The interaction of supercritical turbulent flows with granular sediment beds is challenging to study both experimentally and numerically; this challenging task has hampered the advances in understanding antidunes, the most characteristic bedform of supercritical flows.
This article presents the first numerical attempt to simulate upstream-migrating antidunes with geometrically resolved particles and a liquid–gas interface.
Our simulations provide data at a resolution higher than laboratory experiments, and they can therefore provide new insights into the mechanisms of antidune migration and contribute to a deeper understanding of the underlying physics.
To manage the simulations' computational costs and physical complexity, we employ the cumulant lattice Boltzmann method in conjunction with a discrete element method for particle interactions, as well as a volume of fluid scheme to track the deformable free surface of the fluid.
By reproducing two flow configurations of previous experiments (Pascal et al., Earth Surf. Proc. Land., vol. 46(9), 2021, 1750–1765), we demonstrate that our approach is robust and accurately predicts the antidunes' amplitude, wavelength, and celerity.
Furthermore, the simulated wall-shear stress, a key parameter governing sediment transport, is in excellent agreement with the experimental measurements.
The highly resolved data of fluid and particle motion from our simulation approach open new perspectives for detailed studies of morphodynamics in shallow supercritical flows.

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§ INTRODUCTION

Antidunes are short-wave periodic disturbances that develop on the surface of loose granular beds in response to the interaction with supercritical and near-critical shallow, turbulent flows <cit.>, where criticality is defined by the Froude number.
These types of bedforms arise in fluvial, coastal, and submarine environments and are closely tied to the resulting flow resistance, turbulence, and sediment transport, which are quantities that are of great interest in geology as well as hydraulic and environmental engineering.
Among the bedforms at supercritical flow conditions, antidunes are the shortest in wavelength (on the order of the flow thickness), and they are also the only bedforms that can either migrate upstream, downstream, or even remain stationary.
This work deals with upstream migrating antidunes (UMAs).
The longitudinal profile of an UMA is symmetrical, exhibiting a sinusoidal shape that is mirrored by the undulating free water surface.
Consequently, UMAs are coupled to the fluid surface and are, thus, dependent on the presence of a free surface or an interface where stationary waves can develop.
The movement of UMAs in the direction opposite to the main flow is counterintuitive, and it is believed to result from the rhythmic aggradation of particles on the upstream flank of the bedform and erosion on the lee side.
However, little is known about the migration mechanism in connection to turbulence, bed morphology and sediment transport.

Theoretical studies have made significant advances to understanding the characteristics of antidunes, by defining antidune instability regions <cit.> and identifying hydraulic constraints for antidune migration regimes <cit.>.
Unfortunately, due to the challenging supercritical flow conditions often associated with low submergences, experimental data sets to validate those theories are extremely sparse.
The limited number of systematic experimental data sets with antidune configurations is in part related to the inherent technical challenges to reproduce rapid flows over an erodible bed in laboratory flumes <cit.>, as well as to the difficulties to perform non-intrusive measurements in flows which are generally unstable and shallow.
Numerical simulations of supercritical flows above an erodible bed, as a methodological alternative for the study of antidunes, are challenging because of the strongly varying water surface of rapid flows.
Existing simulations of supercritical bedforms have mainly considered the Saint–Venant shallow-water equations <cit.> and the Reynolds-averaged Navier–Stokes approach to model cyclic steps <cit.> or downstream-migrating antidunes <cit.>, but to the best of the authors' knowledge, simulations of UMAs have not been reported so far.

Recently, <cit.> have successfully simulated downstream migrating ripples in subcritical flows, with particle-resolved direct numerical simulations in which the fluid–particle interaction is resolved by considering the motion of the flow and of each individual particle.
Owing to the subcritical flow conditions in their study, <cit.> could simplify the flow geometry by keeping the fluid depth constant.
In this work we propose to use particle-resolved simulations in conjunction with a deformable fluid surface, to simulate the formation and propagation of UMAs in supercritical flows.
We aim to numerically reproduce an experimental campaign recently reported by <cit.>, who managed to measure the propagation of UMAs with a high spatial and temporal resolution.
For this, we use the lattice Boltzmann method (LBM) to simulate the fluid hydrodynamics <cit.>  and extend it with a volume of fluid scheme <cit.> to track the strongly deformable free fluid surface in supercritical flows.
In turn, we couple the fluid hydrodynamics with the particle dynamics via the momentum-exchange method to simulate the antidune formation with high fidelity.
The parameter choices of <cit.>, with coarse sediment grains and low particle relative submergence, allow for a direct overlap of experimental conditions with simulations.
In this manner, the present work closes the gap between river morphodynamics’ experiments and numerical simulations, and demonstrates the capabilities offered by large-scale simulations of a mobile sediment bed comprising thousands of particles in unidirectional, supercritical turbulent flows, to study coupled bedform and free-surface interactions in great detail.

§ NUMERICAL METHODS

All numerical simulations in this work were performed using the open-source multi-physics software framework  <cit.>.
In what follows, we briefly summarize the simulation's key features, namely the fluid–gas, the particle–particle, and the fluid–particle interactions.
<Ref> presents a sketch of our numerical scheme.

Schematic representation of the simulations, coupling the liquid (l) with a particle (p) and gas (g) phase fig:method, and the simulation setup in its initial condition fig:setup.
<Ref> is a zoom into the computational domain covering only 16 % of its streamwise extent.

§.§ Free-surface lattice Boltzmann method

The LBM <cit.> discretizes the Boltzmann equation from kinetic theory and describes the evolution of particle distribution functions (PDFs) on a uniform computational grid to simulate a fluid flow.
Macroscopically, the LBM approximates the Navier–Stokes equations with second-order accuracy in space $\boldsymbol{x}$ and time $t$.
The cell-local fluid density $\rho_{\text{l}}(\boldsymbol{x},t)$ and velocity $\boldsymbol{u}_{\text{l}}(\boldsymbol{x},t)$ are given by the zeroth- and first-order moments of the cell's PDFs.
In the remainder of this article, all quantities denoted with the superscript LU are specified in a normalized LBM unit system.
We set the cell size $\Delta x^{\text{LU}} = 1$, time step size $\Delta t^{\text{LU}} = 1$, and the reference density of the fluid $\rho^{\text{LU}}_{\text{l},\text{ref}}=1$.
We use the D3Q27 cumulant collision model proposed by <cit.> and set all relaxation rates to unity, except for $\omega^{\text{LU}}$ that defines the kinematic viscosity of the fluid $\nu_{\text{l}}^{\text{LU}} = 1/3 (1/\omega^{\text{LU}} - \Delta t^{\text{LU}}/2)$.
We model no-slip boundary conditions at solid obstacles using the bounce-back rule <cit.>.

The  framework includes an implementation of the free-surface lattice Boltzmann method <cit.>.
The FSLBM is based on a volume of fluid scheme and simulates the interface between two immiscible fluids.
The model assumes that the liquid fluid (higher density) governs the flow dynamics of the system so that the fluid flow in the gaseous fluid (lower density) is neglected.
A free-surface boundary condition captures the motion of the liquid–gas interface.
The boundary condition incorporates the gas pressure $p_{\text{g}}$ and the Laplace pressure $p_{\text{l},\text{L}}(\boldsymbol{x},t)=2\sigma_{\text{l}}\kappa_{\text{l}}(\boldsymbol{x},t)$, where $\sigma_{\text{l}}$ is the surface tension of the liquid, and $\kappa_{\text{l}}(\boldsymbol{x},t)$ is the local curvature of the free surface.

§.§ Particle–particle and fluid–particle coupling

The temporal evolution of the translational and rotational velocity, $\boldsymbol{u}_{\text{p},i}$ and $\boldsymbol{\omega}_{\text{p},i}$, of an individual rigid particle $i$ is described by the Newton–Euler equations for rigid-body motion.
The total force and torque acting on a particle comprises contributions from inter-particle collisions, hydrodynamic interactions, and an external force attributed to the gravitational acceleration $\boldsymbol{g}$.
The inter-particle collisions are modeled via the discrete element method (DEM) specifically designed for a four-way coupled lattice Boltzmann method of particle-laden flows. The model is fully parameterized by the dry coefficient of restitution of $e_{\text{dry}} = 0.97$, the collision time $T_{\text{c}}^{\text{LU}} = 4 d_{50}^{\text{LU}} \Delta t^{\text{LU}} /\Delta x^{\text{LU}}$, the Poisson's ratio $\nu_{\text{p}} = 0.22$, and the friction coefficient $\mu_{\text{p}}=0.5$ to simulate silica beads.
Details on the particle–particle and fluid–particle coupling can be found in <cit.>.

For fluid–particle interactions, we employ a geometrically resolved coupling between the fluid and the particulate phase.
We impose no-slip boundary conditions for the fluid along the particle surface and compute the hydrodynamic forces and torques acting on the particles directly.
This geometrically resolved coupling is achieved by the LBM-specific momentum-exchange method (MEM).
The MEM requires an explicit mapping of the particles onto the underlying computational grid <cit.>.
Following <cit.>, we additionally augment the DEM with lubrication corrections in the normal and tangential directions to model these strong subgrid-scale hydrodynamic forces for situations where gaps between particles become too small to be properly resolved by the LBM mesh.

§ SIMULATION SETUP

§.§ Physical scenario and simulation parameterization

The simulations carried out for this article aim to reproduce the laboratory experiments of <cit.>, who performed their experiments with water and a sediment bed of natural gravel.
The gravel particles had a density of $\rho_{\text{p}} = 2550$ kg/m3 and characteristic sieving diameters of $d_{16}=2.5$ mm, $d_{50}=2.9$ mm, and $d_{84}=3.3$ mm, where the subscripts denote the respective percentile of the particle size distribution.
The mean flow velocity $U_{x,\text{l}} = q_{\text{w}} / h_{0}$ is computed from the measured flow discharge in the surface flow layer $q_{\text{w}}$ and reference flow depth $h_{0}$ (<ref>).
Three dimensionless numbers specify the flow conditions:
the Reynolds number $\mathit{Re} = U_{x,\text{l}} h_{0} / \nu_{\text{l}}$, the Froude number $\mathit{Fr} = U_{x,\text{l}}/\sqrt{g h_{0}}$, and the Weber number $\mathit{We} = \rho_{\text{l}} U_{x,\text{l}}^{2} h_{0} / \sigma_{\text{l}}$, where  $\nu_{\text{l}} = 1 \cdot 10^{-6}$ m2/s and $\sigma_{\text{l}} = 7.2 \cdot 10^{-2}$ kg/s2 are the kinematic viscosity and surface tension of water, respectively.
The values of these numbers are listed in <ref>, where we refer to the simulation setups with the same labels E1 and E4, as chosen by <cit.>.

Label     $\mathit{Re}$     $\mathit{Fr}$     $\mathit{We}$     $\Delta x$ / m     $\Delta t$ / s     $\omega^{\text{LU}}$     $U^{\text{LU}}_{x,\text{l}}$     $h_{\text{0}}$

E1     $3100$     $1.31$     $15.62$     $2.5 \cdot 10^{-4}$     $1.38 \cdot 10^{-5}$     $1.997$     $0.02$     $2.97 d_{50}$

E4     $4800$     $1.45$     $30.25$     $2.5 \cdot 10^{-4}$     $1.08 \cdot 10^{-5}$     $1.998$     $0.02$     $3.59 d_{50}$

Summary of the main simulation conditions.
The labels are chosen as in the reference experiments from <cit.>.

The simulation domain was discretized with a mesh size $\Delta x = 2.5 \cdot 10^{-4}$ m on a uniform computational grid of size $L^{\text{LU}}_x \times L^{\text{LU}}_y \times L^{\text{LU}}_z = 3200 \times 60 \times 160$ cells, corresponding to $0.8\,\text{m} \times 0.015\,\text{m} \times 0.04\,\text{m}$ in the streamwise, spanwise and vertical direction, respectively.
We have used periodic boundary conditions in the $x$- and $y$-directions.
The boundaries in the $z$-direction and on the particle surface were modeled with a no-slip condition.
The sediment bed consisted of $6528$ spherical particles with a mean diameter of $d^{\text{LU}}_{50} = 11.6$, where the actual diameters were sampled from a uniform distribution of the interval $[0.9,1.1] d_{50}$.
We have generated the initially flat particle bed of height $h_{\text{b}} = 4.14 d_{50}$ using a rigid body dynamics simulation for a sedimentation process in dry conditions.
To this end, particles were dropped onto a single layer of fixed particles that were positioned on a horizontal hexagonal grid with small random perturbations in their $z$-positions.
We fix all particles with a vertical center coordinate $z_{\text{p},i}<h_{\text{b,f}} = 1.5 d_{50}$ to generate a rough bottom.
We have carried out sensitivity studies to verify that the chosen number of particles and $h_{\text{b,f}}$ were sufficient to exclude finite-size effects of the domain extent.

The liquid was initialized with a height of $h_{\text{b}}+h_{0}$, where $h_{0}$ represents the initial height of the liquid on top of the sediment bed, and with a parabolic velocity profile on top of the bed.
The hydrostatic density was initialized, such that the liquid pressure was equal to the constant atmospheric gas pressure at the free surface.
The flow was driven in $x$-direction via an externally imposed body force $F_x$, which was controlled during the simulation to maintain the desired bulk flow rate.
The mean fluid velocity was set to $U^{\text{LU}}_{x,\text{l}} = 0.02$, leading to the time step sizes $\Delta t$ specified in <ref>.

§.§ Evaluation quantities and grid resolution study

The temporal and spatial variability of the bed topography is one of the key quantities to characterize the evolution of antidunes.
We follow the approach of <cit.> and compute the $y$-averaged solid-volume fraction $\phi_{\text{b}}(x,z,t)$ for each computational cell in the $x$-, $z$-plane.
The bed height $h_{\text{b}}(x,t)$ is then given as the linearly interpolated $z$-value, for which $\phi_{\text{b}}(x,z,t) = 0.3$.
The bedload transport rate $q_{\text{b}}(t)$ is computed via

\begin{equation}\label{eq:bedload-transport-rate}
q_{\text{b}}(t)=\frac{\sum_{i \in \text{P}} u_{x,\text{p},i}(t) V_{\text{p},i}}{L_{x}L_{y}},
\end{equation}

where P is the set of all particles and $u_{x,\text{p},i}(t)$ is the streamwise particle velocity of particle $i$.

We carried out a grid sensitivity study to verify that our results are independent of numerical parameters.
To this end, we focused on the more turbulent flow conditions of E4 (see <ref>).
In addition to the reference case from <ref> with  $\Delta x_{50,\textit{Medium}} = 2.5 \cdot 10^{-4}$ m as a  medium resolution, we have tested two additional cases with a two-times coarser ($\Delta x_{50,\textit{Coarse}} = 5 \cdot 10^{-4}$ m) and finer resolution ($\Delta x_{50,\textit{Fine}} = 1.25 \cdot 10^{-4}$ m).
These computational grid resolutions lead to $d^{\text{LU}}_{50,\textit{Coarse}} = 5.8$, $d^{\text{LU}}_{50,\textit{Medium}} = 11.6$, and $d^{\text{LU}}_{50,\textit{Fine}} = 23.2$.
For the grid sensitivity study, we shortened the simulation domain by a factor of $10$ in the $x$-direction to keep the computational cost manageable.
We have scaled the time step size $\Delta t$ proportionally to $\Delta x$ in our study with reference $\Delta t_{50,\textit{Medium}} = 1.08 \cdot 10^{-5}$ s, as listed in <ref>.
<Ref> compares the bedload transport rates (<ref>) for the different grid resolutions.
From this study, we conclude that the grid Medium is sufficiently independent of the grid resolution and can be used for further analysis.

Simulated bedload transport rate $q_{\text{b}}(t)$ with different computational grid resolutions using a shortened simulation domain.
The time-averages were computed only in the time-ranges conforming with the lengths of the plotted lines to ensure that the fluid flow was adequately developed.

§ RESULTS

Our particle-resolved simulations provide the position of each particle as well as the location of the water surface in space and time, and full fluid information (<ref>).
Bed elevation perturbations arose spontaneously from the initial flat bed right from the start of the simulations.
From then on, a regular longitudinal pattern of quasi-periodic bedforms, with gentle slopes up- and downstream of the crests, prevailed throughout the simulation time.
The bed patterns were approximately in phase with respect to the free surface and they slowly migrated upstream (see animations in supplementary material).
Therefore, these bedforms can be unequivocally classified as UMAs that compare well to the observations of the experiments of <cit.>.
A data set with such a high resolution as the one generated here numerically, has so far not been available from physical experiments on movable beds under supercritical flows.

Visualization of the simulated velocity in streamwise direction $u_{x,\text{l}}(\boldsymbol{x},t)$.
The undulation of the sediment bed and free surface of the liquid are in phase, conforming to the definition of antidunes.
The figure is a zoom into the computational domain covering only $25$ % of its streamwise extent.

§.§ Bed elevation perturbations

Space–time evolution diagrams of the bed surface, for the simulated and the experimental bedforms, are plotted in <ref>.
Such diagrams result from laterally-averaged bed surface profiles at every 1000 time steps of the simulation, and from footage recorded from the transparent channel walls in the experiments of <cit.>.
From the original experimental data,  we have selected the intervals corresponding to the lower left parts of figures 3a and 3d in this reference, to reproduce a part of their bed elevation plots to compare with the scales of our numerical data.
The alternating blue and yellow (dark and light) diagonal strips denote the troughs and crests of the bedforms, respectively.
The experimental and numerical diagrams with the evolution of the bottom elevation in the spatial–temporal domain are in good qualitative agreement.
A similar upstream migration trend of the bedforms is clearly visible due to the negative slope of the strips.
In some regions the strips bend, indicating acceleration/deceleration, or even stationarity of the bedforms.
Particularly for run E1, these regions are related to local perturbations that migrate downstream. Note that similar perturbations can be identified in both experimental and simulation plots.
Overall, the simulated bedforms appear stable and compare well to their experimental counterparts.

As pointed out by <cit.>, the definition of a dominant bedform migration speed seems inappropriate given the nonuniformity of the bedform celerities observed.
The same reasoning is valid for the amplitude and the wavelength of the antidunes.
In our simulations, the amplitude of the simulated bed undulations ranged approximately from $1$ to $3$ times the particle median grain size $d_{50}$, and the wavelength ranged from about $10$ to $15$ times the water depth $h_0$.
These values are in good quantitative agreement with the experimental data, where the dune amplitudes ranged from one particle median grain size to the mean flow depth ($\approx 3d_{\text{50}}$), and typical wavelengths varied from $0.05$ to $0.15$ m ($\approx 6h_{0}$ to $15h_{0}$).

To obtain a more precise comparison of the numerical and experimental data in terms of their bedform size and fluctuations, we have followed <cit.> and computed  the power spectral density (PSD) from the square of the two-dimensional discrete Fourier transform of $h_{\text{b}}(x,t)$ with respect to the streamwise position $x$ and time $t$.
We have normalized the PSD by the total number of samples available for $x$ and $t$.
The results are shown in <ref>, along with the experimental PSD.
Although there is not a perfect match between the spectra for the experimental and numerical bedforms, the range of wavelengths and periods agree reasonably well.
For the numerical data, the range of periods and wavelengths in the spectra is slightly narrower than in the experiments, which can be attributed to the much shorter data series from the simulations ($45$ s simulation time versus more than $4600$ s in E1, and $2800$ s in E4 in the experiments).

We compared experimental and simulated results further by obtaining the spectra in the celerity–wavelength domain (<ref>).
The celerity $c = \lambda/T$, that is, the movement speed of the dunes, was computed as the ratio between wavelength $\lambda$ and period $T$ from the spectra in <ref>.
Similarly to the PSD in the $\lambda-T$ plane, a fairly good agreement was found between the simulations and the experiments. More precisely, the celerities in the simulations range from $2$ to $12$ mm/s in E1 (experiment: $\approx 2-15$ mm/s) and from $2$ to $18$ mm/s in E4 (experiment: $\approx 5-30$ mm/s).
It is important to note that the simulations reproduced the trends observed experimentally, that for a given wavelength, celerity increases with bedload transport rate, while for a constant bedload transport rate, celerity increases with increasing antidune wavelength.
This latter trend is opposite to that commonly observed for morphodynamics in subcritical flows <cit.>.

point meta max=0.005,
point meta min=-0.005,
colorbar style=
ylabel=$h_{\text{b}}$ / m,
every y tick scale label/.style=
at=(yticklabel* cs:1.075,0),
anchor=near yticklabel

Sediment bed elevation $h_{\text{b}}(x,t)$ for E1 (left) and E4 (right): fig:bed-elevation-experiment-e1 and fig:bed-elevation-simulation-e1 data from <cit.>, and fig:bed-elevation-experiment-e4 and fig:bed-elevation-simulation-e4 data from the numerical simulations.
We consider the system fully developed after $t=30$ s, as illustrated by the dashed black line.

Label     $h_{0}/d_{\text{50}}$     $h_{0}/d_{\text{50}}$     $\bar{h}/d_{\text{50}}$     $\text{tan}\psi$     $\text{tan}\psi$     $\Theta$     $\Theta$     $q_{\text{b}}\ast$     $q_{\text{b}}\ast$

exp.     sim.     sim.     exp.     sim.     exp.     sim.     exp.     sim.

E1     $2.86$     $2.97$     $3.02$     $0.051$     $0.044$     $0.085$     $0.086$     $0.033$     $0.020$

E4     $3.59$     $3.59$     $3.10$     $0.052$     $0.051$     $0.108$     $0.119$     $0.100$     $0.052$

Summary of experimental (exp.) and simulated (sim.) variables, where $\psi$ is the mean bed slope, $\Theta=R_{\text{b}}\text{tan}\psi/[d_{\text{50}}(\rho_{\text{p}}/\rho_1 - 1)]$ is the  Shields number, and $q_{\text{b}}\ast=q_{\text{b}}/[(\rho_{\text{p}}/\rho_1 - 1)gd_{\text{50}}^{\text{3}}]^{\text{1/2}}$ is the Einstein bed load number.

Power spectral density (PSD) of wavelength $\lambda$ and period $T$ for fig:spectral-density-e1 E1  and fig:spectral-density-e4 E4  from <cit.>.
The orange line is the outermost hull of the simulation results' PSD.

Power spectral density contour of celerity $c$ and wavelength $\lambda$ for fig:celerity-e1 E1  and fig:celerity-e4 E4  from <cit.>.

§.§ Hydraulic variables and sediment transport

To further compare the experimental and simulated flow and sediment transport conditions in a quantitative sense, the normalized average water depth, the bottom slope, the Shields number and the Einstein bed load number are defined and summarized in <ref>.
It must be noted that the bottom profile in the simulations was intentionally horizontal, and thus the slope $\psi$ considered in <ref> is the tangent to the acting force imposed in $x$–$z$ direction.
Assuming a steady-uniform flow where the fluid gravitational force is balanced by boundary friction, the average bed shear stress was computed with the depth-slope product $\rho_1 g R_{\text{b}}\text{tan}\psi$. Here, $R_\text{b}$ is the hydraulic radius.

Comparing the numbers of <ref>, the numerical and experimental Shields numbers are very similar.
We note that in our simulations the horizontal boundary conditions were periodic, that is, the domain was not bound by lateral walls as in the physical experiments.
Therefore, for the simulations, the boundary shear stress in the Shields number was computed based on the average water depth ($R_{\text{b}}=\bar{h}$) from $t=30$ s to $t=75$ s; whereas, for the experiments, the boundary shear stress considered the hydraulic radius as a correction for the effect of the lateral wall roughness.
The simulated sediment transport rates were lower than the experimental mean values by a factor of about $2$.
Nevertheless, the increase in sediment transport rate from case E1 to E4 was similar in the simulations and the experiments (roughly an increment by a factor $3$).
<Ref> shows the evolution of the simulated sediment transport rates with time for E1, together with a representative time window of similar duration of the experiments.
Compared to the simulation data, the experimental transport rates show larger fluctuations.
These variations are likely due to sampling, which captured all particles ejected from the downstream end of the channel in the experiment, as opposed to the averaged sediment transport rates over the entire simulation domain.
It is, therefore, remarkable that the simulated and experimental transport rates match well in the last $15$ s of the time series.
Some reasons for the underestimation of the average sediment transport rates in the simulations can be related to the shape of the particles
<cit.>, to experimental uncertainties as for instance, additional moment added by the sediment feeding system, and to the strong non-linear dependency of sediment transport rates on boundary conditions and the wall shear stresses.
Notwithstanding, the comparison between numerical and experimental hydraulic variables and sediment transport rates must be rated as satisfactory, and given the good correspondence between the simulations and the experiments, we conclude that our simulation approach is adequate to investigate UMAs with high fidelity.

Bedload transport rate $q_{\text{b}}(t)$ as measured in the experiment E1 from <cit.> and in the numerical simulations.
Time-averaged values were computed for a time range of $2804$ s and $45$ s for the experiment and simulation, respectively.

§ CONCLUSION

We have performed particle-resolved numerical simulations of supercritical turbulent flows ($\mathit{Fr}>1$) over an erodible granular bed of spherical particles.
Our goal was to numerically reproduce the experimental work of <cit.>, in which upstream migrating antidunes developed on a bed of natural gravel sheared by a shallow-water flow (water depth $\approx3$ times the sediment grain size).
Supercritical shallow flows over erodible beds are extremely unstable due to the strong feedback between the dynamics of the bottom and the wavy free surface.
Our results demonstrate that one can accurately reproduce the bedform dynamics at different supercritical flow conditions by using a massively parallel simulation approach relying on fully coupled fluid–particle–gas interactions.
In our simulations, no suspended load was present and the bed instability was naturally excited by the simulated flow.
Hence, the range of amplitudes, wavelengths, and celerities of the self-generated upstream migrating bedforms were results of the simulations and agree well with the experimental patterns reported by <cit.>.
Particularly, the averaged bed Shields stress from the simulations as a key quantity to predict sediment transport rates matched very well with the experimental values.

The vast amount of data generated by simulations in a non-intrusive manner such as those presented in this work can be of great use to supplement experimental measurements under challenging supercritical flow conditions.
The detailed data of particle and fluid motion available through our simulation approach opens a multitude of possibilities.
The present study, therefore, encourages further simulation campaigns to understand antidune mechanics and the physical controls on bedform initiation and morphodynamics in supercritical flows.

Supplementary data.  Animations of the simulations are available online.

The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (www.lrz.de).

The authors thank the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for funding the projects 408062554, 434946896, 433735254, and 428445330.
This work was supported by the SCALABLE project.
This project has received funding from the European High-Performance Computing Joint Undertaking (JU) under grant agreement No 956000.
The JU receives support from the European Union’s Horizon 2020 research and innovation programme and France, Germany, the Czech Republic.

Declaration of Interests.
The authors report no conflict of interest.

Author ORCID.
C. Schwarzmeier, <https://orcid.org/0000-0002-5756-3985>;
C. Rettinger, <https://orcid.org/0000-0002-0605-3731>;
S. Kemmler, <https://orcid.org/0000-0002-9631-7349>;
J. Plewinski, <https://orcid.org/0000-0002-4815-7643>;
F. Núñez-González, <https://orcid.org/0000-0002-3676-2715>;
H. Köstler, <https://orcid.org/0000-0002-6992-2690>;
U. Rüde, <https://orcid.org/0000-0001-8796-8599>;
B. Vowinckel, <https://orcid.org/0000-0001-6853-7750>

Author contributions.
C. Schwarzmeier: Conceptualization, Methodology, Software, Formal Analysis, Validation, Investigation, Data Curation, Writing – Original Draft, Writing – Review & Editing, Visualization, Supervision, Project Administration;
C. Rettinger: Conceptualization, Methodology, Software, Formal Analysis, Validation, Investigation, Data Curation, Writing – Original Draft, Visualization, Supervision, Project Administration;
S. Kemmler: Software, Validation, Investigation, Writing – Original Draft;
J. Plewinski: Software, Validation, Writing – Original Draft;
F. Núñez-González: Conceptualization, Writing – Original Draft, Writing – Review & Editing;
H. Köstler: Resources, Funding acquisition;
U. Rüde: Resources, Funding acquisition, Writing – Original Draft;
B. Vowinckel: Conceptualization, Writing – Original Draft, Writing – Review & Editing;


# Abundance of Matrices In Gaussian Integers

Aninda Chakraborty

###### Abstract.

In [HLS], N. Hindman, I. Leader and D. Strauss proved the abundance for a
matrix with rational entries. In this paper we proved it for the ring of
Gaussian integers. We showed the result when the matrix is taken with entries
from $\mathbb{Q}\left[i\right]$. The main obstacle is in the field of complex
numbers, no linear order relation exists. We overcome that in a tactful way.

##  1\. Introduction

Some problems in Ramsey Theory can be seen as follows:

Let $M$ be any $r\times s$ matrix, where $r,s\in\mathbb{N}$and all entries of
$M$ are from $\mathbb{N}$. Then, could it be possible for each and every
partition (finite) of $\mathbb{N}$ , one can find some
$\vec{u}\in\mathbb{N}^{s}$ so that all entries from $M\vec{u}$ are in one cell
from the partition?

At first, take van der Waerden’s Theorem. Then the elements from the
arithmetic progression $\\{a,a+d,\ldots,a+ld\\}$ is nothing but entries from

$\left(\begin{array}[]{cc}1&0\\\ 1&1\\\ \vdots&\vdots\\\
1&l\end{array}\right)\cdot\left(\begin{array}[]{c}a\\\ d\end{array}\right).$

So, we have this typical definition:

######  Definition 1.1.

Let $\left(T,+\right)$ be a commutative monoid. Suppose $r,s\in\mathbb{N}$ and
$M$ be any $r\times s$ matrix which has entries in
$\mathbb{N}\cup\left\\{0\right\\}$. Then $M$ is called image partition regular
on $T$ (symbolised as $IPR/T$) if and only if for each $k\in\mathbb{N}$ and
$T=\bigcup_{j=1}^{k}D_{i}$, then there must exist
$j\in\left\\{1,2,\ldots,k\right\\}$ and
$\vec{u}\in\left(T\setminus\left\\{0\right\\}\right)^{s}$ so that $M\vec{u}\in
D_{j}^{r}$.

Let $\left(T,+\right)$ is a commutative monoid, $t\in T$ be any element and
$m\in\mathbb{N}$. Then we denote the $m$ times sum of $t$ with itself by $mt$.

In 1993, N. Hindman and I. Leader has introduced image partition regularity of
matrices in [1]. Next in [HLS] authors provided some new characterizations and
consequences of image partition regularity. One of there characterizations is
that the image partition regular matrices are precisely those that preserve a
certain notion of largeness (“central sets”). In [DJ], authors completely
accoplised the determination of image partition regular matrices over
$\mathbb{Z}\left[i\right]$.

Furstenberg and Glasner have shown in [F1] that for a particular notion of
largeness in a group, namely piecewise syndeticity, if a set $B$ is a large
subset $\mathbb{Z}$, then for any $l\in\mathbb{N}$, the set of length $l$
arithmetic progressions lying entirely in $B$ is large among the set of all
length $l$-aritmetic progressions. In [BH], authors extend this result to
apply to infinitely many notions of largeness in arbitrary semigroups and to
other partition regular structures besides arithmetic progressions. In [HLS],
authors showed the abundance of image parition regular matrices .Also, they
examined for other well known notions of largeness.( Like
IP∗,$\Delta^{*}$,PS∗, piecewise syndetic, thick etc.). To obtain these
results, usual order relation of $\mathbb{N}$ played an important role. We
have extend these results of abunndance of matrices on
$\mathbb{Z}\left[i\right]$, the ring of Gaussian integers. In
$\mathbb{Z}\left[i\right]$, there is no order relation. So, we need to derive
this results using different technique.

First we need to prove this lemma before proceed to the main results:

######  Lemma 1.2.

Let $A$ be a $u\times v$ image partition regular matrix over
$\mathbb{Q}\left[i\right]$. The following are equivalent:

(a) There exists $\vec{s}\in\left(\mathbb{Q}\left[i\right]\right)^{v}$ such
that $A\vec{s}=\vec{1}$.

(b) There exists $l\in\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}$
such that, if $p$ is in the smallest ideal of $\beta\mathbb{Z}\left[i\right]$
and $l\mathbb{Z}\left[i\right]\in p$, then, for every $P\in p$, there exists
$\vec{z}\in\left(\mathbb{Z}\left[i\right]\right)^{v}$such that $A\vec{z}\in
P^{u}$.

###### Proof.

$\left(a\right)\Rightarrow\left(b\right)$. We can choose
$\vec{w}\in\left(\mathbb{Z}\left[i\right]\right)^{v}$ and
$l\in\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}$ such that
$A\vec{w}=\vec{l}$, where
$\vec{l}=\left(l,l,\ldots,l\right)^{T}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{u}$.
Suppose that $p$ is in the smallest ideal of $\beta\mathbb{Z}\left[i\right]$
and that $l\mathbb{Z}\left[i\right]\in p$. Let $P\in p$. There exists a
minimal idempotent $q\in\beta\mathbb{Z}\left[i\right]$ such that $p=p+q$ [HS,
Theorem 2.8 and Lemma 1.30]. Let $P^{\prime}=\left\\{b\in P:-b+P\in
q\right\\}$. Then $P^{\prime}\in p$. Since $l\mathbb{Z}\left[i\right]\in p$ we
can choose $t\in\mathbb{Z}\left[i\right]$ such that
$lt\in\mathbb{Z}\left[i\right]$. Let $Q=-lt+P\in q$. By [DJ, Theorem 3.1(h)],
there exists
$\vec{x}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}$
such that $A\vec{x}\in Q^{v}$ and for every
$i\in\left\\{1,2,\ldots,v\right\\}$. Then
$t\vec{w}+\vec{x}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}$
and $A\left(t\vec{w}+\vec{x}\right)=lt+A\vec{x}\in P^{u}$.

$\left(\text{b}\right)\Rightarrow\left(\text{a}\right)$. We may suppose that
the entries of $A$ are in $\mathbb{Z}\left[i\right]$ as we could replace $A$
by $nA$ for a suitable $n\in\mathbb{N}$.

Suppose that
$\vec{1}\notin\left\\{A\vec{x}:\vec{x}\in\left(\mathbb{Q}\left[i\right]\right)^{v}\right\\}$.
Then there exists $\vec{u}\in\left(\mathbb{Z}\left[i\right]\right)^{v}$ such
that $\vec{u}\cdot A\vec{x}=0$ for every
$\vec{x}\in\left(\mathbb{Q}\left[i\right]\right)^{v}$, but
$\vec{u}.\vec{1}\neq 0$.

Choose a Gaussian prime number
$r\in\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}$ satisfying
$\left|r\right|>\left|l\right|$ and
$\left|r\right|>\left|l\vec{u}\cdot\vec{1}\right|$. Let $q$ be a minimal
idempotent in $\beta\mathbb{Z}\left[i\right]$ and let $p=l+q$. Then
$P=\left\\{t\in\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}:t\equiv
l\left(\mod r\right)\right\\}\in p$($t\equiv l\left(\mod
r\right)\Rightarrow$$t-l$ is divisible by $r$ in $\mathbb{Z}\left[i\right]$)
(by [DJ, Lemma 2.1]). It follows from $\left(\text{b}\right)$ that there
exists $\vec{z}\in\left(\mathbb{Z}\left[i\right]\right)^{v}$ such that
$A\vec{z}\in P^{u}$ and hence that $\left(A\vec{z}\right)_{i}\equiv
l\left(\mod r\right)$ for $i\in\left\\{1,2,\ldots,u\right\\}$. Thus,
$A\vec{z}\equiv l\vec{1}\left(\mod\vec{r}\right)$, a contradiction, as
$\vec{u}\cdot A\vec{z}\equiv 0\left(\mod r\right)$, but $\vec{u}\cdot
l\vec{1}\neq 0\left(\mod r\right)$. ∎

##  2\. Large sets in $\mathbb{Z}\left[i\right]$

There are several notions of largeness that make sense in any semigroup. The
notion of “central” sets is one of these. Among the others are the notions of
“syndetic”, “piecewise syndetic”, “IP”, and “$\Delta$” sets.

######  Definition 2.1.

Let $\left(S,+\right)$ be a commutative semigroup and let $B\subseteq S$.

(a) The set $B$ is syndetic if and only if there exists some
$G\in\mathcal{P}_{f}\left(S\right)=\left\\{H\subseteq S:H\text{ is finite and
nonempty}\right\\}$ such that $S=\cup_{t\in G}-t+B$.

(b) The set $B$ is piecewise syndetic if and only if there exists some
$G\in\mathcal{P}_{f}\left(S\right)$ such that for every
$F\in\mathcal{P}_{f}\left(S\right)$ there exists $x\in S$ such that
$F+x\subseteq\cup_{t\in G}-t+B$.

(c) The set $B$ is an IP set if and only if there exists a sequence
$\left\langle x_{n}\right\rangle_{n=1}^{\infty}$ in $S$ such that
$FS\left(\left\langle x_{n}\right\rangle_{n=1}^{\infty}\right)\subseteq B$,
where $FS\left(\left\langle
x_{n}\right\rangle_{n=1}^{\infty}\right)=\left\\{\sum_{n\in
F}x_{n}:F\in\mathcal{P}_{f}\left(\mathbb{N}\right)\right\\}$ and the sums are
taken in increasing order of indices.

(d) The set $B$ is a $\Delta$ set if and only if there exists a sequence
$\left\langle x_{n}\right\rangle_{n=1}^{\infty}$ in $S$ such that for every
$n,m\in\mathbb{N}$ with $n<m$, $x_{m}\in\left(x_{n}+B\right)$.

Notice that, if $S$ can be embedded in a group $G$, $B$ is a $\Delta$ set if
and only if there is a sequence $\left\langle
x_{n}\right\rangle_{n=1}^{\infty}$ such that
$\left\\{-x_{n}+x_{m}:m,n\in\mathbb{N}\text{ and }n<m\right\\}\subseteq B$.
Notice also that any IP set is a $\Delta$ set. (Given $\left\langle
x_{n}\right\rangle_{n=1}^{\infty}$ with $FS\left(\left\langle
x_{n}\right\rangle_{n=1}^{\infty}\right)\subseteq B$ and given
$n\in\mathbb{N}$, let $y_{n}=\sum_{t=1}^{n}x_{n}$). Given any property
$\mathcal{E}$ of subsets of a set $X$, there is a dual property
$\mathcal{E^{*}}$ defined by specifying that a subset $B$ of $X$ is an
$\mathcal{E^{*}}$ set if and only if $B\cap A\neq\emptyset$ for every
$\mathcal{E}$ set $A$.

######  Definition 2.2.

Let $\left(S,+\right)$ be a commutative semigroup and let $B\subseteq S$. Then
B is a central∗ set if and only $B\cap A\neq\emptyset$ if for every central
set $A$ in $S$. Also, $B$ is a PS∗ set if and only if $B\cap A\neq\emptyset$
for every piecewise syndetic set $A$ in $S$, $B$ is an IP∗ set if and only if
$B\cap A\neq\emptyset$ for every IP set $A$ in $S$, $B$ is a syndetic∗ set if
and only if $B\cap A\neq\emptyset$ for every syndetic set $A$ in $S$, and $B$
is a $\Delta^{*}$ set if and only if $B\cap A\neq\emptyset$ for every set $A$
in $S$.

The concept of “syndetic∗” is more commonly referred to as “thick”, and we
shall follow this practice.

The sets $\Delta$ and $\Delta^{*}$ sets are interesting because they arise as
sets of recurrence, which in turn have significant combinatorial properties.
(See [F].) The other notions discussed above have simple, and useful,
algebraic characterizations in terms of $\beta S$.

######  Lemma 2.3.

Let $\left(S,+\right)$ be a commutative semigroup and let $B\subseteq S$.

(a) $A$ is piecewise syndetic if and only if $\bar{A}\cap K\left(\beta
S\right)\neq\emptyset$.

(b) $A$ is IP if and only if there is some idempotent of $\beta S$ in
$\bar{A}$.

(c) $A$ is syndetic if and only if for every left ideal $L$ of $\beta S$,
$\bar{A}\cap L\neq\emptyset$.

(d) $A$ is central if and only if there is some minimal idempotent of $\beta
S$ in $\bar{A}$.

(e) $A$ is central∗ if and only if every minimal idempotent of $\beta S$ is in
$\bar{A}$.

(f ) $A$ is thick if and only if $A$ contains a left ideal of $\beta S$.

(g) $A$ is IP∗ if and only if every idempotent of $\beta S$ is in $\bar{A}$.

(h) $A$ is PS∗ if and only if $K\left(\beta S\right)\subseteq\bar{A}$.

###### Proof.

Statement (a) is [HS, Theorem 4.40], (b) is [HS, Theorem 5.12], (c) is [BHM,
Theorem 2.9(d)], and (d) is the definition of central. Statements (e), (f),
(g), and (h) follow easily from statements (d), (c), (b), and (a)
respectively. ∎

Now we will prove a lemma we need.

######  Lemma 2.4.

Let $A$ be a $u\times$v image partition regular matrix over
$\mathbb{Q}\left[i\right]$. The following are equivalent:

(a) There exists $\vec{s}\in\mathbb{Q}\left[i\right]^{v}$ such that
$A\vec{s}=\vec{1}$.

(b) There exists $l\in\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}$
such that, if $p$ is in the smallest ideal of $\beta\mathbb{Z}\left[i\right]$
and $l\mathbb{Z}\left[i\right]\in p$, then, for every $P\in p$, there exists
$\vec{z}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)$
such that $A\vec{z}\in P^{u}$.

###### Proof.

(a) $\Rightarrow$ (b). We can choose
$\vec{u}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}$
and $l\in\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}$ such that
$A\vec{z}=\vec{l}$, where $\vec{l}=\left(ll\ldots
l\right)^{T}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{u}$.
Suppose that $p$ is in the smallest ideal of $\beta\mathbb{Z}\left[i\right]$
and that $l\mathbb{Z}\left[i\right]\in p$. Let $P\in p$. There exists a
minimal idempotent $q\in\beta\mathbb{Z}\left[i\right]$ such that $p=p+q$ [HS,
Theorem 2.8 and Lemma 1.30]. Let $P^{\prime}=\left\\{v\in P:-v+P\in
q\right\\}$. Then $P^{\prime}\in p$. Since $l\mathbb{Z}\left[i\right]\in p$ we
can choose $m\in\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}$ such that
$lm\in P^{\prime}$. Let $Q=-lm+P\in q$. By [DJ, Theorem 3.1(h)], there exists
$\vec{z}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}$
such that $A\vec{z}\in Q^{v}$ and $\left|z_{i}+mu_{i}\right|>0$ for every
$i\in\left\\{1,2,\ldots,v\right\\}$. (The fact that the entries of $\vec{z}$
can be chosen to be arbitrarily large follows from [DJ, Lemma 2.3]. For every
$r\in\mathbb{N}$,
$\left\\{\vec{x}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}:\left|x_{i}\right|>r\text{
for all }i\in\left\\{1,2,\ldots,v\right\\}\right\\}$ is a member of every
idempotent in $\beta\left(\mathbb{Z}\left[i\right]^{v}\right)$.) Then
$m\vec{z}+\vec{x}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}$
and $A\left(m\vec{z}+\vec{x}\right)=m\vec{l}+A\vec{x}\in P^{u}$.

(b) $\Rightarrow$ (a). We may suppose that the entries of $A$ are in
$\mathbb{Z}\left[i\right]$, as we could replace $A$ by $nA$ for a suitable
$n\in\mathbb{N}$.

Suppose that
$\vec{1}\notin\left\\{A\vec{z}:\vec{z}\in\mathbb{Q}\left[i\right]^{v}\right\\}$.
Then there exists $\vec{u}\in\mathbb{Z}^{v}$ such that $\vec{u}\cdot
A\vec{z}=0$ for every $\vec{z}\in\mathbb{Q}\left[i\right]^{v}$, but
$\vec{u}\cdot\vec{1}\neq 0$.

Choose a prime number $r$ satisfying $r>\left|l\right|$ and
$r>\left|l\vec{u}\cdot\vec{1}\right|$. Let $q$ be a minimal idempotent in
$\beta\mathbb{Z}\left[i\right]$ and let $p=l+q$. Then

$P=\left\\{v\in\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}:v\equiv
l\left(\mod r\right)\right\\}\in p\ \ \ \ \ \text{ (by [Lemma 2.14]).}$

It follows from (b) that there exists $\vec{z}\in\mathbb{Z}\left[i\right]$
such that $A\vec{z}\in P^{u}$ and hence that $A\vec{z}\equiv
l\vec{1}\,\left(\mod r\right)$ . This is a contradiction, as $\vec{u}\cdot
A\vec{x}\equiv 0\,\left(\mod r\right)$, but $\vec{u}\cdot l\vec{1}$ is not
divisible by $r$. ∎

The following theorem relates image partition regular matrices and piecewise
syndetic sets.

######  Theorem 2.5.

Let $A$ be a $u\times v$ image partition regular matrix over
$\mathbb{Q}\left[i\right]$. The following statements are equivalent:

(a) For every piecewise syndetic subset $P$ of
$\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}$, there exists
$\vec{z}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}$
such that $A\vec{z}\in P^{u}$;

(b) There exists $\vec{z}\in\left(\mathbb{Z}\left[i\right]\right)^{v}$such
that $A\vec{z}=\vec{1}$, where $\vec{1}$ denotes the vector in
$\left(\mathbb{Z}\left[i\right]\right)^{u}$ whose entries are all equal to
$1$.

###### Proof.

This follows easily from Lemma 2.3(a) and the proof of (a) $\Leftrightarrow$
(b) in Lemma 2.4, taking $l=1$. ∎

In [BHK] it was shown that if “large” meant any of “$\Delta$”, “IP”,
“central”, “central∗”, “IP∗”, or “$\Delta^{*}$”, and $B$ is a large subset of
$\mathbb{N}$, then for every positive $\alpha\in\mathbb{R}$ and every
$\gamma\in\mathbb{R}$ with $0<\gamma<1$, $\left\\{\left\lfloor\alpha
n+\gamma\right\rfloor:n\in B\right\\}$ is also large (in the same sense). In
[F1] it was shown that if $B$ is a piecewise syndetic subset of
$\mathbb{Z}$,$l\in\mathbb{N}$, and
$AP^{l}=\left\\{\left(a,a+d,\ldots,a+\left(l-1\right)d\right):a,d\in\mathbb{Z}\right\\}$,
the group of length $l$ arithmetic progressions (including the constant ones),
then $B^{l}\cap AP^{l}$ is piecewise syndetic in $AP^{l}$. In [BH] a
systematic study of this latter phenomenon was undertaken. These results apply
in the current context in terms of when the set of images contained in a given
set is large among the set of all images.

######  Theorem 2.6.

Let $A$ be a $u\times v$ matrix with entries from $\mathbb{Q}\left[i\right]$,
let
$I=\left\\{A\vec{z}:\vec{z}\in\left(\mathbb{Z}\left[i\right]\right)^{v}\right\\}\cap\left(\mathbb{Z}\left[i\right]\right)^{u}$,
and let $C\subseteq\mathbb{Z}\left[i\right]$.

(a) If $I\neq\emptyset$, “large” is any of “IP∗”, “$\Delta^{*}$”, “PS*”, or
“central*”, and $C$ is large in $\mathbb{Z}\left[i\right]$, then $I\cap C^{u}$
is large in $I$.

(b) If $\bar{1}\in I$, “large” is any of “piecewise syndetic”, “central”, or
“thick”, and $C$ is large in $\mathbb{Z}\left[i\right]$, then $I\cap C^{u}$ is
large in $I$.

###### Proof.

(a) For IP* and $\Delta$*, [BH, Corollary 2.3] requires only that $I$ be a
subsemigroup of $\left(\mathbb{Z}\left[i\right]\right)^{u}$. For PS* and
central*, [BH, Corollary 2.7] requires in addition that for each
$i\in\left\\{1,2,\ldots,u\right\\}$, the $i$-th projection
$\pi_{i}\left[I\right]$ be piecewise syndetic in $\mathbb{Z}\left[i\right]$.
This trivially holds because, if $z\in\pi_{i}\left[I\right]$, then
$z\mathbb{Z}\left[i\right]\in\pi_{i}\left[I\right]$.

(b) Letting $E=I$, we have that

$\left\\{\begin{array}[]{c}a\\\ a\\\ \vdots\\\
a\end{array}:a\in\mathbb{Z}\left[i\right]\right\\}\subseteq E$

so that [BH, Theorem 3.7] applies. ∎

We shall be concerned for the rest of this section with establishing analogues
of [DJ, Theorem 3.1(i)] for the other notions of largeness. That is, we wish
to determine conditions that guarantee that if a set $C$ is “large” in
$\mathbb{Z}\left[i\right]$, then
$\left\\{\vec{x}\in\left(\mathbb{Z}\left[i\right]\right)^{v}:A\vec{x}\in
C^{u}\right\\}$ is “large” in $\left(\mathbb{Z}\left[i\right]\right)^{v}$.

######  Lemma 2.7.

Let $A$ be a $u\times v$ matrix with entries from $\mathbb{Z}\left[i\right]$,
define
$\varphi:\left(\mathbb{Z}\left[i\right]\right)^{v}\rightarrow\left(\mathbb{Z}\left[i\right]\right)^{u}$
by $\varphi\left(x\right)=A\vec{x}$, and let
$\tilde{\varphi}:\beta\left(\mathbb{Z}\left[i\right]^{v}\right)\rightarrow\left(\beta\mathbb{Z}\left[i\right]\right)^{u}$
be its continuous extension. Then $\tilde{\varphi}$ is a homomorphism and
$K\left(\left(\beta\mathbb{Z}\left[i\right]\right)^{u}\right)=\left(K\left(\beta\mathbb{Z}\left[i\right]\right)\right)^{u}$.

(a) If there exists $\vec{z}\in\left(\mathbb{Z}\left[i\right]\right)^{v}$ such
that $A\vec{z}\in\left(\mathbb{Z}\left[i\right]\right)^{u}$, then
$\tilde{\varphi}\left[\beta\left(\mathbb{Z}\left[i\right]\right)^{v}\right]\cap
K\left(\left(\beta\mathbb{Z}\left[i\right]\right)\right)^{u}\neq\emptyset$.

(b) If for all $\vec{z}\in\mathbb{Z}\left[i\right]^{v}$,
$A\vec{z}\in\mathbb{Z}\left[i\right]^{u}$, then
$\tilde{\varphi}\left[K\left(\beta\left(\mathbb{Z}\left[i\right]^{v}\right)\right)\right]\subseteq
K\left(\left(\beta\mathbb{Z}\left[i\right]\right)\right)^{u}$.

###### Proof.

By [HS, Corollary 4.22] we have that $\tilde{\varphi}$ is a homomorphism, and
by [HS, Theorem 2.23]
$K\left(\left(\beta\mathbb{Z}\left[i\right]\right)^{u}\right)=\left(K\left(\beta\mathbb{Z}\left[i\right]\right)\right)^{u}$.

(a). Since
$\tilde{\varphi}\left[\beta\left(\mathbb{Z}\left[i\right]\right)^{v}\right]=\overline{\left\\{A\vec{z}:\vec{z}\in\left(\mathbb{Z}\left[i\right]\right)^{v}\right\\}}$,
we need to show that

$\overline{\left\\{A\vec{z}:\vec{z}\in\left(\mathbb{Z}\left[i\right]\right)^{v}\right\\}}\cap\left(K\left(\beta\mathbb{Z}\left[i\right]\right)\right)^{u}\neq\emptyset.$

Pick $\vec{z}\in\mathbb{Z}\left[i\right]^{v}$ such that
$A\vec{z}\in\mathbb{Z}\left[i\right]^{u}$. Pick any minimal idempotent $p$ in
$\beta\mathbb{Z}\left[i\right]$. Then by Lemma [DJ, Lemma 2.2]
$\vec{p}=\vec{y}p\in\left(K\left(\beta\mathbb{Z}\left[i\right]\right)\right)^{u}$.
To see that
$\vec{p}\in\overline{\left\\{A\vec{w}:\vec{w}\in\mathbb{Z}\left[i\right]^{v}\right\\}}$,
let $U$ be a neighborhood of $\vec{p}$ and for each
$i\in\left\\{1,2,\ldots,u\right\\}$, pick $D_{i}\in p$ such that
$\times_{i=1}^{u}\overline{y_{i}D_{i}}\subseteq U$. Pick
$a\in\bigcap_{i=1}^{u}D_{i}$. Then $A\left(a\vec{z}\right)=a\vec{y}\in U$.

(b). By part (a),
$\tilde{\varphi}\left[\beta\left(\mathbb{Z}\left[i\right]\right)^{v}\right]\cap
K\left(\left(\beta\mathbb{Z}\left[i\right]\right)\right)^{u}\neq\emptyset$, so
by [HS, Theorem 1.65],
$K\left(\tilde{\varphi}\left[\beta\left(\mathbb{Z}\left[i\right]\right)^{v}\right]\right)=\tilde{\varphi}\left[\beta\left(\mathbb{Z}\left[i\right]\right)^{v}\right]\cap
K\left(\left(\beta\mathbb{Z}\left[i\right]\right)\right)^{u}$ (because
$\tilde{\varphi}\left[\beta\left(\mathbb{Z}\left[i\right]^{v}\right)\right]\subseteq\left(\beta\mathbb{Z}\left[i\right]\right)^{u}$).
Also by [HS, Exercise 1.7.3],
$K\left(\tilde{\varphi}\left[\beta\left(\mathbb{Z}\left[i\right]^{v}\right)\right]\right)=\tilde{\varphi}\left[K\left(\beta\left(\mathbb{Z}\left[i\right]^{v}\right)\right)\right]$.
∎

######  Theorem 2.8.

Let $A$ be a $u\times v$ matrix with entries from $\mathbb{Q}\left[i\right]$
and assume that for all $\vec{z}\in\mathbb{Z}\left[i\right]^{v}$, every entry
of $A\vec{z}$ has both positive real and imaginary parts. If “large” is any of
“IP*”, “$\Delta$*”, or “central*”, and $C$ is large in
$\mathbb{Z}\left[i\right]$, then
$W=\left\\{\vec{z}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}:A\vec{z}\in
C^{u}\right\\}$ is large in $\mathbb{Z}\left[i\right]^{v}$.

###### Proof.

We show first that it suffices to prove the theorem under the additional
assumption that all entries of $A$ are in $\mathbb{Z}\left[i\right]$. Indeed,
suppose we have done so and pick $d\in\mathbb{N}$ such that all entries of
$nA$ are in $\mathbb{Z}\left[i\right]$. We claim that $dC$ is large in
$\mathbb{Z}\left[i\right]$, which we check individually.

Assume first that “large” is “IP*” and let a sequence $\left\langle
z_{n}\right\rangle_{n=1}^{\infty}$ in $\mathbb{Z}\left[i\right]$ be given. By
[HS, Theorem 5.14] and [DJ, Lemma 2.1] pick a sum subsystem $\left\langle
w_{n}\right\rangle_{n=1}^{\infty}$ of $\left\langle
z_{n}\right\rangle_{n=1}^{\infty}$ such that $z\mid y_{n}$ for each $n$. Pick
$a\in C\cap
FS\left(\left\langle\frac{y_{n}}{z}\right\rangle_{n=1}^{\infty}\right)$. Then
$da\in dC\cap FS\left(\left\langle z_{n}\right\rangle_{n=1}^{\infty}\right)$.

Next assume that “large” is “$\Delta$*”. Let a set $B$ in
$\mathbb{Z}\left[i\right]$ be given and choose a sequence $\left\langle
z_{n}\right\rangle_{n=1}^{\infty}\subseteq\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}$
such that for every $n,m\in\mathbb{N}$ with $n<m$, $z_{m}\in z_{n}+B$. (In
particular, for each $n<m,\ \left|z_{n}\right|<\left|z_{m}\right|$.) By
passing to a subsequence, we may presume that for each $n<m$, $z_{n}\equiv
z_{m}\left(\text{mod }d\right)$. Pick $j\in\left\\{1,2,\ldots,d\right\\}$ such
that for each $n\in\mathbb{N}$, $z_{n}+j\equiv 0\left(\text{mod }d\right)$.
Then $\left\langle\frac{z_{n}+j}{d}\right\rangle_{n=1}^{\infty}$ is a sequence
in $\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}$ so pick $n<m$ such
that $\frac{z_{m}+j}{d}-\frac{z_{n}+j}{d}\in C$. Then $z_{m}-z_{n}\in dC$.

Finally assume that “large” is “central*”. Let $p$ be a minimal idempotent in
$\beta\mathbb{Z}\left[i\right]$. By Lemma [DJ, Lemma 2.2], $\frac{1}{d}p$ is a
minimal idempotent so $C\in\frac{1}{d}p$ and consequently $dC\in p$. Since we
have established that $dC$ is large, we have that
$\left\\{\vec{z}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}:dA\vec{z}\in\left(dC\right)^{u}\right\\}$
is large, and

$\left\\{\vec{z}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}:dA\vec{z}\in\left(dC\right)^{u}\right\\}=\left\\{\vec{z}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}:A\vec{z}\in
C^{u}\right\\}$

.

Thus we assume that all entries of $A$ are in $\mathbb{Z}\left[i\right]$.
Define
$\varphi:\mathbb{Z}\left[i\right]^{v}\rightarrow\mathbb{Z}\left[i\right]^{u}$
by $\varphi\left(\vec{z}\right)=A\vec{z}$, and let
$\tilde{\varphi}:\beta\left(\mathbb{Z}\left[i\right]^{v}\right)\rightarrow\left(\beta\mathbb{Z}\left[i\right]\right)^{u}$
be its continuous extension. Assume first that “large” is “IP*”. Let $p$ be an
idempotent in $\beta\left(\mathbb{Z}\left[i\right]^{v}\right)$. We need to
show that $W\in p$. Since $\tilde{\varphi}$ is a homomorphism,
$\tilde{\varphi}\left(p\right)$ is an idempotent in
$\left(\beta\mathbb{Z}\left[i\right]\right)^{u}$ and so $\bar{C}^{u}$ is a
neighborhood of $\tilde{\varphi}\left(p\right)$, and hence $W\in p$ as
required.

Next assume that “large” is “$\Delta$*”. Let $B$ be a set in
$\mathbb{Z}\left[i\right]^{v}$ and pick a sequence
$\left\langle\vec{z}_{n}\right\rangle_{n=1}^{\infty}$ in
$\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}$ such
that for every $n,m\in\mathbb{N}$ with $n<m$, $\vec{z_{m}}\in\vec{z_{n}}+B$.
In particular, for each $n<m$, we have
$\vec{z_{m}}-\vec{z_{n}}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}$.
We need to show that there exists $n<m$ such that $\vec{z_{m}}-\vec{z_{n}}\in
W$.

For each $n\in\mathbb{N}$, let $\vec{y_{n}}=A\vec{z_{n}}$. Notice that, for
$n<m$ we have that all real and imaginary entries of
$A\left(\vec{z_{m}}-\vec{z_{n}}\right)$ are positive, and consequently modulus
of each entry of $\vec{y_{m}}$ is larger than the modulus of corresponding
entry of $\vec{y_{n}}$. By Ramsey’s Theorem ([GRS, Theorem 1.5]), pick an
infinite subset $D_{1}$ of $\mathbb{N}$ such that for all $n<m$ in $D_{1}$,
$y_{m}^{1}-y_{n}^{1}\in C$ or for all $n<m$ in $D_{1}$,
$y_{m}^{1}-y_{n}^{1}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)\setminus
C$. Since $C$ is a $\Delta$* set, the latter alternative is impossible, so the
former must hold. Inductively, given $i\in\left\\{1,2,\ldots,u-1\right\\}$,
choose by Ramsey’s Theorem an infinite subset $D_{i+1}$ of $D_{i}$ such that
for all $n<m$ in $D_{i+1}$, $y_{m}^{i+1}-y_{n}^{i+1}\in C$. Having chosen
$D_{u}$ pick $n<m$ in $D_{u}$. Then $\vec{z_{m}}-\vec{z_{n}}\in W$. Finally
assume that “large” is “central*”, and let $p$ be a minimal idempotent in
$\beta\left(\mathbb{Z}\left[i\right]^{v}\right)$. By Lemma 2.7,
$\tilde{\varphi}\left(p\right)$ is an idempotent and
$\tilde{\varphi}\left(p\right)\in\left(K\left(\beta\mathbb{Z}\left[i\right]\right)\right)^{u}$
so that $\tilde{\varphi}\left(p\right)\bar{C}^{u}$. ∎

The requirement in Theorem 2.8 that for all , every
$\vec{z}\in\mathbb{Z}\left[i\right]^{v}$ entry of $A\vec{z}$ be positive may
not be omitted ( see [HLS]).

######  Theorem 2.9.

Let $A$ be a $u\times v$ matrix with entries from $\mathbb{Z}\left[i\right]$
and assume that for all
$\vec{z}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}$,
every entry of
$A\vec{z}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{u}$.
If $C$ is PS* in $\mathbb{Z}\left[i\right]$, then
$W=\left\\{\vec{z}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}:A\vec{z}\in
C^{u}\right\\}$ is PS* in $\mathbb{Z}\left[i\right]^{v}$.

###### Proof.

Define
$\varphi:\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}\rightarrow\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}$
by $\varphi\left(\vec{z}\right)=A\vec{z}$, and let
$\tilde{\varphi}:\beta\left(\mathbb{Z}\left[i\right]^{v}\right)\rightarrow\left(\beta\mathbb{Z}\left[i\right]\right)^{u}$
be its continuous extension. Let $p\in
K\left(\beta\left(\mathbb{Z}\left[i\right]^{v}\right)\right)$. By Lemma 2.7,
$\tilde{\varphi}\left(p\right)\in\left(K\left(\beta\mathbb{Z}\left[i\right]\right)\right)^{u}$
and thus $\tilde{\varphi}\left(p\right)\in\bar{C}^{u}$ and thus $W\in p$ as
required. ∎

######  Theorem 2.10.

Let A be a $u\times v$ image partition regular matrix with entries from
$\mathbb{Z}\left[i\right]$. If $C$ is thick in $\mathbb{Z}\left[i\right]$,
then
$W=\left\\{\vec{z}\in\left(\mathbb{Z}\left[i\right]\setminus\left\\{0\right\\}\right)^{v}:A\vec{z}\in
C^{u}\right\\}$ is thick in $\mathbb{Z}\left[i\right]^{v}$.

###### Proof.

Since $C$ is thick, pick a left ideal $L$ of $\beta\mathbb{Z}\left[i\right]$
such that $L\subseteq\bar{C}$. Pick by [HS, Corollary 2.6] a minimal
idempotent $p\in L$. Define
$\varphi:\mathbb{Z}\left[i\right]^{v}\rightarrow\mathbb{Z}\left[i\right]^{u}$
by $\varphi\left(\vec{z}\right)=A\vec{z}$, and let
$\tilde{\varphi}:\beta\left(\mathbb{Z}\left[i\right]^{v}\right)\rightarrow\left(\beta\mathbb{Z}\left[i\right]\right)^{u}$
be its continuous extension. Let $\bar{p}=\left(p,p,\ldots p\right)^{T}$ and
pick by [DJ, Lemma 2.3] a minimal idempotent
$q\in\beta\left(\mathbb{Z}\left[i\right]^{v}\right)$ such that
$\tilde{\varphi}\left(q\right)=p$. (By [DJ, Theorem 3.1(h)], $p$ satisfies the
hypotheses of [DJ, Theorem 2.3].) We claim that
$\tilde{\varphi}\left[\beta\left(\mathbb{Z}\left[i\right]^{v}\right)+q\right]\subseteq\bar{C}^{u}$
so that $\beta\left(\mathbb{Z}\left[i\right]^{v}\right)+q\subseteq\bar{W}$ as
required. To this end, let
$r\in\beta\left(\mathbb{Z}\left[i\right]^{v}\right)$ and let
$i\in\left\\{1,2,\ldots,u\right\\}$. Then
$\pi_{i}\circ\tilde{\varphi}\left(r+q\right)=\pi_{i}\left(\tilde{\varphi}\left(r\right)\right)+p\in\beta\mathbb{Z}\left[i\right]+p\subseteq
L\subseteq\bar{C}$. ∎

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