\quad n\in\mathbb{N}.$

We call this construction the method of alternating projections. The hoped for
result of this method is that the sequence $(\xi_{n})_{n\in\mathbb{N}}$
converges to a point in the intersection $C_{1}\cap\dots\cap C_{N}$. Following
this idea, von Neumann proved in [8] that at least there is convergence when
$X$ is a Hilbert space, $N=2$, and $C_{1}$ and $C_{2}$ are closed linear
subspaces. A simple geometric proof of von Neumann’s theorem is provided by
Kopecká and Reich in [6]. In [3] Halperin showed that for periodic $x$ von
Neumann’s result holds for any finite number of sets. Sakai was in [9] able to
extend this to quasi-periodic sequences.

######  Definition 1.1.

We call a sequence $x=(x_{n})_{n\in\mathbb{N}}\in K$ quasi-periodic if

$\exists m\in\mathbb{N}\colon\forall
k\in\mathbb{N}\colon\\{x_{k},x_{k+1},\dots,x_{k+m-1}\\}=I.$

The smallest such $m$ we call the quasi-period of $x$.

This condition imposes a uniform bound on the distance of occurrences of
indices in $x$, similar to periodicity.

Kopecká, Müller, and Paszkiewicz showed in [4, 5] that not every $x\in K$
induces a converging sequence $(\xi_{n})_{n\in\mathbb{N}}$. They constructed a
counterexample in an infinite-dimensional Hilbert space of three closed linear
subspaces such that for any $0\neq\xi_{0}\in X$ there is a sequence $x\in K$
where the method of alternating projections does not strongly converge. This
motivates the following question:

> How large is the set of sequences $x\in K$ for which
> $(\xi_{n})_{n\in\mathbb{N}}$ is strongly convergent?

Melo, da Cruz Neto, and de Brito studied this question within the context of
measure. They studied $K$ as a probability space $(K,\mathbb{P})$ by using the
product measure of $\mathbb{P}_{I}$ over the index set $\mathbb{N}$, where
$\mathbb{P}_{I}(\\{1\\})=\dots=\mathbb{P}_{I}(\\{N\\})=\frac{1}{N}$. In
Proposition 4.3 of [7] they showed that the up until now considered sets of
periodic and quasi-periodic sequences are null sets with respect to
$\mathbb{P}$. Furthermore, they introduced a more general notion than quasi-
periodic sequences: the notion of quasi-normal sequences.

######  Definition 1.2 (Definition 4.2 in [7]).

We call a sequence $(x_{n})_{n\in\mathbb{N}}\in K$ quasi-normal if there
exists an $L\in\mathbb{N}$ and a sequence of disjoint blocks
$(\mathcal{R}_{k})_{k\in\mathbb{N}}$ of consecutive elements of
$(x_{n})_{n\in\mathbb{N}}$ with $L$ terms, where each block $\mathcal{R}_{k}$
contains every element of $I$ so that there exists a function
$f\colon\mathbb{N}\to(0,\infty)$ with $\lim_{r_{k}\to\infty}f(r_{k})=\infty$
such that

$\sum_{k\in\mathbb{N}}\frac{1}{r_{k}\cdot f(r_{k})}=\infty,$

where $x_{r_{k}}$ is the first element of the block $\mathcal{R}_{k}$ and
$(r_{k})_{k\in\mathbb{N}}$ is an increasing sequence. We denote the set of
quasi-normal sequences by $\mathcal{N}$.

This type of sequences is interesting, as they guarantee strong convergence of
the projection sequence $(\xi_{n})_{n\in\mathbb{N}}$ if the sequence
$(\xi_{n})_{n\in\mathbb{N}}$ has at least one accumulation point; see Theorem
4.1 of [7]. In fact, the statement there even works for Hadamard spaces.
Additionally, Proposition 4.2 of [7] states that quasi-normal sequences form a
full measure subset of $K$.

In this paper, we consider topological and metric notions of large and small
subsets instead of measure theoretic versions. More precisely, we consider the
question whether the set of sequences $x\in K$ leading to a strongly
convergent projection sequence $(\xi_{n})_{n\in\mathbb{N}}$ is a large subset
in a topological and metric sense and, in particular, if the quasi-normal
sequences already form such a large subset. Specifically, we are interested in
the topological notions of meager and dense $G_{\delta}$ subsets and the
stronger notions of $\sigma$-porous and co-$\sigma$-porous subsets as a metric
notion of small and large subsets, respectively; see e.g. [11].

######  Definition 1.3 (($\sigma$-)porous subset).

A subset $A$ of a metric space $(X,d)$ is called porous at $x\in A$ if there
are $\varepsilon_{0}>0$ and $\alpha>0$ such that for every
$\varepsilon\in(0,\varepsilon_{0})$ there is a point $y\in X\setminus A$ with
$\rho(x,y)<\varepsilon$ and $B(y,\alpha\varepsilon)\cap A=\emptyset$ or, put
differently,

$A\textnormal{ porous at }x:\Leftrightarrow$
$\exists\varepsilon_{0}>0:\exists\alpha>0:\forall\varepsilon\in(0,\varepsilon_{0}):\exists
y\in X\setminus A:\rho(x,y)<\varepsilon\wedge B(y,\alpha\varepsilon)\cap
A=\emptyset.$

The set $A$ is called porous if it is porous at all its points. A subset of
$X$ is called $\boldsymbol{\sigma}$-porous if it is a countable union of
porous sets. We call a set co-porous or co-$\boldsymbol{\sigma}$-porous if its
complement is porous or $\sigma$-porous, respectively.

For $N\in\mathbb{N}$ we equip $I=\\{1,\dots,N\\}$ with the discrete topology
and $K\coloneqq I^{\mathbb{N}}$ with the product topology. It is well known,
that the topology on $K$ is induced by the complete metric

$\displaystyle d(x,y)\coloneqq\max\\{2^{-j}d_{0}(x_{j},y_{j})\colon
j\in\mathbb{N}\\},$

where $d_{0}$ denotes the discrete metric on $I$. Note that for $x\in K$ and
$j\in\mathbb{N}$ we have

$B(x,2^{-j})=\\{y\in K\colon y_{1}=x_{1}\wedge\dots\wedge y_{j}=x_{j}\\}.$ (1)

Using these topological notions of large and small subsets, one would hope for
an analogous result to that of Melo, da Cruz Neto, and de Brito stating that
almost all projection orders induce a strongly convergent projection series,
or that the quasi-normal sequences $\mathcal{N}$ form a large subset. In the
measure theoretic case this was shown by defining a stronger condition
resulting in a subset $\mathcal{N}_{0}$ of the quasi-normal sequences and
showing that already this set has full measure. The subset $\mathcal{N}_{0}$
is defined as the set of all $x\in K$ satisfying the assumption of the
following proposition.

######  Proposition 1.4 (Proposition 4.1 in [7]).

Let $(x_{n})_{n\in\mathbb{N}}\in K$. Suppose that there exists an
$L\in\mathbb{N}$, a sequence of disjoint blocks
$(\mathcal{R}_{k})_{k\in\mathbb{N}}$ of consecutive elements of
$(x_{n})_{n\in\mathbb{N}}$ with $L$ terms, where each block $\mathcal{R}_{k}$
has every element of $I$. For each $k\in\mathbb{N}$ let $\mathcal{S}_{k}$ be
the block formed by the elements between $\mathcal{R}_{k-1}$ and
$\mathcal{R}_{k}$, which may eventually be empty. Thus, the sequence
$(x_{n})_{n\in\mathbb{N}}$ can be seen as follows:

$\mathcal{S}_{1}\mathcal{R}_{1}\mathcal{S}_{2}\mathcal{R}_{2}\dots\mathcal{R}_{k-1}\mathcal{S}_{k}\mathcal{R}_{k}\dots$

Let $|\mathcal{S}_{k}|$ be the number of elements of this block and $c>0$ a
constant. If for all $k\in\mathbb{N}$ we have

$\sum_{i=1}^{k}|\mathcal{S}_{i}|\leq ck,$

then the sequence $(x_{n})_{n\in\mathbb{N}}$ is quasi-normal.

However, from a topological point of view, as the author showed in [10], the
equivalent statement to the measure theoretic result of $\mathcal{N}_{0}$
being large does not hold, in fact, the set $\mathcal{N}_{0}$ turns out to be
meager. Apparently, the strategy of Melo, da Cruz Neto, and de Brito cannot be
transferred to the topological setting.

The current work instead shows directly that the set of quasi-normal sequences
is a large subset and analyzes exactly which aspects of $\mathcal{N}_{0}$ make
it a small subset. Proving co-$\sigma$-porosity of $\mathcal{N}$ could however
not be achieved. Instead we were able to prove a similar result using a weaker
version of porosity, namely the notion of $\phi$-lower porosity where the size
of the “holes” is still controlled, but it no longer needs to be linear with
respect to the distance. This weaker notion of porosity can be found for
example in Definition 2.1 in [11].

######  Definition 1.5 (($\sigma$-)$\phi$-porous set).

Let $(X,\rho)$ be a metric space, $A\subseteq X$, $x\in A$, $\eta>0$ and
$\phi\colon(0,\eta)\to(0,\infty)$ be an increasing function. Given $r>0$, we
define

$\gamma(x,r,A)\coloneqq\sup\\{s>0\colon\exists y\in X\colon B(y,s)\subseteq
B(x,r)\setminus A\\},$

where we use the convention that $\sup\emptyset=-\infty$. The set $A$ is
called $\phi$-lower porous at $x$ if

$\liminf_{r\to 0}\frac{\phi(\gamma(x,r,A))}{r}>0.$

A set $A$ is called $\phi$-lower porous if it is $\phi$-lower porous at all
points $x\in A$. Since we do not use the notion of $\phi$-upper porosity, we
refer to $\phi$-lower porous sets simply as $\phi$-porous sets. Again, we call
a set $\sigma$-$\phi$-porous, if it is a countable union of $\phi$-porous
sets. Also, we call a set co-$\psi$-porous or co-$\sigma$-$\phi$-porous if it
is the complement of a $\phi$-porous or $\sigma$-$\phi$-porous set,
respectively.

We will heavily use the following characterization of $\phi$-porous sets due
to M. Dymond.

######  Lemma 1.6 (Lemma 2.2 in [2]).

Let $(X,\rho)$ be a metric space without isolated points, $A\subseteq X$,
$x\in A$, $\eta>0$ and

$\phi\colon(0,\eta)\to(0,\infty)$

be a strictly increasing, concave function with $\lim_{t\to 0}\phi(t)=0$. Then
$A$ is $\phi$-porous at $x$ if and only if there are an $\varepsilon_{0}>0$
and an $\alpha\in(0,1)$ such that for every
$\varepsilon\in(0,\varepsilon_{0})$ there is a $y\in X\setminus A$ such that
$0<\rho(x,y)\leq\varepsilon$ and $B(y,\phi^{-1}(\alpha\varepsilon))\cap
A=\emptyset$, or put differently

$A\,\phi\textnormal{-porous at }x:\Leftrightarrow$
$\exists\varepsilon_{0}>0:\exists\alpha\in(0,1):\forall\varepsilon\in(0,\varepsilon_{0}):\exists
y\in X\setminus A:\leavevmode\hbox to13.4pt{\vbox
to13.4pt{\pgfpicture\makeatletter\hbox{\hskip 6.7pt\lower-6.7pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.5pt}{0.0pt}\pgfsys@curveto{6.5pt}{3.58989pt}{3.58989pt}{6.5pt}{0.0pt}{6.5pt}\pgfsys@curveto{-3.58989pt}{6.5pt}{-6.5pt}{3.58989pt}{-6.5pt}{0.0pt}\pgfsys@curveto{-6.5pt}{-3.58989pt}{-3.58989pt}{-6.5pt}{0.0pt}{-6.5pt}\pgfsys@curveto{3.58989pt}{-6.5pt}{6.5pt}{-3.58989pt}{6.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.33334pt}{-2.77779pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\wedge$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\begin{cases}\rho(x,y)\leq\varepsilon\\\
B(y,\phi^{-1}(\alpha\varepsilon))\cap A=\emptyset\end{cases}$.

This relatively new characterization not only proved to be useful in [2] but
also in [1].

Using this characterization we were able to prove our main result, Theorem
2.8, which states that the set of quasi-normal sequences is contained in a
co-$\sigma$-$\phi$-porous subset of $K$ with respect to a metric generating
the product topology on $K$.

Furthermore, in Section 3 we are able to exactly pinpoint why
$\mathcal{N}_{0}$ is a meager subset and formulate similar, more general
subsets that are residual. Lastly, in Section 4 we show that almost all
sequences $x\in K$ contain all indices in $I$ infinitely often.

##  2 Main Result

In order to prove our main result, Theorem 2.8, we will first simplify the
rather complicated Definition 1.2 of the quasi-normal sequences $\mathcal{N}$.
Following some preparation, we are able to state Lemma 2.4, which gives us a
characterization of quasi-normal sequences that eliminates the use of the
function $f$ of the quasi-normal definition and also uses a constructive way
of defining a partition instead of merely demanding the existence of a certain
partition.

###  2.1 Simplifying the Quasi-Normal Definition

######  Proposition 2.1.

A sequence $(x_{n})_{n\in\mathbb{N}}\in K$ is quasi-normal if and only if
there exists an $L\in\mathbb{N}$ and a sequence of disjoint blocks
$(\mathcal{R}_{k})_{k\in\mathbb{N}}$ of consecutive elements of
$(x_{n})_{n\in\mathbb{N}}$ with $L$ terms, where each block $\mathcal{R}_{k}$
contains every element of $I$ such that

$\sum_{k\in\mathbb{N}}\frac{1}{r_{k}}=\infty,$

where $x_{r_{k}}$ is the first element of the block $\mathcal{R}_{k}$ and
$(r_{k})_{k\in\mathbb{N}}$ is an increasing sequence.

###### Proof.

We first show that every quasi-normal sequence satisfies the conditions of the
proposition. Let $x$ be quasi-normal and let $(r_{k})_{k\in\mathbb{N}}$ and
$f$ be the respective quantities from the definition of quasi-normality. In
particular, we have that $\lim_{t\to\infty}f(t)=\infty$ and

$\sum_{k\in\mathbb{N}}\frac{1}{r_{k}\cdot f(r_{k})}=\infty.$

Now, since $f$ tends towards infinity, we may define the smallest index
$k_{f}$ after which all $f(r_{j})$ for $j\geq k_{f}$ are greater or equal to
$1$,

$k_{f}=\min\\{k\in\mathbb{N}\colon\forall j\geq k\colon f(r_{j})\geq 1\\}.$

Then we have

$\displaystyle\infty=\sum_{k\in\mathbb{N}}\frac{1}{r_{k}f(r_{k})}=\sum_{k=1}^{k_{f}-1}\frac{1}{r_{k}f(r_{k})}+\sum_{k=k_{f}}^{\infty}\frac{1}{r_{k}f(r_{k})}\leq\sum_{k=1}^{k_{f}-1}\frac{1}{r_{k}f(r_{k})}+\sum_{k=k_{f}}^{\infty}\frac{1}{r_{k}},$

and in particular that $\sum_{k\in\mathbb{N}}\frac{1}{r_{k}}=\infty$.

For the other direction let $x\in K$ and $(\mathcal{R}_{k})_{k\in\mathbb{N}}$
be a partition with starting indices $(r_{k})_{k\in\mathbb{N}}$ satisfying the
condition of the proposition. Since
$\sum_{k\in\mathbb{N}}\frac{1}{r_{k}}=\infty$, we can choose a sequence
$(k_{l})_{l\in\mathbb{N}}$ of positive integers such that $k_{1}=1$ and

$\forall l\in\mathbb{N}\colon\sum_{k=k_{l}}^{k_{l+1}-1}\frac{1}{r_{k}}\geq l.$

Note that this is equivalent to

$\forall l\in\mathbb{N}\colon\sum_{k=k_{l}}^{k_{l+1}-1}\frac{1}{r_{k}\cdot
l}\geq 1.$

Now, define $f$ by $f(r_{k})=l$ for $k_{l}\leq k<k_{l+1}$ and observe that it
is divergent. We then have

$\sum_{k\in\mathbb{N}}\frac{1}{r_{k}f(r_{k})}=\sum_{l\in\mathbb{N}}\sum_{k=k_{l}}^{k_{l+1}-1}\frac{1}{r_{k}f(r_{k})}\geq\sum_{l\in\mathbb{N}}1=\infty.$

Hence, $x$ is quasi-normal. ∎

Now that we have simplified the definition of the quasi-normal sequences, we
are able to define a unique partition type that will maximize the sum in the
previous proposition.

######  Definition 2.2 (Greedy $L$-partition).

Let $L\in\mathbb{N}$ with $L\geq|I|$ be given. Furthermore, let $x\in K$. Let

$r_{1}\coloneqq\min\\{r\in\mathbb{N}\colon\\{x_{r},x_{r+1},\dots,x_{r+L-1}\\}=I\\},$

and for $k\in\mathbb{N}$, $k\geq 2$ let

$r_{k}\coloneqq\min\\{r\in\mathbb{N}\colon
r>r_{k-1}+L-1\wedge\\{x_{r},x_{r+1},\dots,x_{r+L-1}\\}=I\\}$

provided that these minima exist. This results in the blocks

$\mathcal{R}_{k}=(x_{r_{k}},\dots,x_{r_{k+L-1}}),\quad k\in\mathbb{N}.$

We call this partition of $x$ the greedy $L$-partition. Note that by
definition every block $\mathcal{R}_{k}$ contains all elements of $I$. If any
of the above minima does not exist, we say that the greedy $L$-partition of
$x$ does not exist. In this case, let $\tilde{k}$ be the largest
$k\in\mathbb{N}$ for which $r_{k}$ is well defined. We then say that the
greedy $L$-partition of $x$ exists up until at most block $\tilde{k}$. Since
for given $x$ and $L$ the partition is fully determined by the starting
indices $(r_{k})_{k\in\mathbb{N}}$ of $(\mathcal{R}_{k})_{k\in\mathbb{N}}$, we
will interchangeably instead of the partition itself refer to
$(r_{k})_{k\in\mathbb{N}}$ as the greedy $L$-partition of $x$.

The next proposition shows in what sense the greedy $L$-partition is greedy.

######  Proposition 2.3.

Let $L\in\mathbb{N}$ with $L\geq N$ be given. Furthermore, let $x\in K$ and
$(r_{k})_{k\in\mathbb{N}}$ be the indices of the greedy $L$-partition of $x$,
and $(\tilde{r}_{k})_{k\in\mathbb{N}}$ a partition of $x$ different from the
former with disjoint blocks of length $L$ in which all elements of $I$ occur.
Then we have

$\sum_{k\in\mathbb{N}}\frac{1}{r_{k}}\geq\sum_{k\in\mathbb{N}}\frac{1}{\tilde{r}_{k}}.$

If the right hand side converges we even get a strict inequality.

###### Proof.

As all $r_{k}$ are chosen minimally, we have that $r_{k}\leq\tilde{r}_{k}$. To
further elaborate on this we use induction: Consider $r_{1}$ and
$\tilde{r}_{1}$. Here it is clear that $r_{1}\leq\tilde{r_{1}}$, since $r_{1}$
marks the beginning of the first possible block of length $L$ containing all
elements of $I$. Now assume that for $n\leq m\in\mathbb{N}$ we have
$r_{n}\leq\tilde{r}_{n}$. Then $r_{m}$ marks the beginning of the $m$-th block
of the greedy $L$-partition. This block extends up to index $r_{m+L-1}$. Now
since $r_{m}\leq\tilde{r}_{m}$ we have that the $m$-th block in the partition
$(\tilde{r}_{k})_{k\in\mathbb{N}}$ extends up to index $\tilde{r}_{m}+L-1$ and
thus the same or farther than the $m$-th block of $(r_{k})_{k\in\mathbb{M}}$.
Since in the greedy $L$-partition $(r_{k})_{k\in\mathbb{N}}$ the $(m+1)$-th
block is chosen to begin after index $r_{m}+L-1$ with minimal distance, it may
already start in a part that the $m$-th block of
$(\tilde{r}_{k})_{k\in\mathbb{N}}$ still occupies. If this is the case, then
clearly $r_{m+1}\leq\tilde{r}_{m+1}$. In the case that the $(m+1)$-th block of
$(r_{k})_{k\in\mathbb{N}}$ begins after the $m$-th block of
$(\tilde{r}_{k})_{k\in\mathbb{N}}$, then by the greedy nature of
$(r_{k})_{k\in\mathbb{N}}$ we again have that $r_{m+1}\leq\tilde{r}_{m+1}$. It
follows that indeed $r_{k}\leq\tilde{r}_{k}$ for all $k\in\mathbb{N}$.

Since $(r_{k})_{k\in\mathbb{N}}\neq(\tilde{r}_{k})_{k\in\mathbb{N}}$, we know
that for at least one $k\in\mathbb{N}$ we have that $r_{k}<\tilde{r}_{k}$.
This directly implies the result. ∎

The previous proposition again allows us to reformulate the definition of
quasi-normal sequences.

######  Lemma 2.4.

A sequence $x=(x_{n})_{n\in\mathbb{N}}\in K$ is quasi-normal if and only if
there exists an $L\in\mathbb{N}$ such that the greedy $L$-partition
$(r_{k})_{k\in\mathbb{N}}$ of $x$ exists and

$\sum_{k\in\mathbb{N}}\frac{1}{r_{k}}=\infty.$

###  2.2 Proving the Main Theorem

We are going to show that the set of quasi-normal sequences $\mathcal{N}$
contains a co-$\sigma$-$\phi$-porous subset, that is, that the complement
$K\setminus\mathcal{N}$ is contained in a $\sigma$-$\phi$-porous subset. For
this let us for $L\in\mathbb{N}$ with $L\geq N$ and $M\in\mathbb{N}$ define
the set

$\mathcal{A}_{L,M}\coloneqq\left\\{x\in K\colon\textnormal{greedy
$L$-partition $(r_{k})_{k\in\mathbb{N}}$ of $x$ exists
and}\sum_{k\in\mathbb{N}}\frac{1}{r_{k}}<\frac{1}{L}\log(M)\right\\}$

and for $k\in\mathbb{N}$ the set

$\mathcal{B}_{L,k}\coloneqq\\{x\in K\colon\textnormal{greedy $L$-partition of
$x$ exists up until at most block $k$}\\}.$

A sequence is not quasi-normal if and only if for all $L\geq N$ either the
greedy $L$-partition $(r_{k})_{k\in\mathbb{N}}$ exists and produces a sum
$\sum_{k=1}^{\infty}\frac{1}{r_{k}}$ that is finite or if the greedy
$L$-partition does not exist. Therefore, we may write

$K\setminus\mathcal{N}=\bigcap_{\begin{subarray}{c}L\in\mathbb{N}\\\ L\geq
N\end{subarray}}\left(\bigcup_{M\in\mathbb{N}}\mathcal{A}_{L,M}\cup\bigcup_{k\in\mathbb{N}}\mathcal{B}_{L,k}\right).$
(2)

We observe that $K\setminus\mathcal{N}$ is contained in a
$\sigma$-$\phi$-porous subset if already the set

$\bigcup_{M\in\mathbb{N}}\mathcal{A}_{N,M}\cup\bigcup_{k\in\mathbb{N}}\mathcal{B}_{N,k}$

is $\sigma$-$\phi$-porous or rather if
$\mathcal{A}_{M}\coloneqq\mathcal{A}_{N,M}$ and
$\mathcal{B}_{k}\coloneqq\mathcal{B}_{N,k}$ are $\phi$-porous for all
$M,k\in\mathbb{N}$.

Let us choose $\phi\colon(0,1)\to(0,1)$ implicitly by defining its inverse

$\phi^{-1}(t)=t^{\frac{1}{t}}.$

Writing $\phi$ explicitly is unnecessarily difficult, which is why we choose
to only give $\phi$ implicitly. Note that $\phi^{-1}$ is continuous and
strictly increasing on $(0,1)$ since for every $t\in(0,1)$ we have that

$\mathrm{D}\phi^{-1}(t)=-t^{\frac{1}{t}-2}(\log(t)-1)>0.$

Also we note that $\lim_{t\to 0}\phi^{-1}(t)=0$ and $\phi^{-1}(1)=1$. We
conclude that $\phi^{-1}$ is bijective on $(0,1)$, and hence, that $\phi^{-1}$
indeed implicitly defines a function $\phi\colon(0,1)\to(0,1)$. Since for
$t<\frac{1}{3}$ we have

$\mathrm{D}^{2}\phi^{-1}(t)=t^{\frac{1}{t}-4}\left(-3t+\log^{2}(t)+2(t-1)\log(t)+1\right)>0,$

we know that $\phi^{-1}$ is convex on $(0,\frac{1}{3})$, and hence, that
$\phi$ is concave. Furthermore, since $\phi^{-1}$ is convex and since
$\phi^{-1}(t)<t$ for $t<1$ we conclude that also $\phi$ is strictly
increasing. Lastly, we note that since $\lim_{t\to 0}\phi^{-1}(t)=0$ we have
$\lim_{s\to 0}\phi(s)=0$. These observations show that $\phi$ is a function
suitable for $\phi$-porosity.

######  Lemma 2.5.

For every $M\in\mathbb{N}$ the set $\mathcal{A}_{M}$ is $\phi$-porous.

###### Proof.

Let $x\in\mathcal{A}_{M}$ and let us choose

$\varepsilon_{0}\coloneqq\min\left\\{\frac{1}{8M},2^{-3N-1}\right\\},\quad\alpha=1,\quad\text{and}\quad\varepsilon\in(0,\varepsilon_{0}).$

Choose $j=\lceil-\log_{2}(\varepsilon)\rceil$ and note that then
$2^{-j}<\varepsilon\leq 2^{-j+1}$. We now define $y\in K$ by

$y\coloneqq(x_{1},x_{2},\dots,x_{j},\underbrace{1,\dots,1}_{N},1,2,\dots,N,1,2,\dots,N,\dots).$

By (1) we have $d(x,y)<\varepsilon$. Observe that the greedy $N$-partition
$(r^{y}_{k})_{k\in\mathbb{N}}$ of $y$ exists and that
$\sum_{k\in\mathbb{N}}\frac{1}{r^{y}_{k}}=\infty$, since somewhere after
$x_{j}$ all blocks are densely packed without spaces in between. Hence,
$y\notin\mathcal{A}_{M}$. Now set

$m\coloneqq\left\lfloor-\log_{2}(\varepsilon)\frac{1}{\varepsilon}\right\rfloor.$

Note that we have that

$\phi^{-1}(\alpha\varepsilon)=\varepsilon^{\frac{1}{\varepsilon}}=2^{\log_{2}(\varepsilon)\frac{1}{\varepsilon}}\leq
2^{-\left\lfloor-\log_{2}(\varepsilon)\frac{1}{\varepsilon}\right\rfloor}=2^{-m},$

and therefore, that $B(y,\phi^{-1}(\alpha\varepsilon))\subseteq B(y,2^{-m})$.
Now assume that $z\in B(y,\phi^{-1}(\alpha\varepsilon))$. If the greedy
$N$-partition of $z$ does not exist we immediately conclude that
$z\notin\mathcal{A}_{M}$. In the following we assume that the greedy
$N$-partition $(r^{z}_{k})_{k\in\mathbb{N}}$ of $z$ does exist. Since

$\displaystyle m$
$\displaystyle\geq-\log_{2}(\varepsilon)\frac{1}{2\varepsilon}\geq-\log_{2}(\varepsilon)\frac{1}{2^{-3N}}\geq\lceil-\log_{2}(\varepsilon)\rceil
2^{3N-1}$
$\displaystyle>\lceil-\log_{2}(\varepsilon)\rceil(1+2N)\geq\lceil-\log_{2}(\varepsilon)\rceil+2N=j+2N.$

and due to (1) we have that the leading entries of $z$ are

$(z_{i})_{i=1}^{m}=(\underbrace{x_{1},\dots,x_{j},\underbrace{1,\dots,1}_{N},\underbrace{1,2,\dots,N,1,2,\dots,N}_{\text{at
least one full block}},\dots}_{m\text{ entries}}).$

Next we show that $\sum_{k\in\mathbb{N}}\frac{1}{r^{z}_{k}}$ is too large for
$z$ to belong to $\mathcal{A}_{M}$ by estimating a lower bound on it using
only the $r^{z}_{k}$ corresponding to blocks that fit entirely in the section
of consecutive blocks $1,2,\dots,N$. By construction, we know that a block
must begin at index $j+N+1$ and all other such blocks follow without spaces
immediately after. For the sum we have

$\displaystyle\sum_{k\in\mathbb{N}}\frac{1}{r^{z}_{k}}$
$\displaystyle\geq\sum_{\begin{subarray}{c}k\in\mathbb{N}\\\ j+N+1\leq
r^{z}_{k}\\\ r^{z}_{k}+N-1\leq
m\end{subarray}}\frac{1}{r^{z}_{k}}=\sum_{k=0}^{\left\lfloor\frac{m-(j+N)}{N}\right\rfloor-1}\frac{1}{j+N+1+kN}=\sum_{k=0}^{\left\lfloor\frac{m-(j+N)}{N}\right\rfloor-1}\frac{1}{j+1+(k+1)N}$
$\displaystyle>\int_{t=0}^{\left\lfloor\frac{m-(j+N)}{N}\right\rfloor-1}\frac{1}{j+1+(t+1)N}=\frac{1}{N}\log\left(1+\frac{\left(\left\lfloor\frac{m-(j+N)}{N}\right\rfloor-1\right)N}{1+j+N}\right)$
$\displaystyle\geq\frac{1}{N}\log\left(1+\frac{\left(\frac{m-(j+N)}{N}-2\right)N}{1+j+N}\right)=\frac{1}{N}\log\left(1+\frac{m-j-3N}{1+j+N}\right)$
$\displaystyle\geq\frac{1}{N}\log\left(1+\frac{m-j-3N}{1+j+3N}\right)=\frac{1}{N}\log\left(\frac{m}{1+j+3N}\right)$
$\displaystyle\geq\frac{1}{N}\log\left(\frac{m}{2j}\right)=\frac{1}{N}\log\left(\frac{\left\lfloor-\log_{2}(\varepsilon)\frac{1}{\varepsilon}\right\rfloor}{2\lceil-\log_{2}(\varepsilon)\rceil}\right)\geq\frac{1}{N}\log\left(\frac{-\log_{2}(\varepsilon)\frac{1}{2\varepsilon}}{4(-\log_{2}(\varepsilon))}\right)$
$\displaystyle=\frac{1}{N}\log\left(\frac{1}{8\varepsilon}\right)>\frac{1}{N}\log(M),$

which implies that $z\notin\mathcal{A}_{M}$. We conclude that
$\mathcal{A}_{M}$ is $\phi$-porous. ∎

######  Lemma 2.6.

For every $k\in\mathbb{N}$ the set $\mathcal{B}_{k}$ is porous.

###### Proof.

Let $x\in\mathcal{B}_{k}$

$\varepsilon_{0}=1,\quad\alpha=2^{-N-(k+1)N},\quad\text{and}\quad\varepsilon\in(0,\varepsilon_{0}).$

Choose $j=\lceil-\log_{2}(\varepsilon)\rceil$. We now define $y\in K$ by

$y\coloneqq(x_{1},\dots,x_{j},\underbrace{1,\dots,1}_{N},1,2,\dots,N,1,2,\dots,N,\dots).$

Note that the greedy $N$-partition
$HindawiPublishingCorporation(r^{y}_{k})_{k\in\mathbb{N}}$ of $y$ exists, and
hence, $y\notin\mathcal{B}_{L,k}$. Then for every $z\in
B(y,\alpha\varepsilon)\subseteq B(y,2^{-j-N-(k+1)N})$ we have that $z$ has the
leading entries

$(z_{i})_{i=1}^{j+N+(k+1)N}=(x_{1},\dots,x_{j},\underbrace{1,\dots,1}_{N},\underbrace{\overbrace{1,2,\dots,N},\dots,\overbrace{1,2,\dots,N}}_{k+1\textnormal{
many blocks }}).$

No block in the greedy $N$-partition of $z$ can begin in the section of
consecutive ones, since then $N$ cannot occur in such a block. It follows that
the blocks in the greedy $N$-partition after the section of consecutive ones
must be chosen in the way they are marked above. Therefore, the greedy
$N$-partition of $z$ exists at least up to block $k+1$, and hence,
$z\notin\mathcal{B}_{k}$. We conclude that the set $\mathcal{B}_{k}$ is
porous. ∎

######  Remark 2.7.

A similar proof can be carried out for all sets $\mathcal{A}_{L,M}$ and
$\mathcal{B}_{L,k}$, but as mentioned above, we only need the proved case.

######  Theorem 2.8.

The set $\mathcal{N}$ contains a co-$\sigma$-$\phi$-porous subset.

###### Proof.

By Equation (2) we see that $K\setminus\mathcal{N}$ is contained in a subset
of all $\mathcal{A}_{M}$ and $\mathcal{B}_{k}$ for $M,k\in\mathbb{N}$, which
are $\phi$-porous and porous, respectively. This is the case by Lemma 2.5 and
Lemma 2.6. ∎

######  Corollary 2.9.

The set of quasi-normal sequences $\mathcal{N}$ contains a dense $G_{\delta}$
subset.

##  3 Some Extensions

In this section we take a closer look at why the construction of the set
$\mathcal{N}_{0}$ gives us a meager set.

###  3.1 A more General Subset of Quasinormal Sequences

Let us first elaborate why the sequences from Proposition 1.4 were chosen,
i.e., we look at the proof of the proposition as given in [7].

###### Proof of Proposistion 1.4.

Let $x\in K$ be such that it satisfies the conditions of the proposition. We
denote by $x_{r_{k}}$ the first element of the block $\mathcal{R}_{k}$.
Observe that

$r_{k}=1+(k-1)L+\sum_{i=1}^{k}|\mathcal{S}_{i}|.$

Now consider a function $f\colon\mathbb{N}\to(0,\infty)$, where $f(r_{k})=\log
k$. We have by imposing the bound $\sum_{i=1}^{k}|\mathcal{S}_{i}|\leq ck$ for
some $c>0$ that

$\displaystyle\sum_{k=1}^{\infty}\frac{1}{r_{k}f(r_{k})}$
$\displaystyle=\sum_{k=1}^{\infty}\frac{1}{\left(1+(k-1)L+\sum_{i=1}^{k}|\mathcal{S}_{i}|\right)\cdot\log
k}$ $\displaystyle\geq\sum_{k=1}^{\infty}\frac{1}{(1+(k-1)L+ck)\cdot\log k}$
$\displaystyle\geq\sum_{k=1}^{\infty}\frac{1}{(L+c)k\cdot\log k}$
$\displaystyle=\infty,$

which implies that $x$ is quasi-normal. ∎

######  Remark 3.1.

The proof essentially uses two features: First, it chooses a single function
for $f$ and also this function depends on $x$ and the chosen partition in the
way that $f$ only depends on $k$. Second, a bound on
$\sum_{i=1}^{k}|\mathcal{S}_{i}|$ is imposed to ensure the divergence of the
entire series. The following discussion will show, that the second feature,
i.e., the bound cannot be chosen such that the resulting subset is large in a
topological sense.

In the following discussion we are going to show that no matter what bound on
$\sum_{i=1}^{k}|\mathcal{S}_{i}|$ is chosen, we still have that the set of
sequences satisfying this condition is small.

Let $(p_{c}:\mathbb{N}\to(0,\infty))_{c\in C}$ be a countable family of
increasing functions. We use the family $(p_{c})_{c\in C}$ as a replacement
for the linear bound $\sum_{i=1}^{k}|\mathcal{S}_{i}|\leq ck$, i.e., now we
prescribe

$\sum_{i=1}^{k}|\mathcal{S}_{i}|\leq p_{c}(k)$

for some $c\in C$. In order for this new bound to also produce quasi-normal
sequences, there must for every $x\in K$ exist some
$f\colon\mathbb{N}\to(0,\infty)$ with $\lim_{t\to\infty}f(t)=\infty$ such that

$\sum_{k=1}^{\infty}\frac{1}{(1+(k-1)L+p_{c}(k))\cdot f(r_{k})}=\infty,$

see the proof of Proposition 1.4. In the following discussion we adapt the
strategy from [10].

######  Definition 3.2.

Let $L\in\mathbb{N}$ and $c\in C$. We then define

$\displaystyle P_{L,c}(x)\leftrightarrow/$ $\displaystyle\exists\textnormal{ a
representation
}\mathcal{S}_{1}\mathcal{R}_{1}\mathcal{S}_{2}\mathcal{R}_{2}\dots=x\textnormal{
such that }$ $\displaystyle\leavevmode\hbox to13.4pt{\vbox
to13.4pt{\pgfpicture\makeatletter\hbox{\hskip 6.7pt\lower-6.7pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.5pt}{0.0pt}\pgfsys@curveto{6.5pt}{3.58989pt}{3.58989pt}{6.5pt}{0.0pt}{6.5pt}\pgfsys@curveto{-3.58989pt}{6.5pt}{-6.5pt}{3.58989pt}{-6.5pt}{0.0pt}\pgfsys@curveto{-6.5pt}{-3.58989pt}{-3.58989pt}{-6.5pt}{0.0pt}{-6.5pt}\pgfsys@curveto{3.58989pt}{-6.5pt}{6.5pt}{-3.58989pt}{6.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.33334pt}{-2.77779pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\wedge$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\begin{cases}\forall
k\in\mathbb{N}\colon|\mathcal{R}_{k}|=L\\\ \forall
k\in\mathbb{N}\colon\mathcal{R}_{k}\textnormal{ contains all elements of }I\\\
\forall k\in\mathbb{N}\colon\sum_{i=1}^{k}|\mathcal{S}_{i}(x)|\leq
p_{c}(k)\end{cases}$

and

$P(x)\leftrightarrow/\exists L\in\mathbb{N}\colon\exists c>0\colon
P_{L,c}(x).$

Let $\mathcal{N}_{0}$ denote the set of all sequences $x\in K$ satisfying the
assertion of Proposition 1.4, that is,

$\mathcal{N}_{0}\coloneqq\\{x\in K\colon P(x)\\}.$

. Let us for $L\in\mathbb{N}$ and $c>0$ define the set

$\displaystyle\mathcal{N}_{L,c}\coloneqq\\{x\in K\colon P_{L,c}(x)\\}.$

Similarly to the case of the linear bound let us set

$\mathcal{N}_{0}\coloneqq\bigcup_{L\in\mathbb{N}}\bigcup_{c\in
C}\mathcal{N}_{L,c}.$

######  Theorem 3.3.

For every $L\in\mathbb{N}$ and $c\in C$ the set $\mathcal{N}_{L,c}$ is nowhere
dense.

The proof of this theorem is provided in Section 3.2.

######  Corollary 3.4.

The set $\mathcal{N}_{0}$ is meager.

This shows that no matter what bound on $\sum_{i=1}^{\infty}|\mathcal{S}_{i}|$
is chosen, and no matter what $f$ is chosen, the strategy of Melo, da Cruz
Neto, and de Brito does not work in a topological context.

###  3.2 Proof of Theorem 3.3

######  Lemma 3.5.

Let $L\in\mathbb{N},c>0,x\in\mathcal{N}_{L,c}$, and $n\in\mathbb{N}$.
Furthermore, let

$y\coloneqq(x_{1},\dots,x_{n},1,1,\dots)$

and

$m\coloneqq p_{c}\left(L\left\lfloor\frac{n}{L}\right\rfloor+1\right)+2L+2.$

Then

$B(y,2^{-n-m})\cap\mathcal{N}_{L,c}=\emptyset.$

###### Proof.

Let us fix some arbitrary $z\in B(y,2^{-n-m})$ and assume that
$z\in\mathcal{N}_{L,c}$. Note that the ball may be written as

$B(z,2^{-n-m})=\\{x_{1}\\}\times\cdots\times\\{x_{n}\\}\times\underbrace{\\{1\\}\times\cdots\times\\{1\\}}_{m}\times\prod_{k=n+m+1}^{\infty}K.$

Since $z\in\mathcal{N}_{L,c}$, we can find a partition
$z=\mathcal{S}_{1}\mathcal{R}_{1}\mathcal{S}_{2}\mathcal{R}_{2}\dots$
satisfying all conditions in the definition of $\mathcal{N}_{L,c}$. Let us fix
this partition. Now, let $s_{k}$ denote the index in $z$ of the leftmost entry
of $\mathcal{S}_{k}$. By $i$ we denote the smallest index $j$ such that
$s_{j}\geq n+1$, i.e.,

$i\coloneqq\min\\{j\in\mathbb{N}\colon s_{j}\geq n+1\\}.$

This retrieves the beginning of the first block $\mathcal{S}_{i}$ that starts
in or after the section of $z$ with the consecutive ones. Since
$\mathcal{R}_{i-1}$ has to contain all elements of $I$ and has a length of $L$
it can at most extend $L-(N-1)$ places into the section of consecutive ones.
Since $m>L-(N-1)$ we conclude that in fact $i$ is the index of the leftmost
index of the first $\mathcal{S}_{k}$ to start within the section of
consecutive ones, and not after.

Next, we will find an upper bound on $i$. All $\mathcal{R}_{k}$ have the same
length $L$, but the length of the $\mathcal{S}_{k}$ is arbitrary. Hence, $i$
will be the largest if $\mathcal{S}_{1},\dots,\mathcal{S}_{i-1}=()$. For this
reason we follow that

$i\leq L\left\lfloor\frac{n}{L}\right\rfloor+1.$ (3)

We will now establish a lower bound on $|\mathcal{S}_{i}|$. The block
$\mathcal{S}_{i}$ ends one place before $\mathcal{R}_{i}$ begins. Since
$\mathcal{R}_{i}$ is of length $L$ and contains all elements of $I$, it can
extend at most $L-(N-1)$ places into the section of consecutive ones from the
right. This together with the identical condition at the beginning of the
section of $m$ consecutive ones gives us that

$|\mathcal{S}_{i}|\geq m-2(L-(N-1)).$

By the definition of $m$ and by (3) we have

$\displaystyle|\mathcal{S}_{i}|$ $\displaystyle\geq m-2(L-(N-1))$
$\displaystyle=p_{c}\left(L\left\lfloor\frac{n}{L}\right\rfloor+1\right)+2L+2-2(L-(N-1))$
$\displaystyle=p_{c}\left(L\left\lfloor\frac{n}{L}\right\rfloor+1\right)+2N$
$\displaystyle\geq p_{c}(i)+2N$ $\displaystyle>p_{c}(i).$

Hence,

$\sum_{k=1}^{i}|\mathcal{S}_{k}|\geq|\mathcal{S}_{i}|>p_{c}(i)$

and, since the partition
$\mathcal{S}_{1}\mathcal{R}_{1}\mathcal{S}_{2}\mathcal{R}_{2}\dots$ was
arbitrary, we have $z\notin\mathcal{N}_{L,c}$. Since $z\in B(y,2^{-n-m})$ was
arbitrary, we have shown the assertion. ∎

In the previously considered subset $\mathcal{N}_{0}$ of quasi-normal
functions, the function $f$ was chosen as $f(r_{k})=\log(k)$. This means that
the function $f$ also indirectly depends on a chosen partition $r$ and the
fixed $L$ the partition uses. To accommodate and also include such functions
$f$ that somehow are of the same type, but may depend on $r$, we instead of a
single function define a family $(f^{L}_{r})$ of functions as more accurately
stated in the following lemma. It states that also in the case of using some
given $f$ in the sum, the greedy $L$-partition is greedy.

######  Lemma 3.6.

Let $L\in\mathbb{N}$ with $L\geq N$ be given and let

$\mathcal{P}_{L}\coloneqq\\{(r_{k})_{k\in\mathbb{N}}\in\mathbb{N}^{\mathbb{N}}\colon\forall
k\in\mathbb{N}\colon r_{k+1}-r_{k}\geq L\\}$

denote the set of possible $L$-partitions. Furthermore, let

$f=(f^{L}_{r}\colon\mathbb{N}\to(0,\infty))_{\begin{subarray}{c}N\leq
L\in\mathbb{N}\\\ r\in\mathcal{P}_{L}\end{subarray}}$

be a parametrized family of functions that for all $L\in\mathbb{N}$ with
$L\geq N$ fulfills the following two properties:

  1. (i)

$\forall
r\in\mathcal{P}_{L}\colon\displaystyle\lim_{t\to\infty}f^{L}_{r}(t)=\infty$

  2. (ii)

$\forall r,\tilde{r}\in\mathcal{P}_{L}\colon\forall
k\in\mathbb{N}\colon(r_{k}\leq\tilde{r}_{k}\implies f^{L}_{r}(r_{k})\leq
f^{L}_{\tilde{r}}(\tilde{r}_{k}))$

Now, let for $x\in K$ and $r\coloneqq(r_{k})_{k\in\mathbb{N}}$ denote the
indices of the greedy $L$-partition of $x$, and
$\tilde{r}\coloneqq(\tilde{r}_{k})_{k\in\mathbb{N}}$ a partition of $x$
different from the former with disjoint blocks of length $L$ in which all
elements of $I$ occur. Then we have

$\sum_{k\in\mathbb{N}}\frac{1}{r_{k}f^{L}_{r}(r_{k})}\geq\sum_{k\in\mathbb{N}}\frac{1}{\tilde{r}_{k}f^{L}_{\tilde{r}}(\tilde{r}_{k})}.$

If the right hand side converges we even get a strict inequality.

###### Proof.

The proof of the statement is identical to the one of Proposition 2.3. We just
use the additional fact that $f$ is increasing in the sense of (ii) and that
$t\mapsto\frac{1}{t}$ is monotonically decreasing for $t>0$. ∎

######  Remark 3.7.

The second property in the previous lemma is inspired by the original case
where $f^{L}_{r}(r_{k})=\log k$, and therefore, also fulfills
$f^{L}_{r}(r_{k})\leq f^{L}_{\tilde{r}}(\tilde{r}_{k})$ for $k\in\mathbb{N}$.

We are going to show, that a similar construction to the one of
$\mathcal{N}_{0}$, but without using the bound
$\sum_{i=1}^{k}|\mathcal{S}_{i}|\leq p_{c}(k)$, leads to a topologically large
set. In essence, this tells us that the part where the attempt of showing that
$\mathcal{N}_{0}$ is topologically large breaks at the point of prescribing
the bound $p_{c}(k)$ and not due to prescribing only a certain type of
function $f$.

We therefore choose some family
$f=(f^{L}_{r}\colon\mathbb{N}\to(0,\infty))_{\begin{subarray}{c}N\leq
L\in\mathbb{N}\\\ r\in\mathcal{P}\end{subarray}}$ with the following three
properties:

  1. (F1)

$\forall
r\in\mathcal{P}_{L}\colon\displaystyle\lim_{t\to\infty}f^{L}_{r}(t)=\infty$

  2. (F2)

$\forall r,\tilde{r}\in\mathcal{P}_{L}\colon\forall
k\in\mathbb{N}\colon(r_{k}\leq\tilde{r}_{k}\implies f^{L}_{r}(r_{k})\leq
f^{L}_{\tilde{r}}(\tilde{r}_{k}))$

  3. (F3)

$\forall r\in\mathcal{P}_{N}\colon\forall
k\in\mathbb{N}\colon\sum_{i\in\mathbb{N}}f^{L}_{r}(k+iN)=\infty$

Then we define the property

$\displaystyle P_{L,f}(x)\leftrightarrow/$ $\displaystyle\exists\textnormal{ a
representation
}\mathcal{S}_{1}\mathcal{R}_{1}\mathcal{S}_{2}\mathcal{R}_{2}\dots=x\textnormal{
such that }$ $\displaystyle\leavevmode\hbox to13.4pt{\vbox
to13.4pt{\pgfpicture\makeatletter\hbox{\hskip 6.7pt\lower-6.7pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.5pt}{0.0pt}\pgfsys@curveto{6.5pt}{3.58989pt}{3.58989pt}{6.5pt}{0.0pt}{6.5pt}\pgfsys@curveto{-3.58989pt}{6.5pt}{-6.5pt}{3.58989pt}{-6.5pt}{0.0pt}\pgfsys@curveto{-6.5pt}{-3.58989pt}{-3.58989pt}{-6.5pt}{0.0pt}{-6.5pt}\pgfsys@curveto{3.58989pt}{-6.5pt}{6.5pt}{-3.58989pt}{6.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.33334pt}{-2.77779pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\wedge$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\begin{cases}\forall
k\in\mathbb{N}\colon|\mathcal{R}_{k}|=L\\\ \forall
k\in\mathbb{N}\colon\mathcal{R}_{k}\textnormal{ contains all elements of }I\\\
\forall
k\in\mathbb{N}\colon\sum_{k\in\mathbb{N}}\frac{1}{r_{k}f^{L}_{r}(r_{k})}=\infty\end{cases},$

where $r$ denotes the greedy $L$-partition of $x$. We then define the set

$\mathcal{N}_{f}\coloneqq\\{x\in K\colon\exists L\in\mathbb{N}\colon L\geq
N\wedge P_{L,f}(x)\\}.$

We are going to show that for every $f$ satisfying the conditions of Lemma
3.6, the complement $K\setminus\mathcal{N}_{f}$ is contained in a meager
subset. For this let us for $L\in\mathbb{N}$ with $L\geq N$ and
$M\in\mathbb{N}$ define the set

$\mathcal{A}_{L,M}\coloneqq\left\\{x\in K\colon\textnormal{greedy
$L$-partition $(r_{k})_{k\in\mathbb{N}}$ of $x$ exists
and}\sum_{k\in\mathbb{N}}\frac{1}{r_{k}f^{L}_{r}(r_{k})}<M\right\\}$

and for $k\in\mathbb{N}$ the set

$\mathcal{B}_{L,k}\coloneqq\\{x\in K\colon\textnormal{greedy $L$-partition
exists up until at most block $k$}\\}.$

Technically, both sets $\mathcal{A}_{L,M}$ and $\mathcal{B}_{L,k}$ should
additionally carry the index $f$, but for the sake of better readability we
drop it here.
A sequence is not in $\mathcal{N}_{f^{L}_{\cdot}}$ if and only if for all
$L\geq N$ either the greedy $L$-partition $(r_{k})_{k\in\mathbb{N}}$ exists
and produces a sum $\sum_{k=1}^{\infty}\frac{1}{r_{k}f^{L}_{r}(r_{k})}$ that
is finite or if the greedy $L$-partition does not exist. Therefore, we may
write

$K\setminus\mathcal{N}_{f}=\bigcap_{\begin{subarray}{c}L\in\mathbb{N}\\\ L\geq
N\end{subarray}}\left(\bigcup_{M\in\mathbb{N}}\mathcal{A}_{L,M}\cup\bigcup_{k\in\mathbb{N}}\mathcal{B}_{L,k}\right).$
(4)

We observe that $K\setminus\mathcal{N}_{f}$ is contained in a meager subset if
already the set

$\bigcup_{M\in\mathbb{N}}\mathcal{A}_{N,M}\cup\bigcup_{k\in\mathbb{N}}\mathcal{B}_{N,k}$

is meager or rather if $\mathcal{A}_{N,M}$ and $\mathcal{B}_{N,k}$ are nowhere
dense for all $M,k\in\mathbb{N}$.

######  Lemma 3.8.

For every $M\in\mathbb{N}$ the set $\mathcal{A}_{N,M}$ is nowhere dense.

###### Proof.

Let $x\in\mathcal{A}_{N,M}$ and $\varepsilon>0$. Choose $j\in\mathbb{N}$ such
that $2^{-j}<\varepsilon\leq 2^{-j+1}$. We now define $y\in K$ by

$y\coloneqq(x_{1},x_{2},\dots,x_{j},1,2,\dots,N,1,2,\dots,N,\dots).$

By (1) we have $d(x,y)<\varepsilon$. Observe that the greedy $N$-partition
$(r^{y}_{k})_{k\in\mathbb{N}}$ of $y$ exists and that
$\sum_{k\in\mathbb{N}}\frac{1}{r^{y}_{k}f^{L}_{r^{y}}(r^{y}_{k})}=\infty$,
since somewhere after $x_{j}$ all blocks are densely packed without spaces in
between. Hence, $y\notin\mathcal{A}_{N,M}$. Now set $m\in\mathbb{N}$ such that

$\sum_{\begin{subarray}{c}k\in\mathbb{N}\\\
r^{y}_{k}+N-1<m\end{subarray}}\frac{1}{r^{y}_{k}f^{L}_{r^{y}}(r^{y}_{k})}>M$
(5)

and let $n\in\mathbb{N}$ denote the last index $k$ used in the sum, i.e.,

$n=\max\\{k\in\mathbb{N}\colon r^{y}_{k}+N-1<m\\}.$

Then by (1) we have for every $z\in B(y,2^{-m})$ that

$z_{1}=y_{1}\wedge\dots\wedge z_{r^{y}_{n}+N-1}=y_{r^{y}_{n}+N-1}.$

If the greedy $N$-partition of $z$ does not exist, we have that
$z\notin\mathcal{A}_{N,M}$. In the case that the greedy $N$-partition
$(r^{z}_{k})_{k\in\mathbb{N}}$ of $z$ does exist, we also have
$z\notin\mathcal{A}_{N,M}$ since $z$ and $y$ match up until index
$r^{y}_{n}+N-1$, and hence, the first $n$ blocks in the greedy $L$-partition
of $z$ and $y$ agree. Therefore, by (5) we have

$\sum_{k\in\mathbb{N}}\frac{1}{r^{z}_{k}f^{L}_{r^{z}}(r^{z}_{k})}\geq\sum_{k=1}^{n}\frac{1}{r^{y}_{k}f^{L}_{r^{y}}(r^{y}_{k})}>M.$

We conclude that $\mathcal{A}_{N,M}$ is nowhere dense. ∎

######  Lemma 3.9.

For every $k\in\mathbb{N}$ the set $\mathcal{B}_{N,k}$ is nowhere dense.

###### Proof.

Let $x\in\mathcal{B}_{N,k}$ and $\varepsilon>0$. Choose $j\in\mathbb{N}$ such
that $2^{-j}<\varepsilon\leq 2^{-j+1}$. We now define $y\in K$ by

$y\coloneqq(x_{1},\dots,x_{j},\underbrace{1,\dots,1}_{N},1,2,\dots,N,1,2,\dots,N,\dots).$

Note that the greedy $N$-partition $(r^{y}_{k})_{k\in\mathbb{N}}$ of $y$
exists, and hence, $y\notin\mathcal{B}_{N,k}$. Then for every $z\in
B(y,2^{-j-N-(k+1)N})$ we have that $z$ has the leading entries

$(z_{i})_{i=1}^{j+N+(k+1)N}=(x_{1},\dots,x_{j},\underbrace{1,\dots,1}_{N},\underbrace{\overbrace{1,2,\dots,N},\dots,\overbrace{1,2,\dots,N}}_{k+1\textnormal{
many blocks }}).$

No block in the greedy $N$-partition of $z$ can begin in the section of
consecutive ones, since then $N$ cannot occur in such a block. It follows that
the blocks in the greedy $N$-partition after the section of consecutive ones
must be chosen in the way they are marked above. Therefore, the greedy
$N$-partition of $z$ exists at least up to block $k+1$, and hence,
$z\notin\mathcal{B}_{N,k}$ ∎

######  Remark 3.10.

A similar proof can be carried out for all sets $\mathcal{A}_{L,M}$ and
$\mathcal{B}_{L,k}$, but as mentioned above, we only need the proved case.

######  Theorem 3.11.

The set $\mathcal{N}_{f}$ contains a co-meager subset of $K$.

###### Proof.

By Equation (2) we have that $K\setminus\mathcal{N}_{f}$ is contained in a
meager subset if all $\mathcal{A}_{M}$ and $\mathcal{B}_{N,k}$ for
$M,k\in\mathbb{N}$ are nowhere dense. This is the case by Lemma 3.8 and Lemma
3.9. ∎

######  Corollary 3.12.

The set of quasi-normal sequences $\mathcal{N}_{f}$ contains a dense
$G_{\delta}$ subset.

##  4 On Sequences with Finite Index Occurrence

For the method of alternating projections to work, one needs to use sequences
$x\in K$ where every index in $I$ appears infinitely often. Since the previous
sections show that almost all sequences are quasi-normal, it is clear that
almost all sequences have to contain every index infinitely often. However, at
least with the previously given proofs, it seems that there are qualitatively
more sequences with infinite index appearance than there are sequences that
lead to convergence. This may be suspected as the following theorem shows that
the sequences with infinite index occurrence form a co-$\sigma$-porous subset,
but the quasi-normal sequences only have been proven to form a
co-$\sigma$-$\phi$-porous subset. This tells us that we qualitatively lose
some sequences, as $\phi$-porosity is a weaker notion of porosity. If indeed
the quasi-normal sequences are non-co-$\sigma$-porous this would stand in
contrast to the measure theoretic case, where the quasi-normal sequences and
the ones with infinite index occurrence both have full measure, and
qualitatively no sequences are lost when making this transition. Investigating
if the quasi-normals are indeed non-co-$\sigma$-porous might be an interesting
extension of the presented results.

######  Theorem 4.1.

The set

$\mathcal{F}\coloneqq\\{x\in K\colon\exists n\in I\colon\exists
M\in\mathbb{N}\colon|\\{k\in\mathbb{N}\colon x_{k}=n\\}|<M\\}$

of all sequences in $K$ in which some index $n\in I$ appears only finitely
often is co-$\sigma$-porous.

###### Proof.

By defining

$\mathcal{F}_{n,M}\coloneqq\\{x\in K\colon|\\{k\in\mathbb{N}\colon
x_{k}=n\\}|<M\\}$

we may write

$\mathcal{F}=\bigcup_{n\in\mathbb{N}}\bigcup_{M\in\mathbb{N}}\mathcal{F}_{n,M}.$

We show that for all $n,M\in\mathbb{N}$ the set $\mathcal{F}_{n,M}$ is porous.
For this let $x\in\mathcal{F}_{n,M}$, let

$\varepsilon_{0}\coloneqq
1,\quad\alpha=2^{-M-1},\quad\text{and}\quad\varepsilon\in(0,\varepsilon_{0})$

and choose $j\in\mathbb{N}$ such that $2^{-j}<\varepsilon\leq 2^{-j+1}$. We
now define

$y=(x_{1},\dots,x_{j},n,n,n,\dots).$

By (1) we have that $d(x,y)<\varepsilon$. By the same relation we observe that
all $z\in B(y,\alpha\varepsilon)\subseteq B(y,2^{-j-M-1})$ have at least $M$
many occurrences of $n$. Hence,
$B(y,\alpha\varepsilon)\cap\mathcal{F}_{n,M}=\emptyset$, and therefore,
$\mathcal{F}_{n,M}$ is porous. This directly implies the desired result, as
$\mathcal{F}$ may be written as a countable union of porous sets. ∎

Acknowledgements:  Special thanks go to my supervisor Eva Kopecká for many
valuable discussions and suggestions.

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