Original Article 11affiliationtext: Image Analysis, German Center for
Neurodegenerative Diseases (DZNE), Bonn, Germany 22affiliationtext: Population
Health Sciences, German Center for Neurodegenerative Diseases (DZNE), Bonn,
Germany 33affiliationtext: Institute for Medical Biometry, Informatics and
Epidemiology (IMBIE), Faculty of Medicine, University of Bonn, Bonn, Germany
44affiliationtext: A.A. Martinos Center for Biomedical Imaging, Massachusetts
General Hospital, Boston MA, USA  55affiliationtext: Department of Radiology,
Harvard Medical School, Boston MA,USA Santiago Estrada, DZNE, Venusberg-Campus
1 BLDG 99 , 53127 Bonn, Germany<EMAIL_ADDRESS>Diet-BB Competence
Cluster in Nutrition Research, Federal Ministry of Education and Research
(BMBF), Grant Numbers: 01EA1410C and FKZ: 01EA1809C ; JPI HDHL on Biomarkers
for Nutrition and Health,BMBF, Grant Number: 01EA1705B; National Institutes of
Health (NIH),Grant Number: R01NS083534 and R01LM012719

# FatSegNet: A Fully Automated Deep Learning Pipeline for Adipose Tissue
Segmentation on Abdominal Dixon MRI

Santiago Estrada  Ran Lu  Sailesh Conjeti  Ximena Orozco-Ruiz  Joana Panos-
Willuhn  Monique M.B. Breteler  Martin Reuter

###### Abstract

Purpose: Introduce and validate a novel, fast, and fully automated deep
learning pipeline (FatSegNet) to accurately identify, segment, and quantify
visceral and subcutaneous adipose tissue (VAT and SAT) within a consistent,
anatomically defined abdominal region on Dixon MRI scans.

Method: FatSegNet is composed of three stages: (i) Consistent localization of
the abdominal region using two 2D-Competitive Dense Fully Convolutional
Networks (CDFNet), (ii) Segmentation of adipose tissue on three views by
independent CDFNets, and (iii) View-aggregation. FatSegNet is validated by: 1)
comparison of segmentation accuracy (sixfold cross-validation), 2) test-retest
reliability, 3) generalizability to randomly selected manually re-edited
cases, and 4) replication of age and sex effects in the Rhineland Study - a
large prospective population cohort.

Results: The CDFNet demonstrates increased accuracy and robustness compared to
traditional deep learning networks. FatSegNet Dice score outperforms manual
raters on VAT (0.850 vs. 0.788),and produces comparable results on SAT (0.975
vs. 0.982). The pipeline has excellent agreement for both test-retest (ICC VAT
0.998 and SAT 0.996) and manual re-editing (ICC VAT 0.999 and SAT 0.999).

Conclusion: FatSegNet generalizes well to different body shapes, sensitively
replicates known VAT and SAT volume effects in a large cohort study, and
permits localized analysis of fat compartments. Furthermore, it can reliably
analyze a 3D Dixon MRI in $\sim$ 1 min, providing an efficient and validated
pipeline for abdominal adipose tissue analysis in the Rhineland Study.

###### keywords:

Subcutaneous adipose tissue, Visceral adipose tissue, Dixon MRI, Neural
networks, Deep learning, Semantic segmentation

##  1 Introduction

The excess of body fat depots is an increasing major public health issue
worldwide and an important risk factor for the development of metabolic
disorders and reduced quality of life [1, 2]. While the body mass index (BMI)
is a widely used indicator of adipose tissue accumulation in the body, it does
not provide information on fat distribution [3] – neither with respect to
different fat tissue types nor with respect to deposit location. Different
compartments of adipose tissue are associated with different
physiopathological effects [4, 5]. Abdominal adipose tissue (AAT), composed of
subcutaneous and visceral adipose tissue (SAT and VAT), has long been
associated with an increased risk of chronic cardiovascular diseases, glucose
impairment, and dyslipidemia [6, 7]. Recently, several studies have indicated
a stronger relation between the accumulation of VAT with an adverse metabolic
and inflammatory profile compared to SAT [8, 9]. Therefore, an accurate and
independent measurement of VAT and SAT volumes (VAT-V and SAT-V) is of
significant clinical and research interest.

Currently, the gold standard for measuring VAT-V and SAT-V is the manual
segmentation of abdominal fat images from Dixon magnetic resonance (MR) scans
– a very expensive and time-consuming process. Thus, especially for large
studies, automatic segmentation methods are required. However, achieving good
accuracy is challenging due to complex AAT structures, a wide variety of VAT
shapes, large anatomical differences across subjects, and the inherent
properties of the Dixon images: low intensity contrast between adipose tissue
classes, inhomogeneous signals, and potential organ motion. So far, those
limitations impeded the wide-spread implementation of automatic and semi-
automatic techniques based on intensity and shape features, such as fuzzy-
clustering [10], k-means clustering [11], graph cut [12, 13] active contour
methods [14], and statistical shape models [15].

Recently, fully convolutional neural networks (F-CNNs) [16, 17] have been
widely adopted in the computer vision community for pixel/voxel-wise image
segmentation in an end-to-end fashion to overcome above-mentioned challenges.
With these methods there is no need to extract manual features, divide images
into patches, or implement sliding window techniques. F-CNNs can automatically
extract intrinsic features and integrate global context to resolve local
ambiguities thereby improving the results of the predicted models [17]. Langer
et al. [18] proposed a three channel UNet for AAT segmentation, which is a
conventional architecture for 2D medical image segmentation [19]. While this
method showed promising results, we demonstrate that our network architecture
outperforms the traditional UNet for segmenting AAT on our images with a wide
range of anatomical variation. More recent architectures such as the SD-Net
[20] and Dense-UNet, a densely connected network [21], have the potential to
improve generalizability and robustness by encouraging feature re-usability
and strengthening information propagation across the network [21]. In prior
work, we introduced a competitive dense fully convolutional network (CDFNet)
[22] as a new 2D F-CNN architecture that promotes feature selectivity within a
network by introducing maximum attention through a maxout activation unit
[23]. The maxout boosts performance by allowing the creation of specialized
sub-networks that target a specific structure during training [24]. Therefore,
this approach facilitates the learning of more complex structures [22, 24]
with the added benefit of reducing the number of training parameters relative
to the aforementioned networks.

In this paper, we propose FatSegNet, a novel fully automated deep learning
pipeline based on our CDFNet architecture to localize and segment VAT and SAT
on abdominal Dixon MR images from the Rhineland Study, an ongoing large
population-based cohort study [25, 26]. To constrain AAT segmentations to a
consistent anatomically defined region, the proposed pipeline consists of
three stages:

  1. 1.

Localization of the abdominal region using a semantic segmentation approach by
implementing CDFNet models on sagittal and coronal planes; we use the lumbar
vertebrae positions as reference points for selecting the region of interest.

  2. 2.

Segmentation of VAT and SAT within the abdominal region through 2D CDFNet
models on three different planes (axial, sagittal and coronal).

  3. 3.

A view-aggregation stage where the previous generated label maps are combined
to generate a final 3D segmentation.

We initially evaluate and compare the individual stages of the pipeline with
other deep learning approaches in a sixfold cross-validation. We show that the
proposed network architecture (CDFNet) improves segmentation performance and
simultaneously reduces the number of required training parameters in step 1
and 2. After asserting segmentation accuracy, we evaluate the whole pipeline
(FatSegNet) with respect to robustness and reliability against two independent
test sets: a manually edited and a test-retest set. Finally, we present a case
study on unseen data comparing the VAT-V and SAT-V calculated from the
FatSegNet segmentations against BMI to replicate age and sex effects on these
volumes in a large cohort.

##  2 Methods

###  2.1 Data

####  2.1.1 MR imaging acquisition

MR image acquisition was performed at two different sites both with identical
3T Siemens MAGNETOM Prisma MR scanners (Siemens Healthcare, Erlangen,
Germany). The body coil was used for signal reception of a three-dimensional
two-point Dixon sequence (acquisition time =12 $s$, echo time TE1=1.23 $ms$,
TE2=2.46 $ms$, repetition time TR=4.12 $ms$, axial field of view =500
$mm\times$ 437 $mm$, flip angle =6 degrees, left-right readout bandwidth =750
$Hz\\-/pixel$, partial Fourier factor 6/8 $\times$ 5/8). Based on a preceding
moving-table abdominal localizer, the field-of-view was centered on the middle
of the third lumbar vertebra (L, L3). Data were acquired during a single
breath-hold in supine position with arms placed at the sides. The image
resolution was finally interpolated from 2.0 $mm\times$ 2.7 $mm\times$ 10.0
$mm$ to 2.0 $mm\times$ 2.0 $mm\times$ 5.0 $mm$ (matrix size =256 $\times$ 224
$\times$ 72).

Figure 1: MR Dixon images and ground truth from two subjects with different
BMI (obese (upper), normal (lower). a) Fat images: axial plane. b) Initial
manual segmentation (blue: SAT, green: VAT, orange: bone and surrounding
structures). c) Ground truth with additional synthetic class (red: other-
tissue) and filled-in bone structures (orange). d) Fat images: coronal plane.
e) Ground truth for localization of region of interest (red: thoracic region,
white: abdominal region (region of interest), blue: pelvic region).

####  2.1.2 Datasets

The Rhineland Study is an ongoing population-based prospective cohort
(https://www.rheinland-studie.de/) which enrolls participants aged 30 years
and above at baseline from Bonn, Germany. The study is carried out in
accordance with the recommendations of the International Council for
Harmonisation (ICH) Good Clinical Practice (GCP) standards (ICH-GCP). Written
informed consent was obtained from all participants in accordance with the
Declaration of Helsinki.
The first 641 subjects from the Rhineland Study with BMI and abdominal MR
Dixon scans are included. The sample presents a mean age of 54.2 years (range
30 to 95) and 55.2% of the subjects are women. The BMI of the participants
ranges from 17.2 to 47.7 kg/m2 with a mean of 25.2 kg/m2. Subjects were
stratified into two subsets: 38 scans were manually annotated for training and
testing; the remaining 603 subjects were segmented using the proposed
pipeline. After visual inspection, 16 subjects were excluded due to poor image
quality or extreme motion artifacts (e.g. potentially caused by breathing).
Thus, 587 participants were used for the case study analysis and a subset of
50 subjects were randomly selected for manual corrections of the predicted
label maps. This manually edited set and an independent test-retest set of 17
healthy young volunteers were used to assess reliability of the automated
segmentation and volume estimates.

Ground Truth Data: 38 subjects were randomly selected from sex and BMI strata
to ensure a balanced population distribution. These scans were manually
annotated by two trained raters without any semi-automated support such as
thresholding, which can reduce accuracy in the ground truth and lead to
overestimation of the performance of the proposed automated method.

Specific label schemes were created for each individual task of the pipeline.
For localizing the abdominal region, raters divided the scans into three
different blocks defined by the location of the vertebrae as follows: the
abdominal region (from lower bound of twelfth thoracic vertebra (Th12) to the
lower bound of L5), the thoracic region (all above the lower bound of Th12),
and the pelvic region (everything below the lower bound of L5), as illustrated
in Fig. 1 e). For AAT segmentation, 60 slices per subject were manually
labeled into three classes: SAT, VAT, and bone with neighbouring tissues. The
bone was labeled to prevent bone marrow from being misclassified as adipose
tissue. In order to improve spatial context and prevent misclassification of
the arms, the dataset was complemented by a synthetic class defined as “other
tissue” that was composed of any soft tissue inside the abdomen cavity that is
not VAT or SAT. The manual annotations are illustrated in Fig. 1 b) and c).
Furthermore, four subjects were labeled by both raters to evaluate the inter-
rater variability.

Test-Retest Data: 17 additional subjects were recruited with the exclusive
purpose of measuring the acquisition protocol reliability. The group presents
a mean age of 25.5 years (range: 20 to 31) and 65.0 % of the participants are
women; all of them have a normal BMI (BMI < 25 kg/m2). Subjects were scanned
in two consecutive sessions. Before starting the second session, subjects were
removed from the scanner and re-positioned.

###  2.2 FatSegNet Pipeline

The FatSegNet is to be deployed as a post-processing adipose analysis pipeline
for the abdominal Dixon MR images acquired in the Rhineland Study. Therefore,
it should meet the following requirements: 1) be fully automated, 2) segment
the different adipose tissue types within the anatomically defined abdominal
region, and 3) be robust to body type variations and generalizable in presence
of high population heterogeneity. Following the prior conditions, we designed
FatSegNet as a fully automated deep learning pipeline for adipose segmentation
(Fig. 2).

Figure 2: Proposed FatSegNet Pipeline for segmenting AAT. The pipeline is
divided into three stages: First, localization of abdominal region. Then,
tissue segmentation on the abdominal region and finally, view-aggregation.
Both local and global volume estimates of individual structures are calculated
on the final prediction.

The proposed pipeline consists of three stages: (i)the abdominal region is
localized by averaging bounding boxes from two abdominal segmentation maps
generated by CDFNets on the sagittal and coronal view. For each view a
bounding box is set to the full image width. The height is extracted by
localizing the highest and lowest slice with at least 85% of none background
voxels classified as abdominal region. Highest and lowest slice position are
averaged across the views. (ii) Afterward, adipose tissue is segmented within
the abdominal region by three CDFNets on different views (axial, coronal, and
sagittal) with standardized input sizes (zero padding). (iii) Finally, a view-
aggregation network merges the predicted label maps from the previous stage
into a final segmentation; the implemented multi-view scheme is designed to
improve segmentation of structures that are not clearly visible due to poor
lateral resolution. This 2.5D strategy produces a fully automated pipeline to
accurately segment adipose tissue inside a consistent anatomically defined
abdominal region.

####  2.2.1 Pipeline components

Figure 3: Proposed network architecture : Competitive Dense Fully
Convolutional Network (CDFNet), with 4 competitive dense blocks (CDB) on each
encoder and decoder path and 4 competitive unpool blocks (CUB) between them.
CDB and CUB induce local and global competition within the network. Note - the
output filters for all convolutional layers in CUB, CDB, and Bottleneck were
standardized to 64 channels.

Competitive Dense Fully Convolutional Network (CDFNet): For the segmentation
task we introduce the CDFNet architecture due to its robustness and
generalizability properties. The proposed network improves feature selectivity
and, thus, boosts the learning of fine-grained anatomies without increasing
the number of learned parameters [22]. We implemented the CDFNet by suitably
adopting the Dense-UNet architecture proposed by Roy et al. [27] and extending
it towards competitive learning via maxout activations [24].

The Dense-UNet proposed in [27] follows the usual dumb-bell like architecture
with four dense-block encoders, four dense-block decoders and one bottleneck
layer. Each dense-block is based on short-range-skip-connections between
convolutional layers as introduced for densely-connected neural networks [28];
the dense connection approach stacks multiple convolutional layers in sequence
and the input of a layer is iteratively concatenated with the outputs of the
previous layers. This type of connectivity improves feature reusability,
increases information propagation, and alleviates vanishing gradients [28].
The architecture additionally incorporates the traditional long-range skip-
connections between all encoder and decoder blocks of the same spatial
resolution as introduced by Ronnenberger et al. [19] which improves gradient
flow and spatial information recovery.

Within the Dense-UNet, the information aggregation through these connections
is performed by concatenation layers. Such a design increases the size of the
output feature map along the feature channels, which in turn results in the
need to learn filters with a higher number of parameters. Goodfellow et al.
introduced the idea of competitive learning through maxout activations [23],
which was adapted by Liao and Carneiro [24] for competitive pooling of multi-
scale filter outputs. Both [23] and [24] proved that the use of maxout
competitive units boosts performance by creating a large number of dedicated
sub-networks within a network that learns to target specific sub-tasks and
reduces the number of required parameters significantly, which in turn can
prevent over-fitting.

The maxout is a simple feed-forward activation function that chooses the
maximum value from its inputs [23]. Within a CNN, a maxout feature map is
constructed by taking the maximum across multiple input feature maps for a
particular spatial location. The proposed CDFNet uses competitive layers
(maxout activation) instead of concatenation layers. Our preliminary results
[22] demonstrate that these competitive units promote the formation of
dedicated local sub-networks in each of the densely connected blocks within
the encoder and the decoder paths. This encourages sub-modularity through a
network-in-network design that can learn more efficiently. Towards this, we
propose two novel architectural elements targeted at introducing competition
within the short- and long-range connections, as follows:

  * •

Local Competition - Competitive Dense Block (CDB): By introducing maxout
activations within the short-range skip-connections of each of the densely
connected convolutional layers (at the same resolution), we encourage local
competition during learning of filters. The multiple convolution layers in
each block prevent filter co-adaptation.

  * •

Global Competition - Competitive Un-pooling Block (CUB): We introduce a maxout
activation between a long-range skip-connection from the encoder and the
features up-sampled from the prior lower-resolution decoder block. This
promotes competition between finer feature maps with smaller receptive fields
(skip connections) and coarser feature maps from the decoder path that spans
much wider receptive fields encompassing higher contextual information.

In brief, the proposed CDFNet comprises a sequence of four CDBs, constituting
the encoder path (down-sampling block), and four CDBs constituting the decoder
path (up-sampling block), which is joined via a bottleneck layer. The
bottleneck consists of a 2D convolutional layer followed by a Batch
Normalization. The skip-connections from each of the encoder blocks feed into
the CUB that subsequently forwards features into the corresponding decoder
block of the same resolution as illustrated in Fig. 3.

View-Aggregation Network The proposed view-aggregation network is designed to
regularize the prediction for a given voxel by considering spatial information
from the coronal, axial, and sagittal view. The network, therefore, merges the
probability maps of the three different CDFNets from the previous stage by
applying a ($3\times 3\times 3$) 3D-convolution (30 filters) followed by a
Batch Normalization. Then a ($1\times 1\times 1$) 3D-convolution is employed
to reduce the feature maps to the desired number of classes (n=5). The final
prediction probabilities are obtained via a concluding softmax layer (as
illustrated in supporting information Fig. S1). Our approach learns to weigh
each view differently on a voxel level, compared to standard hard-coded global
view-aggregation schemes. Such hard-coded weighting schemes can be suboptimal
when working with anisotropic voxels sizes (e.g., here 2 $mm$ $\times$ 2 $mm$
$\times$ 5 $mm$) as resolution differences impose a challenge when combining
the spatial information from the finer (within-plane) and coarser (across
slice) resolutions. Additionally, in the presence of high variance of
abdominal body shapes across subjects segmentation benefits from data-driven
approaches that can flexibly adopt weights to individual situations and even
spatial locations, which are not possible if hard-coded global weights are
being used.

###  2.3 Experimental setup

For training and testing the pipeline, we perform a sixfold cross-validation
subject-space split on the ground truth dataset. For each fold, 32 subjects
are used for training and 6 held out for testing; the test sets splits are
approximately balanced based on their BMI classification (underweight
[BMI<18.5 kg/m2], normal [18.5 $\leq$ BMI<25 kg/m2], overweight [25 $\leq$
BMI<30 kg/m2], and obese [BMI $\geq$ 30 kg/m2]). This selection process
ensures that all BMI categories are used for bench-marking the cross-
validation models. Additionally, a final model is implemented using 33
subjects for training holding out 5 subjects spanning different BMI levels for
a final performance sanity-check (visual quality check and stability of Dice
score). Given the limited ground truth data, for all models a validation set
to assets convergence during training was created by randomly separating 15%
of the slices from the corresponding training set. This allows evaluating
performance and generalizability on a completely separate test set.

Baselines and comparative methods: We validate the FatSegNet by comparing the
performance of each stage of the pipeline against the cross-validation test
sets using Dice score index (DSC) to measure similarity between the prediction
and the ground truth. Let M (ground truth) and P (prediction) denote the
labels binary segmentation, the Dice score index is defined as

$\displaystyle DSC=\frac{2\cdot\left|M\cap
P\right|}{\left|M\right|+\left|P\right|}$ (1)

Where $|M|$ and $|P|$ represents the number of elements in each segmentation,
and $|M\cap P|$ the number of common elements. Therefore, the DSC ranges from
0 to 1 and a higher DSC represents a better agreement between segmentations.

Additionally, we benchmark the proposed CDFNet models for abdominal region
localization and AAT delineation with state-of-the-art segmentation F-CNNs
such as UNet [19], SD-Net [20], and Dense-UNet [27]. We use the probability
maps generated from the aforementioned networks to train the view-aggregation
model and measure performance with and without view-aggregation. The proposed
view-aggregation performance for each FCNNs is compared against two non-data-
driven (hard-coded) methods: equally balanced weights for all views, and axial
focus weights (accounting for higher in-plane resolution, axial=0.5,
coronal=0.25, sagittal=0.25). Finally, to permit a fair comparison, all
benchmark networks follow the same architecture of four encoder blocks, four
decoders blocks, and one bottleneck layer as illustrated in Fig. 3 with an
input image size of 224$\times$ 256\. Note, significant differences between
our proposed methods and comparative baselines are evaluated by a Wilcoxon
signed-rank test [29] after multiple comparisons correction using a one-sided
adaptive FDR [30].

The aforementioned models are implemented in Keras [31] with a TensorFlow
back-end using an NVIDIA Titan Xp GPU with 12 GB RAM and the following
parameters: batch size of 8, momentum set to 0.9, constant weight decay of
10-06, and an initial learning rate of 0.01 decreased by a order of 10 every
20 epochs. The models are trained for 60 epochs with an early-stopping
criterion (no relevant changes on the validation loss after the last 8 epochs
– convergence was observed around 50 epochs). A composite loss function of
median frequency balanced logistic loss and Dice loss [20] is used. This loss
function emphasizes the boundaries between classes and supports learning of
unbalanced classes such as VAT. Finally, online data augmentation
(translation, rotation and global scaling) is performed to increase training
set size and improve the networks generalizability. Note, the FatSegNet
implementation is available at https://github.com/reuter-lab/FatSegNet.

Pipeline reliability: We assess the FatSegNet reliability by comparing the
difference of VAT-V and SAT-V across sessions for each subject of the test-
retest and manually edited set. Given a predicted label map and $N_{i}(l)$ the
number of voxels classified as $l$ (VAT or SAT) in session $i$ (test-retest,
or manual-automated), the absolute percent difference (APD ($l$)) of a label
volume measures variability across sessions. It is defined as

$\displaystyle
APD(l)=\frac{2\cdot\left|N_{1}(l)-N_{2}(l)\right|}{N_{1}(l)+N_{2}(l)}\cdot
100$ (2)

Additionally, we calculate the agreement of total VAT-V and SAT-V between
sessions by an intra-class correlation (ICC) using a two-way fixed, absolute
agreement and single measures ICC(A,1) [32].

Case study analysis on the Rhineland Study: We compare the volumes of
abdominal adipose tissue (AAT-V, SAT-V, and VAT-V) generated from FatSegNet
with BMI on the unseen dataset. A fast quality control is performed to
identify drastic failure cases. The differences among BMI groups are evaluated
with a one-way analysis of variance (ANOVA) with subsequent Tukey’s honest
significant difference (HSD) post-hoc comparisons. The associations of volumes
of abdominal adipose tissue and BMI are assessed using partial correlation and
linear regression after accounting for age, sex, and height of the abdominal
region. Separate linear regression analyses are performed to explore the
effect of age on SAT-V and VAT-V in men and women. All the statistical
analyses are performed in R [33].

##  3 Results

###  3.1 Method Validation

Localization of abdominal region: For assessing the performance of abdominal
region detection after creation of an average bounding box from the coronal
and sagittal views the average Dice overlap (sixfold cross-validation) was
calculated, as illustrated on the supporting information Fig. S2. We observe
that all models perform extremely well on the relatively easy task of
localizing the desired abdominal region (DSC > 0.96). There is no significant
difference between the models; however, we use our CDFNet because it requires
substantially less parameters (see Table 1) compared to the UNet and Dense-
UNet.

Segmentation of AAT: In Table 1, we present the average Dice score (sixfold
cross-validation) for VAT and SAT for each individual view as well as for the
view-aggregation model. Here, we observe that all methods work extremely well
for SAT segmentation. Nevertheless, our proposed CDFNet outperforms the UNet
and SD-Net on all single-view models and, when compared with the Dense-UNet,
there is significant improvement in the sagittal and coronal views. For the
more challenging task of VAT recognition, which is a more fine-grained
compartment with large shape variation, the proposed CDFNet outperforms the
SD-Net on all single planes; when compared with Dense-UNet and U-Net, there is
only significant improvement in the axial and coronal plane. Nonetheless,
CDFNet achieves this performance with $\sim$30 %( Dense-UNet) and $\sim$80
%(UNet) less parameters, demonstrating that the proposed architecture improves
feature selectivity and simplifies network learning. Furthermore, fewer
parameters can help decrease over-fitting error, especially when training with
limited annotated data, and thus improve generalizability.

Table 1: Mean (and standard deviation) Dice scores (cross-validation) of the
FCNN models for abdominal adipose tissue segmentation. We show FDR corrected
significance indicators of Wilcoxon signed-rank test [29] comparing the
proposed CDFNet vs. benchmark FCNNs

| Subcutaneous (SAT) | Visceral (VAT)
---|---|---
Models (PRM)† | Axial | Coronal | Sagittal | V. Aggregation | Axial | Coronal | Sagittal | V. Aggregation
UNet ($\sim$20 M) | 0.965 (0.029)∗ | 0.960 (0.034)∗ | 0.960 (0.035)∗ | 0.972 (0.019)∗ | 0.810 (0.111)∗ | 0.804 (0.113)∗ | 0.820 (0.101) | 0.837 (0.095)∗
SD-Net ($\sim$ 1,5M) | 0.969 (0.027)∗ | 0.954 (0.040)∗ | 0.956 (0.034)∗ | 0.972 (0.020)∗ | 0.820 (0.097)∗ | 0.812 (0.099)∗ | 0.822 (0.091)∗ | 0.843 (0.081)∗
Dense-UNet ($\sim$ 3,3M) | 0.972 (0.025)∗ | 0.959 (0.037)∗ | 0.963 (0.029)∗ | 0.975 (0.019)∗ | 0.824 (0.091)∗ | 0.814 (0.097)∗ | 0.827 (0.090)∗ | 0.847(0.080)∗
Proposed ($\sim$2,5M) | 0.970 (0.025) | 0.966 (0.029) | 0.966 (0.027) |  0.975(0.018) | 0.826 (0.095) | 0.826 (0.085) | 0.824 (0.092) |  0.850(0.076)
Inter-rater variability | 0.982 (0.018) | 0.788 (0.060)

  * •

† The approximately number of learn parameters reported is for the models
without the View-Aggregation Network

  * •

∗ Statistical difference using a one-sided adaptive FDR multiple comparison
correction [30] at a level of 0.05

Note, that Dice scores increase and difference of pairwise comparisons is
slightly reduced after the view-aggregation (Table 1), showing that this steps
helps all individual networks to reach a better performance by introducing
spatial information from multiple views and regularizing the prediction maps.
The proposed data-driven aggregation scheme outperforms (DSC) the hard-coded
models for SAT and with statistically significance for VAT as shown in Table
2. Furthermore, learned weights are spatially varying and can adjust to
subject-specific anatomy, which in turn can improve generalizability. We
empirically observe that the aggregation model smoothes the label maps
slightly, resulting in visually more appealing boundaries. It also
significantly reduces the arms from being misclassified as adipose tissue
which can otherwise be observed in different views, especially on overweight
and obese subjects, where arms are located closer to the abdominal cavity, as
seen supporting information Fig. S3.

Finally it should be highlighted, that all single-view and the view-
aggregation models achieve similarly excellent results on the SAT segmentation
compared to inter-rater variability and outperform the manual raters for the
more challenging VAT segmentation by a margin.

Table 2:  Mean (and standard deviation) Dice scores (cross-validation) of
hard-coded balanced weights, hard-coded axial focus weights, and the proposed
view-aggregation for abdominal adipose tissue segmentation. We show FDR
corrected significance indicators of Wilcoxon signed-rank test [29] comparing
the proposed data-driven aggregation scheme vs. each hard-coded method.

| Subcutaneous (SAT) | Visceral (VAT)
---|---|---
Single-View Model | Balanced | Axial Focus | Proposed | Balanced | Axial Focus | Proposed
UNet | 0.970 (0.026) | 0.970 (0.026) |  0.972 (0.019) | 0.830 (0.098)∗ | 0.829 (0.099)∗ |  0.837 (0.095)
SD-Net | 0.970 (0.026)∗ | 0.972 (0.025)∗ |  0.972 (0.020) | 0.839 (0.084)∗ | 0.838 (0.085)∗ |  0.843 (0.082)
Dense-UNet | 0.973 (0.025) | 0.974 (0.024)∗ |  0.975 (0.019) | 0.841 (0.081)∗ | 0.840 (0.082)∗ |  0.847 (0.080)
CDFNet | 0.972 (0.025)∗ | 0.973 (0.024) |  0.975 (0.018) | 0.844 (0.077)∗ | 0.841 (0.080)∗ |  0.850 (0.076)

  * •

∗ Statistical difference using a one-sided adaptive FDR multiple comparison
correction [30] at a level of 0.05

FatSegNet reliability: Table 3 presents the reliability metrics evaluated on
the test-retest and the manually edited test set. The proposed pipeline
presents only a small absolute percent volume difference (APD) for VAT and
SAT, and excellent agreement between the predicted and corrected segmentation
maps. It must be noted, that APD is larger for both tissue types in the test-
retest setting as it also includes variance from acquisition noise (e.g.
motion artefacts, non-linearities based on different positioning) in addition
to potential variances of the processing pipelines. Nevertheless, we observe
excellent agreement (ICC) between sessions for the test-retest dataset for
both adipose tissue types.

Table 3: Mean absolute percent difference (APD) and interclass correlation
agreement (ICC(A,1)) for the volumes estimates of VAT and SAT across sessions
of the manually edited and test-retest set.

| Manually Edited Set | Test-Retest Set
---|---|---
Metric | SAT-V | VAT-V | SAT-V | VAT-V
ICC [95% CI] | 0.999 [0.999 - 1.000] | 0.999 [0.994 - 0.999] | 0.996 [0.986 - 0.999] | 0.998 [0.995 - 0.999]
APD (SD) | 0.149 % (0.424) | 1.398 % (0.963) | 3.254 % (2.524) | 2.957 % (2.600)

Figure 4:  Examples of FatSegNet predictions and excluded cases on the
Rhineland Study. (a-b) Subjects with different body shapes and accurate
segmentations. (c-d) Excluded subjects from the case study due to extreme
motion noise (c), or low image contrast quality (d).

###  3.2 Case Study: Analysis of Rhineland Study Data

The characteristics of the study population: After visual quality inspection,
16 scans were flagged due to image artefacts, such as motion or low contrast
(see Fig. 4 c) and d) for two examples). The characteristics of the remaining
587 participants with valid data on BMI and volumes of abdominal adipose
tissue are presented in supporting information Table S1. The mean (SD) age of
the subjects is 54.2 (13.3) years, and 54.7% are women. 311 (53.0%) subjects
are normal weight, 209 (35.6%) overweight, and 67 (11.4%) obese. We observed a
BMI increase with age ($\beta$ = 0.03, p=0.007), and a borderline significance
of age difference among BMI groups (p=0.052, ANOVA). Obvious differences are
observed in AAT-V, VAT-V, and SAT-V across BMI groups (p<0.001, ANOVA). VAT-V
to SAT-V ratio is higher in overweight and obese participants compared to
those with normal weight (p<0.001), but there is no difference between
overweight and obese (p=0.505).

The association between abdominal adipose tissue volumes and BMI: BMI shows a
strong positive correlation with AAT-V and SAT-V (AAT-V: r = 0.88, p<0.001;
SAT-V: r=0.85, p<0.001), but only a moderate correlation with VAT-V (r=0.65,
p<0.001) after adjusting for age, sex, and abdominal region height. As
illustrated in Fig. 5, both SAT-V and VAT-V are positively associated with BMI
after accounting for age, sex, and abdominal region height (p<0.001). The
accumulation of SAT-V is higher than VAT-V as BMI increases.

Influence of age and sex on VAT-V and SAT-V: The influence of age and sex on
VAT-V and SAT-V follows different patterns (as illustrated in Fig. 6). Men
tend to have lower SAT and higher VAT compared to women (p<0.001). VAT-V
significantly increase with age in both men and women. Conversely, SAT-V is
weakly associated with age in women ($\beta$ = 0.02, p=0.012), but not in men
($\beta$ = -0.01, p=0.337).

Figure 5: Association of BMI with SAT-Volume and VAT-Volume Figure 6: The
association between age and SAT-Volume and VAT-Volume in men and women.

##  4 Discussion

In our study, we established, validated, and implemented a novel deep learning
pipeline to segment and quantify the components of abdominal adipose tissue,
namely, VAT-V, SAT-V, and AAT-V on a fast acquisition abdominal Dixon MR
protocol for subjects from the Rhineland Study, a large population-based
cohort. The proposed pipeline is fully automated and requires approximately 1
min for analyzing a subject’s whole volume. Moreover, since the pipeline is
based on deep learning models, it can be easily updated and retrained as the
study progresses and new manual data are generated - which can further improve
overall pipeline robustness and generalizability, providing a pragmatic
solution for a population-based study.

The proposed pipeline, termed FatSegNet implements a three-stage design with
the CDFNet architecture at the core for localizing the abdominal region and
segmenting the AAT. The introduction of our CDFNet inside the pipeline boosts
the competition among filters to improve feature selectivity within the
networks. CDFNet introduces competition at a local scale by substituting
concatenation layers with maxout activations that prevent filter co-adaptation
and reduce the overall network complexity. It also induces competition at a
global scale through competitive unpooling. This network design, in turn, can
learn more efficiently.

Figure 7: Examples of FatSegNet predictions on the Rhineland Study. (a-f)
Accurate automatic segmentation of different body shapes. Extreme cases: a)
arms are in front of the abdominal cavity,and f) deviated spine.

For the first stage of the pipeline, i.e. localization of the abdominal
region, all FCNNs can successfully determine the upper and lower limit of the
abdominal region from a segmentation prediction map. However, our CDFNet
requires significantly fewer parameters compared to the traditional UNet and
Dense-UNet. Furthermore, the localization block is able to identify the
abdominal region correctly even in cases with scoliosis (curved spine) as
illustrated in Fig. 7 f). For the more challenging task of segmenting AAT, we
demonstrate that CDFNet recovers VAT significantly better than traditional
deep learning variants that rely on concatenation layers. Additionally, each
individual CDFNet view model outperforms manual raters for segmenting the
complex VAT and accomplishes equivalent results on SAT. The selection of an
inhomogeneous BMI testing set ensures that our method is evaluated for
different body types and avoids biases, as better segmentation performance can
be achieved on subjects with high content of AAT compared to lean subjects
[34, 35]. Moreover, images from individuals with high AAT could be accompanied
by other types of issues, such as fat shadowing (Fig. 7 d), or arms located in
close proximity to the abdominal cavity (Fig. 7 a), d) and e). These issues
are mitigated by our view-aggregation model that regularizes the predicted
segmentation by combining the spatial context from different views ultimately
improving segmentation of tissue boundaries. Moreover, this approach
automatically prevents misclassification of arms whereas previous deep
learning AAT segmentation methods required manual removal of the upper
extremities in a pre-processing step [18]. Note, that we prefer the 2D over a
full 3D approach in this work. A full 3D network architecture has more
parameters, requiring significantly more expert annotated training data (full
3D cases) and/or artificial data augmentation, which could increase the chance
of overfitting – in addition to increased GPU memory requirements.

As demonstrated on the Rhineland Study data, the proposed pipeline exhibits
high robustness and generalizability across a wide range of age, BMI, and a
variety of body shapes as seen in Fig. 7 and Fig. 4 a) and b). FatSegNet
successfully identifies the AAT in different abdomen morphologies, spine
curvatures, adipose shadowing, arms positioning, or intensity inhomogeneities.
Furthermore, the pipeline has a high test-retest reliability between the
calculated volumes of VAT and SAT without the need of any image pre-processing
(bias-correction, image registration, etc.) or manual selection of a slice or
region. Furthermore, the manually edited test set demonstrates a high
similarity of automated and manual labels and excellent agreement of volume
estimates. However, as is usual with any automated method, segmentation
reliability decreases when input images have low quality as illustrated in
Fig. 4 c), and d) where the scans present severe motion/breathing artifacts or
very low image contrast. In order to detect these problematic images in large
studies, an automated or manual quality control protocol should be implemented
before passing images to automated pipelines.

In accordance with previous studies on smaller data sets [13, 36], our data
showed a lower correlation of BMI with VAT-V than with AAT-V and SAT-V. We
also observed a sex difference of the SAT-V and VAT-V accumulation as
previously reported [37, 38]: men were more likely to have higher VAT-V and
lower SAT-V compared to women. Moreover, we further explored the association
between age with SAT-V and VAT-V and found an obvious age effect on the
accumulation of VAT-V in both men and women, and a weak age effect on SAT-V in
women but not in men. This discrepancy was previously observed by Machann
et.al. [37], who assessed the body composition using MRI in 150 healthy
volunteers aged 19 to 69 years. They reported a strong correlation between
VAT-V and age both in men and women, whereas SAT-V only slightly increased
with age in women. The fact that our results replicate these previous findings
on a large unseen dataset corroborates stability and sensitivity of our
pipeline.

In conclusion, we have developed a fully automated post-processing pipeline
for adipose tissue segmentation on abdominal Dixon MRI based on deep learning
methods. While reducing the number of required parameters, the pipeline
outperforms other deep learning architectures and demonstrates high
reliability. Furthermore, the proposed method was successfully deployed in a
large population-based cohort, where it replicated well known SAT-V and VAT-V
age and sex associations and demonstrated generalizability across a large
range of anatomical differences, both with respect to body shape and fat
distribution.

##  5 acknowledgements

We would like to thank the Rhineland Study group for supporting the data
acquisition and management, as well as Mohammad Shahid for his support on the
deployment of the FatSegNet method into the Rhineland Study processing
pipeline. Furthermore we acknowledge Tony Stöcker and his team for developing
and implementing the Dixon MRI sequence used in this work. This work was
supported by the Diet-Body-Brain Competence Cluster in Nutrition Research
funded by the Federal Ministry of Education and Research (BMBF), Germany
(grant numbers 01EA1410C and FKZ: 01EA1809C), by the JPI HDHL on Biomarkers
for Nutrition and Health (HEALTHMARK), BMBF (grant number 01EA1705B), the NIH
R01NS083534, R01LM012719, and an NVIDIA Hardware Award.

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## Supporting Information

Figure S1: View-aggregation Network The proposed network is composed of a
initial 3D convolution layer with 30 channels, followed by a batch
normalization and a 3D convolutional layer for reducing the feature map
dimensionality into the number of classes(n=5). Figure S2: Step 1: Abdominal
region localization. Dice scores box-plot: Average Dice score (cross-
validation) of the abdominal region detection comparing the Proposed CDFNet
vs. other FCNN architectures. The Dice scores are calculated on the average
abdominal region generated from the average bounding boxes of the sagittal and
coronal model. There is no significant difference between models, nonetheless,
the proposed method achieves the same performance with $\sim$30 % and $\sim$80
% less parameters compared to Dense-UNet and UNet, respectively. Figure S3:
Comparison of single view model (left) vs. view-aggregation (right): AAT
predictions of two unseen subjects: a) normal subject, b) obese subject. View-
aggregation avoids arm-misclassification (red boxes) and improves SAT (purple
box). Table S1:  Case study analysis on the Rhineland Study data .
Characteristics of the participants (n=587) showing mean (SD) for continuous
and counts (PCT) for categorical variables


# Divisibility of Integer Laurent Polynomials,
Homoclinic Points, and Lacunary Independence

Douglas Lind  Douglas Lind: Department of Mathematics, University of
Washington, Seattle, Washington 98195, USA<EMAIL_ADDRESS>and
Klaus Schmidt  Klaus Schmidt: Mathematics Institute, University of Vienna,
Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria<EMAIL_ADDRESS>Dedicated to the memory of Professor K. R. Parthasarthy

###### Abstract.

Let $f$, $p$, and $q$ be Laurent polynomials with integer coefficients in one
or several variables, and suppose that $f$ divides $p+q$. We establish
sufficient conditions to guarantee that $f$ individually divides $p$ and $q$.
These conditions involve a bound on coefficients, a separation between the
supports of $p$ and $q$, and, surprisingly, a requirement on the complex
variety of $f$ called atorality satisfied by many but not all polynomials.

Our proof involves a related dynamical system and the fundamental dynamical
notion of homoclinic point. Without the atorality assumption our methods fail,
and it is unknown whether our results hold without this assumption.

###### Key words and phrases:

Algebraic action, lacunary independence, homoclinic point, atoral polynomial

###### 2020 Mathematics Subject Classification:

Primary: 37A15, 13F20, 37A44; Secondary: 37B40, 13F20

##  1\. Introduction

We begin with some notation and terminology, and then state our main results.
In the next section we describe the dynamical context that underlies our
proofs.

Let $d\geqslant 1$, and let $R_{d}=\mathbb{Z}[x_{1}^{\pm 1},\dots,x_{d}^{\pm
1}]$ denote the ring of Laurent polynomials with integer coefficients in $d$
commuting variables. We write $f\in R_{d}$ as
$f=\sum_{\mathbf{n}\in\mathbb{Z}^{d}}f_{\mathbf{n}}\mathbf{x}^{\mathbf{n}}$,
where $\mathbf{x}=(x_{1},\dots,x_{d})$, $\mathbf{n}=(n_{1},\dots,n_{d})$,
$\mathbf{x}^{\mathbf{n}}=x_{1}^{n_{1}}\cdots x_{d}^{n_{d}}$, and
$f_{\mathbf{n}}\in\mathbb{Z}$ with $f_{\mathbf{n}}=0$ for all but finitely
many $\mathbf{n}$. The units in $R_{d}$ are those elements of the form
$\pm\,\mathbf{x}^{\mathbf{n}}$. An element of $R_{d}$ is _irreducible_
provided that it is neither a unit nor a product of two non-units in $R_{d}$.
We let $\langle f\rangle=fR_{d}$ denote the principal ideal in $R_{d}$
generated by $f$. The _support_ of $f$ is defined as
$\operatorname{supp}f=\\{\mathbf{n}\in\mathbb{Z}^{d}\colon f_{\mathbf{n}}\neq
0\\}$, and we set
$\|f\|_{\infty}=\max_{\mathbf{n}\in\mathbb{Z}^{d}}|f_{\mathbf{n}}|$. The
_adjoint_ $f^{*}$ of $f$ is defined by $f^{*}_{\mathbf{n}}=f_{-\mathbf{n}}$.
Let $\|\mathbf{n}\|=\max_{1\leqslant j\leqslant d}\\{|n_{j}|\\}$. For nonempty
subsets $\mathcal{S}$ and $\mathcal{T}$ of $\mathbb{Z}^{d}$ we define their
distance to be
$\operatorname{dist}(\mathcal{S},\mathcal{T})=\min_{\mathbf{m}\in\mathcal{S},\mathbf{n}\in\mathcal{T}}\|\mathbf{m}-\mathbf{n}\|$.

Maps $v\colon\mathbb{Z}^{d}\to\mathbb{R}$ are denoted by
$v=(v_{\mathbf{n}})_{\mathbf{n}\in\mathbb{Z}^{d}}$. We let
$\ell^{1}(\mathbb{Z}^{d},\mathbb{R})$ be the space of all $v$ for which
$\|v\|_{1}=\sum_{\mathbf{n}\in\mathbb{Z}^{d}}|v_{\mathbf{n}}|<\infty$. This
space is equipped with the convolution product

$(v\cdot
w)_{\mathbf{n}}=\sum_{\mathbf{m}\in\mathbb{Z}^{d}}v_{\mathbf{m}}w_{\mathbf{n}-\mathbf{m}}=\sum_{\mathbf{m}\in\mathbb{Z}^{d}}v_{\mathbf{n}-\mathbf{m}}w_{\mathbf{m}}.$
(1.1)

If we view each
$f=\sum_{\mathbf{n}\in\mathbb{Z}^{d}}f_{\mathbf{n}}\mathbf{x}^{\mathbf{n}}$ as
the element
$(f_{\mathbf{n}})_{\mathbf{n}\in\mathbb{Z}^{d}}\in\ell^{1}(\mathbb{Z}^{d},\mathbb{Z})\subset\ell^{1}(\mathbb{Z}^{d},\mathbb{R})$,
we obtain an embedding
$R_{d}\hookrightarrow\ell^{1}(\mathbb{Z}^{d},\mathbb{R})$ in which
multiplication of Laurent polynomials agrees with the convolution product
(1.1) in $\ell^{1}(\mathbb{Z}^{d},\mathbb{R})$.

In this paper we investigate divisibility of elements of $R_{d}$ by a given
irreducible element $f\in R_{d}$. Somewhat surprisingly, our results hinge on
a property of $f$ called _atorality_. Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$
denote the additive torus, and $\SS=\\{z\in\mathbb{C}\colon|z|=1\\}$ be its
multiplicative counterpart.

######  Definition 1.1.

The _unitary variety_ of $f\in R_{d}$ is defined to be

$\mathsf{U}(f)=\\{(t_{1},\dots,t_{d})\in\mathbb{T}^{d}\colon f(e^{2\pi
it_{1}},\dots,e^{2\pi it_{d}})=0\\}.$

We remark that this is an additive version of the more usual definition of
unitary variety as the intersection of the complex variety of $f$ with
$\SS^{d}$.

According to the discussion in [Lind-atoral]*§1, the unitary variety
$\mathsf{U}(f)$ is a real semialgebraic set. By the cell decomposition theorem
for such sets, it is a finite disjoint union of cells of various dimensions.
We can therefore define its dimension $\dim\mathsf{U}(f)$ to be the maximal
dimension of these cells, and this is known to be the same for all possible
cell decompositions. If $\dim\mathsf{U}(f)=d$, then $f$ vanishes on an open
subset of $\SS^{d}$, and by holding all variables but one fixed, it is easy to
show that $f=0$.

######  Definition 1.2.

A nonzero Laurent polynomial $f\in R_{d}$ is _atoral_ if
$\operatorname{dim}\mathsf{U}(f)\leqslant d-2$. This includes the possibility
that $\mathsf{U}(f)=\varnothing$. If $\dim\mathsf{U}(f)=d-1$ then $f$ is
called _toral_.

These notions were introduced by Agler, Starkus, and McCarthy [Agler-toral] in
the context of several complex variables.

######  Examples 1.3.

(1) Every $f\in R_{d}$ with $\mathsf{U}(f)=\varnothing$ is atoral. The
dynamical consequences of this case were extensively studied in [Lind-
homoclinic].

(2) If $f\in R_{1}$, then $f$ is atoral if and only if
$\mathsf{U}(f)=\varnothing$, i.e., $f$ has no roots of absolute value 1. For
instance, $x^{2}-x-1$ is atoral, while $x^{4}-x^{3}-x^{2}-x+1$ is toral [Lind-
atoral]*Exam. 4.2.

(3) If $f(x,y)=1+x+y\in R_{2}$, then
$\mathsf{U}(f)=\\{(\frac{1}{3},\frac{2}{3}),(\frac{2}{3},\frac{1}{3})\\}$ with
dimension 0, and so $f$ is atoral.

(4) If $f(x,y)=3+x+y+x^{-1}+y^{-1}\in R_{2}$, then according to [Lind-
atoral]*Exam. 4.2

$\mathsf{U}(f)=\bigl{\\{}(s,t)\in\mathbb{T}^{2}:t=\pm\arccos\bigl{(}\tfrac{3}{2}-\cos
2\pi s\bigr{)},\ -\tfrac{1}{6}\leqslant s\leqslant\tfrac{1}{6}\bigr{\\}},$

a smooth closed curve of dimension 1. Hence $f$ is toral.

(5) If $f(x,y,z)=1+x+y+z\in R_{3}$, then $\mathsf{U}(f)$ is a union of three
1-dimensional circles [Lind-atoral]*Exam. 4.6, and so $f$ is atoral.

(6) A consequence of [Lind-atoral]*§2 is that if $f\in R_{d}$ is irreducible
and if $f^{*}$ is not a unit times $f$, then $f$ is atoral. Thus it is very
easy to create atoral examples.

Being atoral turns out to be equivalent to an invertibility property in
$\ell^{1}(\mathbb{Z}^{d},\mathbb{R})$ established in [Lind-atoral]*Prop. 2.2
and Rem. 2.7.

######  Theorem 1.4.

Let $0\neq f\in R_{d}$. Then $f$ is atoral if and only if there exists an
element $f^{\\#}\in\ell^{1}(\mathbb{Z}^{d},\mathbb{R})$ such that
$f^{\\#}\cdot f\in R_{d}\smallsetminus\langle f\rangle$.

The proof consists of using atorality to find an $h\in
R_{d}\smallsetminus\langle f\rangle$ so that $h/f$ is smooth enough on
$\SS^{d}$ to have absolutely convergent Fourier series, and then using the
resulting Fourier coefficients to define $f^{\\#}$. The element $f^{\\#}$ is
obviously not unique. We shall call any such $f^{\\#}$ a _quasi-inverse_ of
$f$.

For every nonempty subset $\mathcal{S}\subset\mathbb{Z}^{d}$ and integer
$H\geqslant 1$ we let

$\mathcal{P}(\mathcal{S},H)=\\{p\in
R_{d}\colon\operatorname{supp}(p)\subseteq\mathcal{S}{\text{ and
}}\|p\|_{\infty}\leqslant H\\}.$ (1.2)

The following result was proved in [Fan-bohr]*Thm. 7.3.

######  Theorem 1.5 (Gap Theorem).

Let $f\in R_{d}$ be irreducible and atoral. For every $H\geqslant 1$ there
exists a number $M=M(f,H)\geqslant 1$ with the following property. Suppose
that $\mathcal{S}$, $\mathcal{T}$ are nonempty subsets of $\mathbb{Z}^{d}$
with $\operatorname{dist}(\mathcal{S},\mathcal{T})\geqslant M$, and that
$p\in\mathcal{P}(\mathcal{S},H)$ and $q\in\mathcal{P}(\mathcal{T},H)$. If $f$
divides $p+q$, then $f$ divides $p$ and $f$ divides $q$.

This theorem has two obvious corollaries.

######  Corollary 1.6.

Let $f\in R_{d}$ be irreducible and atoral. For every $H\geqslant 1$ there
exists a number $M=M(f,H)\geqslant 1$ with the following property. Suppose
that $(\mathcal{S}_{i})_{i\in I}$ is a finite or infinite collection of
nonempty subsets of $\mathbb{Z}^{d}$ with
$\operatorname{dist}(\mathcal{S}_{i},\mathcal{S}_{j})\geqslant M$ whenever
$i\neq j$. Let $F$ be an arbitrary finite subset of $I$ and let
$p_{i}\in\mathcal{P}(\mathcal{S}_{i},H)$ for every $i\in F$. If $f$ divides
$\sum_{i\in F}p_{i}$, then $f$ divides every $p_{i}$ for $i\in F$.

The second corollary is closely related to the notion of lacunary independence
[Lind-lacunary] and its dynamical consequences.

######  Corollary 1.7.

Let $f\in R_{d}$ be irreducible and atoral. For every finite nonempty subset
$\mathcal{F}\subset R_{d}$ there exists a number $M=M(f,\mathcal{F})\geqslant
1$ with the following property. For every finite subset
$\mathcal{S}\subset\mathbb{Z}^{d}$ with $\|\mathbf{m}-\mathbf{n}\|\geqslant M$
for $\mathbf{m}\neq\mathbf{n}$ in $\mathcal{S}$, and for every choice of
$p^{(\mathbf{n})}\in\mathcal{F}$ for $\mathbf{n}\in\mathcal{S}$, if $f$
divides the sum
$\sum_{\mathbf{n}\in\mathcal{S}}\mathbf{x}^{\mathbf{n}}p^{(\mathbf{n})}$ then
$f$ divides each $p^{(\mathbf{n})}$ for $\mathbf{n}\in\mathcal{F}$.

Roughly speaking, this says that in the quotient ring $R_{d}/\langle
f\rangle$, multiplying a fixed finite subset by sufficiently widely spaced
monomials results in independent subsets of the quotient ring as an additive
group, hence the terminology lacunary independence.

Although these statements about divisibility of polynomials are purely
algebraic, they arose in a dynamical context which we discuss in the next
section.

##  2\. Algebraic $\mathbb{Z}^{d}$-actions

An _algebraic $\mathbb{Z}^{d}$-action_ is an action of $\mathbb{Z}^{d}$ by
continuous automorphisms of a compact metrizable abelian group. Our
divisibility results are proved using a particular kind of algebraic
$\mathbb{Z}^{d}$-action called a _principal action_.

Let $\mathbb{T}^{\mathbb{Z}^{d}}$ denote the compact abelian group consisting
of all $t=(t_{\mathbf{n}})_{\mathbf{n}\in\mathbb{Z}^{d}}$ with
$t_{\mathbf{n}}\in\mathbb{T}$ for every $\mathbf{n}\in\mathbb{Z}^{d}$. Define
the _shift action_ $\sigma$ on $\mathbb{T}^{\mathbb{Z}^{d}}$ by
$\sigma^{\mathbf{m}}(t)_{\mathbf{n}}=t_{\mathbf{n}+\mathbf{m}}$. Every
$g=\sum_{\mathbf{n}\in\mathbb{Z}^{d}}\in R_{d}$ defines a homomorphism
$g(\sigma)=\sum_{\mathbf{m}\in\mathbb{Z}^{d}}g_{\mathbf{n}}\sigma^{\mathbf{n}}$
from $\mathbb{T}^{\mathbb{Z}^{d}}$ to itself. Fix such a $g$, and consider the
closed, shift-invariant subgroup $X_{g}$ of $\mathbb{T}^{\mathbb{Z}^{d}}$
defined by

$X_{g}=\ker
g(\sigma)=\Bigl{\\{}t\in\mathbb{T}^{\mathbb{Z}^{d}}\colon\sum_{\mathbf{n}\in\mathbb{Z}^{d}}t_{\mathbf{m}+\mathbf{n}}g_{\mathbf{n}}\equiv
0\pmod{1}\text{ for all $\mathbf{m}\in\mathbb{Z}^{d}$}\Bigr{\\}}.$ (2.1)

Let $\alpha_{g}$ denote the restriction of $\sigma$ to $X_{g}$. The dynamical
system $(X_{g},\alpha_{g})$ is called the _principal algebraic
$\mathbb{Z}^{d}$-action_ corresponding to $g\in R_{d}$. When $g=0$ this
reduces $X_{g}=\mathbb{T}^{\mathbb{Z}^{d}}$ and $\alpha_{g}=\sigma$.

In order to use analytical ideas, we linearize $(X_{g},\sigma_{g})$ as
follows. Let $\ell^{\infty}(\mathbb{Z}^{d},\mathbb{R})$ denote the space of
all $v=(v_{\mathbf{n}})$ for which $v_{\mathbf{n}}\in\mathbb{R}$ and
$\|v\|_{\infty}=\sup_{\mathbf{n}\in\mathbb{Z}^{d}}|v_{\mathbf{n}}|<\infty$.
Define the shift action $\overline{\sigma}$ on
$\ell^{\infty}(\mathbb{Z}^{d},\mathbb{R})$ by
$(\overline{\sigma}^{\mathbf{m}}v)_{\mathbf{n}}=v_{\mathbf{n}+\mathbf{m}}$.
For every $w\in\ell^{1}(\mathbb{Z}^{d},\mathbb{R})$ we can define a bounded
shift-commuting linear operator $w(\overline{\sigma})$ on
$\ell^{\infty}(\mathbb{Z}^{d},\mathbb{R})$ by
$w(\overline{\sigma})=\sum_{\mathbf{m}\in\mathbb{Z}^{d}}w_{\mathbf{m}}\overline{\sigma}^{\mathbf{m}}$.
Let $w^{*}$ denote the adjoint element $w^{*}_{\mathbf{n}}=w_{-\mathbf{n}}$.
Then

$\bigl{(}w(\overline{\sigma})v\bigr{)}_{\mathbf{n}}=\sum_{mzd}w_{\mathbf{m}}(\overline{\sigma}^{\mathbf{m}}v)_{\mathbf{n}}=\sum_{\mathbf{m}\in\mathbb{Z}^{d}}w_{\mathbf{m}}v_{\mathbf{n}+\mathbf{m}}=(v\cdot
w^{*})_{\mathbf{n}}.$

Define the surjective homomorphism
$\eta\colon\ell^{\infty}(\mathbb{Z}^{d},\mathbb{R})\to\mathbb{T}^{\mathbb{Z}^{d}}$
by $\eta(v)_{\mathbf{n}}=v_{\mathbf{n}}\pmod{1}$. If $w\in R_{d}$, then
$w(\sigma)$ is well defined on $\mathbb{T}^{\mathbb{Z}^{d}}$, and $\eta\circ
w(\overline{\sigma})=w(\sigma)\circ\eta$.

An element $t\in X_{g}$ is called a _summable homoclinic point_ if there
exists a $v\in\ell^{1}(\mathbb{Z}^{d},\mathbb{R})$ such that $\eta(v)=t$. Such
points provide a powerful way to localize behavior in algebraic
$\mathbb{Z}^{d}$-actions (see [Lind-homoclinic] and [Lind-atoral] for many
applications), and are the essential ingredient to prove the results in §1.

##  3\. Proof of the Gap Theorem

Since the proof of Theorem 1.5 appearing in [Fan-bohr]*Theorem 7.3 is quite
short, but still illustrates the effective use of dynamical ideas in proving
algebraic results, we include it here for the reader’s convenience.

###### Proof of Theorem 1.5.

Suppose that $f\in R_{d}$ is irreducible and atoral. Put $g=f^{*}$, and define
the principal algebraic $\mathbb{Z}^{d}$-action $(X_{g},\alpha_{g})$ as above.
By Theorem 1.4 we may fix a quasi-inverse
$f^{\\#}\in\ell^{1}(\mathbb{Z}^{d},\mathbb{R})$ of $f$ such that $f^{\\#}\cdot
f=h\in R_{d}\smallsetminus\langle f\rangle$. Then

$0=\eta(h)=\eta(f^{\\#}\cdot f)=\eta(g(\overline{\sigma})\cdot
f^{\\#})=g(\sigma)\eta(f^{\\#}),$

so that $\eta(f^{\\#})\in X_{g}$. If $\eta(f^{\\#})$ were equal to zero, then
$f^{\\#}$ would be a polynomial, violating Theorem 1.4. Hence $\eta(f^{\\#})$
is a nonzero summable homoclinic point in $X_{g}$.

Since $f^{\\#}\in\ell^{1}(\mathbb{Z}^{d},\mathbb{R})$, for every $H\geqslant
1$ there is an $R=R(f,f^{\\#},H)$ such that

$\sum_{\|\mathbf{n}\|\geqslant
R}|f^{\\#}_{\mathbf{n}}|<\frac{1}{2H\|f\|_{1}}.$

For every nonempty subset $\mathcal{S}\subset\mathbb{Z}^{d}$, let
$B_{R}(\mathcal{S})=\\{\mathbf{n}\in\mathbb{Z}^{d}\colon\operatorname{dist}(\mathbf{n},\mathcal{S})\leqslant
R\\}$. We put $M=3R$, and show that this separation satisfies the conclusion
of the theorem.

Suppose that $\mathcal{S}$ and $\mathcal{T}$ are nonempty subsets of
$\mathbb{Z}^{d}$ with $\operatorname{dist}(\mathcal{S},\mathcal{T})\geqslant
M=3R$, and let $\mathcal{U}=\mathcal{S}\cup\mathcal{T}$. Let
$p\in\mathcal{P}(\mathcal{S},H)$ and $q\in\mathcal{P}(\mathcal{T},H)$, and put
$r=p+q\in\mathcal{P}(\mathcal{U},H)$. We assume that $f$ divides $r$, and will
prove that $f$ divides both $p$ and $q$ individually.

Since $f$ divides $r$, there is a $k\in R_{d}$ such that $r=k\cdot f$. Then

$r\cdot f^{\\#}=(k\cdot f)\cdot f^{\\#}=k\cdot(f\cdot f^{\\#})=k\cdot h\in
R_{d}.$

Furthermore, we claim that

$\operatorname{supp}(r\cdot f^{\\#})\subset
B_{R}(\mathcal{U})=B_{R}(\mathcal{S})\cup B_{R}(\mathcal{T}).$

For if $\mathbf{n}\notin B_{R}(\mathcal{U})$ then
$r_{\mathbf{n}-\mathbf{m}}=0$ whenever $\|\mathbf{m}\|\leqslant R$. Hence

$|(r\cdot
f^{\\#})_{\mathbf{n}}|=\Bigl{|}\sum_{\mathbf{m}\in\mathbb{Z}^{d}}r_{\mathbf{n}-\mathbf{m}}f^{\\#}_{\mathbf{m}}\Bigr{|}\leqslant\|r\|_{\infty}\sum_{\|\mathbf{m}\|\geqslant
R}|f^{\\#}_{\mathbf{m}}|<H\cdot\frac{1}{2H\|f\|_{1}}\leqslant\frac{1}{2}.$

Since $r\cdot f^{\\#}=k\cdot h\in R_{d}$ has integer coefficients, this shows
that $\operatorname{supp}(r\cdot f^{\\#})\subset B_{R}(\mathcal{U})$.

Now let $u$ be the restriction of the integer polynomial $r\cdot f^{\\#}$ to
$B_{R}(\mathcal{S})$, so that $u\in R_{d}$ as well. We claim that $\|u-p\cdot
f^{\\#}\|_{\infty}<\frac{1}{2\|f\|_{1}}$.

Indeed, if $\mathbf{n}\in B_{R}(\mathcal{S})$, then
$\operatorname{dist}(\mathbf{n},\mathcal{T})>R$, so that

$\Bigl{|}(u-p\cdot f^{\\#})_{\mathbf{n}}\Bigr{|}=\Bigl{|}(q\cdot
f^{\\#})_{\mathbf{n}}\bigr{|}=\Bigl{|}\sum_{\mathbf{m}\in\mathcal{T}}q_{\mathbf{m}}f^{\\#}_{\mathbf{n}-\mathbf{m}}\Bigr{|}\leqslant\|q\|_{\infty}\sum_{\|\mathbf{k}\|\geqslant
R}|f^{\\#}_{\mathbf{k}}|<H\cdot\frac{1}{2H\|f\|_{1}}.$

On the other hand, if $\mathbf{n}\notin B_{R}(\mathcal{S})$, then

$|(u-p\cdot f^{\\#})_{\mathbf{n}}|=|(p\cdot
f^{\\#})_{\mathbf{n}}|=\Bigl{|}\sum_{\mathbf{m}\in\mathcal{S}}p_{\mathbf{m}}f^{\\#}_{\mathbf{n}-\mathbf{m}}\Bigr{|}\leqslant\|p\|_{\infty}\sum_{\|\mathbf{k}\|\geqslant
R}|f^{\\#}_{\mathbf{k}}|<H\cdot\frac{1}{2H\|f\|_{1}}.$

Observe that both $p\cdot f^{\\#}\cdot f=p\cdot h$ and $u\cdot f$ lie in
$R_{d}$. Furthermore,

$\|p\cdot f^{\\#}\cdot f-u\cdot f\|_{\infty}\leqslant\|p\cdot
f^{\\#}-u\|_{\infty}\cdot\|f\|_{1}<\frac{1}{2}.$

Hence $(p\cdot f^{\\#}-u)\cdot f=0$. Thus $p\cdot h=p\cdot f^{\\#}\cdot
f=u\cdot f$. Since $f$ is irreducible and does not divide $h$, it must divide
$p$. Then $f$ must also divide $q=r-p$. ∎

In order to interpret Corollary 1.7 for our purposes we first consider the
trivial (but instructive) case $f=0$. In this case the Pontryagin dual
$\widehat{X_{g}}$ of the group $X_{g}=\mathbb{T}^{\mathbb{Z}^{d}}$ in (2.1) is
equal to $R_{d}$ (cf. [Schmidt]*Example 5.2 (1)): every character (i.e.,
continuous group homomorphism) $\chi\colon X_{g}\to\mathbb{T}$ of $X_{g}$ is
of the form

$t\mapsto\chi_{p}(t)=\sum_{\mathbf{n}\in\mathbb{Z}^{d}}t_{\mathbf{n}}p_{\mathbf{n}}\pmod{1}$
(3.1)

for some
$p=\sum_{\mathbf{n}\in\mathbb{Z}^{d}}p_{\mathbf{n}}\mathbf{x}^{\mathbf{n}}\in
R_{d}$. If $p$ and $q$ are elements of $R_{d}$ with disjoint supports, the
corresponding characters are independent of each other; in particular, if
$p+q=0$, then both $p$ and $q$ are equal to zero.

Now assume that $f\neq 0$. In this case the Pontryagin dual of the group
$X_{g}$ in (2.1) is equal to $R_{d}/\langle f\rangle$ ([Schmidt]*Example 5.2
(2)): every character $\chi\in\widehat{X_{g}}$ is again of the form
$\chi_{p}\colon X_{g}\to\mathbb{T}$ in (3.1) for some
$p=\sum_{\mathbf{n}\in\mathbb{Z}^{d}}p_{\mathbf{n}}\mathbf{x}^{\mathbf{n}}\in
R_{d}$, but $\chi_{p}=\chi_{q}$ whenever $p-q\in\langle f\rangle$. In
particular, if $p\in\langle f\rangle$, then the corresponding character of
$X_{g}$ is trivial.

Although the notion of “support” no longer has an obvious meaning for
characters of $X_{g}$, the map $p\mapsto\eta(p\cdot f^{\\#})$ maps an entire
coset of $p$ in $R_{d}/\langle f\rangle$ to a single summable homoclinic point
of $X_{g}$. By bounding the coefficients of $p$ by $H$ and using
$R=R(f,f^{\\#},H)$ from the proof, most of the $\ell^{1}$-mass of $p\cdot
f^{\\#}$ is located within $B_{R}(\operatorname{supp}p)$, and we can think of
this set as the “approximate support” of $p$. If the supports of two
polynomials $p$ and $q$ are widely separated, then so are their approximate
supports, and multiplication of $p\cdot f^{\\#}$ and $q\cdot f^{\\#}$ by $f$
leads to Laurent polynomials $p^{\prime}=p\cdot f^{\\#}\cdot f$ and
$q^{\prime}=q\cdot f^{\\#}\cdot f$ with genuinely disjoint supports, in
analogy with the trivial case.

##  4\. Lacunary Independence and Exponential Recurrence

In this section we discuss some dynamical implications of the results in §1.
Let $\mathcal{A}$ be a finite set, $\nu_{0}$ be a probability measure on
$\mathcal{A}$, $Y=\mathcal{A}^{\mathbb{Z}^{d}}$ equipped with the product
topology, and $\nu=\nu_{0}^{\mathbb{Z}^{d}}$ be the product measure on $Y$.
Then the $\mathbb{Z}^{d}$-shift $\sigma_{Y}$ on $Y$ preserves $\nu$, and
$(Y,\nu,\sigma_{Y})$ is called a _Bernoulli $\mathbb{Z}^{d}$-shift_.

An algebraic $\mathbb{Z}^{d}$-action $\alpha$ on $X$ automatically preserves
the normalized Haar measure $\mu$ on $X$. The action $\alpha$ is called
_Bernoulli_ if there is a Bernoulli $\mathbb{Z}^{d}$-shift
$(Y,\nu,\sigma_{Y})$ and a measure-preserving equivariant map $\pi\colon X\to
Y$. If $\pi$ can be chosen to be continuous off a null set of $X$, then
$\alpha$ is called _finitarily Bernoulli_. For instance, a hyperbolic toral
automorphism generates a $\mathbb{Z}$-action that is finitarily Bernoulli
since it has Markov partitions [Keane-Smorodinsky]. However, if the toral
automorphism has eigenvalues of modulus 1, then although known to be
Bernoulli, it is not known whether it is finitarily Bernoulli. These cases
correspond to toral and atoral polynomials in one variable, respectively.

We address the question of whether analogues of this exist for algebraic
$\mathbb{Z}^{d}$-actions. Our starting point is the fact that if $f\in R_{d}$
is irreducible, then $\alpha_{f}$ is Bernoulli except for the special cases of
so-called generalized cyclotomic polynomials (see [Schmidt]*§23 and [Rudolph-
Schmidt]).

For an integer $R\geqslant 1$, let
$B_{R}=\\{\mathbf{n}\in\mathbb{Z}^{d}:\|\mathbf{n}\|\leqslant R\\}$. For an
integer $M\geqslant 1$, let $B_{R,M}=B_{R}\cap M\mathbb{Z}^{d}$. Let $|\cdot|$
denote cardinality, and observe that

$B_{R}\subset B_{R,M}+B_{M},\text{\quad and hence
\quad}|B_{R,M}|\geqslant\frac{|B_{R}|}{|B_{M}|}.$ (4.1)

As observed in [Keane-Smorodinsky], the following condition is necessary for
an algebraic action to be finitarily Bernoulli.

######  Definition 4.1.

An algebraic $\mathbb{Z}^{d}$-action $\alpha$ on $(X,\mu)$ is called
_exponentially recurrent_ provided that for every proper closed subset $K$ of
$X$ there are constants $c>0$ and $0<\theta<1$ such that for all $R\geqslant
1$ we have that

$\mu\Bigl{(}\bigcap_{\mathbf{n}\in B_{R}}\alpha^{\mathbf{n}}K\Bigr{)}\leqslant
c\,\theta^{|B_{R}|}$ (4.2)

The following shows that this condition is satisfied for some principal
actions.

######  Theorem 4.2.

Let $f\in R_{d}$ be irreducible and atoral. Then the principal algebraic
$\mathbb{Z}^{d}$-action $\alpha_{f}$ is exponentially recurrent with respect
to Haar measure $\mu_{f}$ on $X_{f}$.

###### Proof.

Having already established lacunary independence in Corollary 1.7, this proof
follows the path in [Lind-lacunary]*§2 where $d=1$.

Simplify notation by putting $X=X_{f}$, $\alpha=\alpha_{f}$, and
$\mu=\mu_{f}$. Let $K$ be a proper closed subset of $X$. Then $\mu(K)<1$. Fix
$\eta$ with $\mu(K)<\eta<1$.

Let $\varepsilon>0$ to be determined later. There is an open neighborhood $U$
of $K$ with $\mu(U\smallsetminus K)<\varepsilon$. By Urysohn’s lemma
[E-W]*Lemma A.27 there is a continuous function $r$ on $X$ such that
$0\leqslant r\leqslant 1$ on $X$, $r=1$ on $K$, and $r=0$ on $X\smallsetminus
U$. By the Stone-Weierstrass theorem [E-W]*Theorem 2.40, there is a finite
subset $\mathcal{E}\subset\mathbb{Z}^{d}$, a positive integer $H$, and an
$\mathbb{R}$-linear combination $\phi$ of the characters
$\\{\chi_{p}:p\in\mathcal{F}\coloneqq\mathcal{P}(\mathcal{E},H)\\}\subset\widehat{X_{g}}$
(defined as in (1.2) and (3.1)) such that $\|r-\phi\|_{C(X)}<\varepsilon$. Let
$\psi=\phi+\varepsilon$. Then $0\leqslant\psi\leqslant 1+\varepsilon$ on $X$,
$\psi\geqslant 1$ on $K$, and $\psi<\varepsilon$ on $X\smallsetminus U$. Hence

$\int_{X}\psi\,d\mu\leqslant(1+\varepsilon)\mu(K)+(1+\varepsilon)\mu(U\smallsetminus
K)+\varepsilon\mu(X\smallsetminus U)<\eta$

provided that $\varepsilon$ is small enough.

By Corollary 1.7, there is an $M=M(f,\mathcal{F})$ such that for every
$R\geqslant 1$ the sets
$\\{\mathbf{x}^{\mathbf{n}}\mathcal{F}\colon\mathbf{n}\in B_{R,M}\\}$ are
independent in the sense described there. This independence shows that the
constant term in the product $\prod_{\mathbf{n}\in
B_{R,M}}\mathbf{x}^{\mathbf{n}}\psi$ is the product of the constant terms of
the factors, which are all equal to $\int_{X}\psi\,d\mu$. Hence using (4.1),

$\displaystyle\mu\Bigl{(}\bigcap_{\mathbf{n}\in
B_{R}}\alpha^{\mathbf{n}}K\Bigr{)}$
$\displaystyle\leqslant\mu\Bigl{(}\bigcap_{\mathbf{n}\in
B_{R,M}}\alpha^{\mathbf{n}}K\Bigr{)}\leqslant\int_{X}\Bigl{(}\prod_{\mathbf{n}\in
B_{R,M}}\mathbf{x}^{\mathbf{n}}\psi\Bigr{)}\,d\mu$
$\displaystyle=\prod_{\mathbf{n}\in
B_{R,M}}\int_{X}\mathbf{x}^{\mathbf{n}}\psi\,d\mu<\eta^{|B_{R,M}|}\leqslant\eta^{|B_{R}|/|B_{M}|}=\theta^{|B_{R}|},$

where $\theta=\eta^{1/|B_{M}|}$ satisfies condition (4.2) for an appropriate
constant $c>0$. ∎

We remark that [Lind-lacunary] proves exponential recurrence when $d=1$ even
for toral polynomials, using $p$-adic roots to carry out an analogue of a
geometric argument.

##  5\. Remarks and Questions

The proof of the Gap Theorem 1.5 depends crucially on the assumption that $f$
is atoral, thereby providing a nonzero summable homoclinic point in the
associated dynamical system. If $f$ is toral it is known that no such points
exist, and this method of proof fails from the very beginning. Ironically, for
the toral $f$ in Example 1.3 (4), the associated dynamical system has
uncountably many points whose coordinates tend to 0 at $\infty$, but not fast
enough to be summable (these points are provided by the Fourier series of
measures supported on the positively curved unitary variety $\mathsf{U}(f)$,
see [Lind-homoclinic]*Exam. 7.3).

The extensions of results depending on atorality to the toral case typically
involve delicate — and at present unresolved — Diophantine questions. For
instance, the main result in [Lind-atoral]*Thm. 1.3 on the convergence of the
number of periodic components to entropy uses summable homoclinic points. A
completely different proof has been recently given in [Dimitrov-galois]*App. B
using algebraic number theory to give effective rates of convergence, and in
earlier work [Dimitrov-mahler] this has been partially extended to the toral
case using intricate Diophantine arguments.

######  Question 5.1.

Is the Gap Theorem valid for irreducible toral polynomials in $R_{d}$?

Lacunary independence can be formulated quite generally. If $A$ is an abelian
group, say that a finite collection $\\{E_{j}\colon 1\leqslant j\leqslant
n\\}$ of finite subsets of $A$ is _independent_ if whenever $e_{j}\in E_{j}$
for all $j$ and $\sum_{j=1}^{n}e_{j}=0$, then each $e_{j}=0$. Call an
$R_{d}$-module $A$ _lacunarily independent_ if for every finite subset
$F\subset A$ there is an $M=M(F,A)$ such that if $\mathcal{S}$ is a finite
subset of $\mathbb{Z}^{d}$ with $\|\mathbf{m}-\mathbf{n}\|\geqslant M$ for
distinct $\mathbf{m},\ \mathbf{n}\in\mathcal{S}$, then the sets
$\\{\mathbf{x}^{\mathbf{n}}\cdot F\colon\mathbf{n}\in\mathcal{S}\\}$ are
independent. Corollary 1.7 shows that if $f$ is irreducible and atoral, then
$R_{d}/\langle f\rangle$ is lacunarily independent as an $R_{d}$-module.

######  Question 5.2.

Which $R_{d}$-modules are lacunarily independent? In particular, is
$R_{2}/\langle x-2,y-3\rangle$ lacunarily independent?

Some conditions beyond those in the Gap Theorem are clearly necessary. For
instance, let $\mathfrak{p}$ be the prime ideal $\langle 2,1+x+y\rangle$ in
$R_{2}$. Then in the quotient $R_{2}/\mathfrak{p}$, the Frobenius squaring map
shows that $(1+x+y)^{2^{n}}=1+x^{2^{n}}+y^{2^{n}}$ for all $n\geqslant 1$, so
that $1+x+y$ divides a sum of widely spaced monomials. More interesting is the
case where $f(x)$ and $g(y)$ are irreducible monic polynomials with constant
term $\pm 1$. The algebraic $\mathbb{Z}^{2}$-action associated with
$R_{2}/\langle f(x),g(y)\rangle$ is generated by a pair of commuting toral
automorphisms. Such actions are known to have strong mixing properties that
are restricted versions of lacunary independence (see [Schmidt]*§27), but
proofs of these do not appear to give full lacunary independence.

There is a complete characterization of which algebraic
$\mathbb{Z}^{d}$-actions are Bernoulli [Schmidt]*§23. Although we have shown
that some algebraic $\mathbb{Z}^{d}$-actions satisfy a necessary condition to
be finitarily Bernoulli, we are far from being able to characterize such
actions.

######  Question 5.3.

Which algebraic $\mathbb{Z}^{d}$-actions are finitarily Bernoulli?

## References


boundary conditions are set to $u_{k}(t_{i})=0$, $\lambda(t_{i})=0$, from
which we have $\dot{u}_{k}(t_{i})=k(\lambda(t_{i})-u_{k}(t_{i}))=0$. We set
$\lambda(t_{f})=\lambda_{f}$, thus, we have
$\dot{u}_{k}(t_{f})=k(\lambda_{f}-mt)$. Let us now set $t_{i}=0$ The total
work is, therefore

$W=m^2t_{f}+\frac{k}{2}\left(\lambda_{f}-mt_{f}\right)^2$ (132)

The optimal solution is given by $m^\star=\frac{k\lambda_{f}}{2+kt_{f}}$.
Thus, the optimal protocol can be expressed as

$\lambda_{k}^\star=\frac{\lambda_{f}(1+kt)}{2+kt_{f}}$ (133)

and the optimal work is

$W_{k}^\star=\frac{k\lambda_{f}^2}{2+kt_{f}}$ (134)

Note that we observe initial and final jumps in the protocol, i.e.,

$\displaystyle\Delta\lambda_{k}(t_{i})=\lim_{t\rightarrow
t_{i}^+}\lambda(t)-\lambda(t_{i})\equiv\Delta\lambda_{k}(t_{f})$
$\displaystyle=\lambda_{k}(t_{f})-\lim_{t\rightarrow
t_{f}^-}\lambda_{k}(t)=\frac{\lambda_{f}}{2+kt_{f}}$ (135)

Let us now focus on the case where $\lambda_{k}$ is used when the stiffness
parameter is $\kappa$, which is not necessarily equal to $k$. In this case, we
take the protocol $\lambda_{k}(t)=\frac{\lambda_{f}(1+kt)}{2+kt_{f}}$ for
$t_{i}\leq t\leq t_{f}$, and $\lambda(0)=0$, $\lambda(t_{f})=\lambda_{f}$, and
plug it to the Langevin equation

$\dot{u}_{\kappa}=\kappa(\lambda_{k}-u_{\kappa})$

together with the initial condition $u_{\kappa}(0)=0$. By solving the
differential equation, we get

$\displaystyle
u_{\kappa}(t)=\frac{\lambda_{f}}{(2+kt_{f})}\left(kt+\left(1-\frac{k}{\kappa}\right)(1-e^{-\kappa
t})\right)$ $\displaystyle=u_{k}(t)+\Delta_{k,\kappa}(t)$ (136)

where $\Delta_{k,\kappa}(t)$ denotes the difference between $u_{k}$ and
$u_{\kappa}$ for the same protocol $\lambda_{k}$. Note that $u_{\kappa}$ boils
down to $u_{k}(t)=m^\star t$ for $k=\kappa$, as expected. The velocity is then

$\dot{u}_{\kappa}(t)=\frac{\lambda_{f}}{(2+kt_{f})}\left(k+(\kappa-k)e^{-\kappa
t}\right)$

By plugging into the Eq. (130), we obtain that the work can be expressed as

$\displaystyle W_{\kappa}[\lambda_{k}(t)]$ $\displaystyle=$
$\displaystyle\int_{0}^{t_{f}}\mathrm{d}t\dot{u}^2_{\kappa}(t)+\frac{[\dot{u}^2_{\kappa}]_{t_{f}}^{t_{i}}}{2\kappa}$
(137) $\displaystyle=$
$\displaystyle\int_{0}^{t_{f}}\mathrm{d}t\frac{\lambda_{f}^2}{(2+kt_{f})^2}\left(k+(\kappa-k)e^{-\kappa
t}\right)^2+\frac{\kappa}{2}(\lambda_{f}-u_{\kappa}(t_{f}))^2$
$\displaystyle=$
$\displaystyle\frac{\lambda_{f}^2}{(2+kt_{f})^2}\left(k^2t_{f}+\frac{2k}{\kappa}(\kappa-k)(1-e^{-\kappa
t_{f}})+\frac{(k-\kappa)^2}{2\kappa}(1-e^{-2\kappa t_{f}})\right)$
$\displaystyle+\frac{\lambda_{f}^2}{2(2+kt_{f})^2\kappa}\left((k+\kappa)^2+2(k+\kappa)(\kappa-k)e^{-\kappa
t_{f}}+(k-\kappa)^2e^{-2\kappa t_{f}}\right)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{f}^2}{(2+kt_{f})^2}\left[(2k+k^2t_{f})+\frac{\kappa^2-k^2}{\kappa}+\frac{(k-\kappa)^2}{\kappa}e^{-\kappa
t_{f}}\right]$

Let us now consider that $\kappa$ is distributed by the distribution
$p(\kappa)$. We choose our $k$ determining our protocol to be
$k=\bar{\varkappa}=\int\mathrm{d}\varkappa\,\varkappa\,p(\varkappa)$. Then the
expected work (unadapted) $\bar{W}=\int\mathrm{d}\kappa
W_{\kappa}(\lambda_{\bar{\kappa}}(t))$ is equal to

$\displaystyle
W_{unad.}\equiv\bar{W}=\frac{\lambda_{f}^2\bar{\kappa}}{(2+\bar{\kappa}t_{f})}+\frac{\lambda_{f}^2\bar{\kappa}}{(2+\bar{\kappa}t_{f})^2}\left(\bar{\kappa}-\bar{\kappa}^2\int\mathrm{d}\kappa\frac{1}{\kappa}p(\kappa)\right)$
$\displaystyle+\frac{\lambda_{f}^2}{(2+\bar{\kappa}t_{f})^2}\int\mathrm{d}\kappa\frac{(\bar{\kappa}-\kappa)^2}{\kappa}e^{-\kappa
t_{f}}p(\kappa)=W_{\bar{\kappa}}[\lambda_{\bar{\kappa}}(t)]+W_{diss}$ (138)

On the other hand, if we were able to choose the optimal protocol for each
$\kappa$, the expected work (adapted) would be

$W_{ad.}=\overline{W_{\kappa}[\lambda_{\kappa}(t)]}=\int\mathrm{d}\kappa\frac{\lambda_{f}^2\kappa}{2+\kappa
t_{f}}p(\kappa)$ (139)

Since $\frac{\kappa}{2+\kappa t_{f}}$ is a concave function, we obtain from
the Jensen inequality that

$W_{ad.}\leq W_{unad.}$ (140)

##  Appendix B Brief review of stochastic thermodynamics

In this section we present some additional details of ordinary, no-uncertainty
stochastic thermodynamics, and review some of the main associated results. We
leave $N$ and associated parameters like the inverse temperatures $\beta^\nu$
and chemical potentials $\mu^\nu$ implicit. We write the rate matrix for going
from state $x^{\prime}$ to state $x$ due to stochastic exchanges of heat and /
or particles with reservoir $\nu$ as $K_{x,x^{\prime}}^\nu(t)$. From now on,
we leave the time index implicit. The full rate matrix of the system is
$K_{x,x^{\prime}}=\sum_{\nu=1}^{N}K_{x,x^{\prime}}^{\nu}$, and so the master
equation of the CTMC is
$\dot{p}_{t}(x)=\sum_{x^{\prime}}K_{x,x^{\prime}}p_{t}(x^{\prime})$.

Often in the literature, due to considerations involving time-symmetric
microscale dynamics, we assume that at any (implicit) time $t$ the separate
matrices $K_{x,x^{\prime}}^{\nu}$ each satisfy _local detailed balance_ (LDB)
with respect to the energy level $u(x)$ at that time, for the particle
reservoir $\nu$:

$\frac{K^{\nu}_{x,x^{\prime}}}{K^{\nu}_{x^{\prime}x}}=\frac{\pi^{\nu}_{x}}{\pi^{\nu}_{x^{\prime}}}=e^{-\beta^\nu[(u(x)-u(x^{\prime}))-\mu^\nu(n^\nu(x)-n^\nu(x^{\prime}))]}\,.$
(141)

where $u(x)$ is the energy level of the system in state $x$ and $n^\nu(x)$ is
the number of particles of the type specified by $\nu$ when the system is in
state $x$. (If the reservoir $\nu$ does not exchange particles with the
system, then $\mu^\nu=n^\nu(x)=0$.)

Note that in the absence of chemical reservoirs, the map from rate matrices to
energy functions is single-valued (up to an overall additive constant), but
the inverse map is multi-valued.

#### Ensemble thermodynamics

The ensemble internal energy is written as $U_{t}:=\langle
u_{t}\rangle=\sum_{x}p_{t}(x)u_{t}(x)$. The system exchanges particles with
some of the reservoirs, as well exchanging energy with each of them directly
(e.g., via kinetic molecular collisions). _The first law of thermodynamics_
can be formulated as

$\Delta U_{t}=\pmb{Q}_{t}+\pmb{W}_{t}+\pmb{C}_{t}$ (142)

where $\Delta U_{t}:=U_{t}-U_{0}$ and
$\pmb{Q}_{t}:=\int_{0}^tdt^{\prime}\pmb{Q}_{t^{\prime}}$ is total heat flow
into the system during the interval $[0,t]$,
$\pmb{W}_{t}:=\int_{0}^tdt^{\prime}\pmb{W}_{t^{\prime}}$ is the total work on
the system during that interval, and
$\pmb{C}_{t}:=\int_{0}^tdt^{\prime}\pmb{C}_{t^{\prime}}$ is the total chemical
work during that interval. The heat flow rate $\dot{\pmb{Q}}_{t}$ can be
decomposed into heat flows in from the separate reservoirs, i.e., direct
energy flows in from the separate reservoirs:

$\displaystyle\dot{\pmb{Q}}_{t}$
$\displaystyle=\sum_{\nu}\dot{\pmb{Q}}^\nu_{t}$
$\displaystyle=\sum_{\nu}\sum_{xx^{\prime}}K_{x,x^{\prime}}^{\nu}p_{t}(x)(u_{t}(x)-\mu^\nu
n^\nu(x))\,.$ (143)

For simplicity, we ignore the possibility of more than one distinguishable
type of particle. The mechanical work flow and chemical work flow are defined
as

$\displaystyle\dot{\pmb{W}}_{t}$
$\displaystyle=\sum_{x}p_{t}(x)\dot{u}_{t}(x)$ (144)
$\displaystyle\dot{\pmb{C}}_{t}$
$\displaystyle=\sum_{x}\dot{p}_{t}(x)\sum_{\nu}\mu^\nu n^\nu(x)\,.$ (145)

Shannon entropy at time $t$ is defined as $S_{t}:=\langle
s_{t}\rangle=-\sum_{x}p_{t}(x)\ln p_{t}(x)$. The _second law of
thermodynamics_ can be written as

$\displaystyle\Delta S_{t}=\pmb{\Sigma}_{t}+\pmb{\mathcal{E}}_{t}$ (146)

where $\Delta S_{t}=S_{t}-S_{0}$, $\pmb{\Sigma}_{t}$ is the entropy production
(EP) and $\pmb{\mathcal{E}}_{t}$ is the entropy flow (EF). The EF rate is

$\displaystyle\dot{\pmb{\mathcal{E}}}_{t}=-\sum_{\nu}\sum_{x,x^{\prime}}K^\nu_{x,x^{\prime}}p(x^{\prime})\ln\dfrac{K^\nu_{x,x^{\prime}}}{K^\nu_{x^{\prime},x}}$
(147)

so that the EP rate is

$\displaystyle\dot{\pmb{\Sigma}}_{t}=-\sum_{\nu}\sum_{x,x^{\prime}}K^\nu_{x,x^{\prime}}p(x^{\prime})\ln\dfrac{K^\nu_{x,x^{\prime}}p_{t}(x^{\prime})}{K^\nu_{x^{\prime},x}p_{t}(x)}\,.$
(148)

The second law of thermodynamics is enforced by the fact that for any rate
matrix, EP rate is non-negative, i.e., $\dot{\pmb{\Sigma}}_{t}\geq 0$. When
LDB holds, the EF rate can be expressed in terms of _thermodynamic entropy_ ,
i.e.,

$\displaystyle\dot{\pmb{\mathcal{E}}}_{t}=\sum_{\nu}\beta^\nu\dot{\pmb{Q}}^{\nu}_{t}\,.$
(149)

#### Trajectory thermodynamics

The _trajectory internal energy_ is written as $\pmb{u}_{t}({\pmb{x}})$. The
first law of thermodynamics on the trajectory level for any time $t$ is

$\frac{\mathrm{d}}{\mathrm{d}t}\pmb{u}_{t}({\pmb{x}})=\dot{\pmb{q}}_{t}({\pmb{x}})+\dot{\pmb{w}}_{t}({\pmb{x}})+\dot{\pmb{c}}_{t}({\pmb{x}})$
(150)

where

$\displaystyle\dot{\pmb{q}}_{t}({\pmb{x}})$
$\displaystyle=\sum_{\nu}\dot{\pmb{q}}_{t}^\nu({\pmb{x}})$
$\displaystyle=\sum_{\nu}\sum_{x}\dot{\delta}_{x,{\pmb{x}}(t)}(u_{t}({x})-\mu^\nu
n^\nu(x)),$ (151) $\displaystyle\dot{\pmb{w}}_{t}({\pmb{x}})$
$\displaystyle=\sum_{x}\delta_{x,{\pmb{x}}(t)}\dot{u}_{t}{(x)},$ (152)
$\displaystyle\dot{\pmb{c}}_{t}({\pmb{x}})$
$\displaystyle=\sum_{\nu}\sum_{x}\dot{\delta}_{x,{\pmb{x}}(t)}\mu^\nu
n^\nu(x)$ (153)

are called the trajectory heat, trajectory mechanical work and trajectory
chemical work, respectively.

Trajectory entropy is defined as $s_{t}({\pmb{x}}):=-\ln p_{t}({\pmb{x}}(t))$.
Then time derivative of entropy can be decomposed as

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\pmb{s}_{t}({\pmb{x}})$
$\displaystyle=\dot{\pmb{\sigma}}_{t}({\pmb{x}})+\dot{\pmb{\epsilon}}_{t}({\pmb{x}})$
(154)

where due to LDB, _trajectory EF rate_ is

$\displaystyle\dot{\pmb{\epsilon}}({\pmb{x}})=\sum_{\nu}\beta_{\nu}\dot{\pmb{q}}^\nu({\pmb{x}})=\sum_{x}\dot{\delta}_{x,{\pmb{x}}(t)}\sum_{\nu}\ln\dfrac{K^\nu_{x,x^{\prime}}}{K^\nu_{x^{\prime},x}}\,.$
(155)

ensemble-level versions, i.e.,
$\boldlangle{\dot{\pmb{q}}}^\nu_{t}\boldrangle=\dot{\pmb{Q}}^\nu_{t}$,
$\boldlangle\dot{\pmb{\epsilon}_{t}}\boldrangle=\dot{\pmb{\mathcal{E}}}_{t}$,

##  Appendix C Proof that the $H^*$ solving Eq. 91 is unique

Write the (countable) elements of $X$ as $1,2,\ldots$, ordered so that
$\overline{p}_{1}(t_{i})\geq\overline{p}_{2}(t_{i})\geq\ldots$. For
simplicity, we assume that $\overline{p}_{|X|}(t_{i})>0$, i.e., we assume that
all elements of $X$ have nonzero probability under $\overline{p}(t_{i})$; it
is straightforward to extend the analysis below to the case where
$\overline{p}_{x}(t_{i})=0$ for some $x$, by setting $H^*(x)=\infty$ for any
such $x$.

By inspection, if there is a solution $H^*(x)=H(x)$ for some given
$\overline{p}(t_{i})$ and $\mathrm{d}P_{\alpha}$, then $H(x)+k$ is also a
solution for that $\overline{p}(t_{i})$ and $\mathrm{d}P_{\alpha}$, for any
real number $k$. Accordingly, wolog set $H^*(1)=1$. For use below, write the
maximal element of $\operatorname*{supp}(\mathrm{d}P_{\beta^\alpha})$ as
$\beta_{max}$.

The proof is by iterative construction of $H^*(x)$. To begin, we set
$H^*(i)=i$ for all $i$. We then run a two-step iterative procedure, from $i=2$
to $i=|X|$, changing each element $H^*(i)$ in turn so that

$\displaystyle\dfrac{\int\mathrm{d}P^\alpha
p^{eq}_{H^*,{\beta^\alpha}}(i-1)}{\int\mathrm{d}P^\alpha
p^{eq}_{H^*,{\beta^\alpha}}(i)}$
$\displaystyle=\dfrac{\overline{p}_{i-1}(t_{i})}{\overline{p}_{i}(t_{i})}$
(156)

When this iterative procedure finishes, we will know that Eq. 156 is satisfied
for all $i\geq 2$ for the finishing $H^*$, and so Eq. 91 is met for that
$H^*$.

In the first step of the iterative procedure, if
$\overline{p}_{i}(t_{i})=\overline{p}_{i-1}(t_{i})$, we set $H^*(i)=H^*(i-1)$.
This guarantees that Eq. 156 holds for this particular $i$.

If instead $\overline{p}_{i}(t_{i})<\overline{p}_{i-1}(t_{i})$ for this $i$,
then we execute the second step of the iterative procedure. Consider two
candidate Hamiltonians, $H^+(x)$ and $H^-(x)$, which both equal the current
$H^*(j)$ for all $j\neq i$. We complete their definitions by setting
$H^+(i)=H^*(i)$ and $H^-(i)=\kappa$, where

$\displaystyle\kappa>H^-(i)+\beta_{max}^{-1}\ln\dfrac{\overline{p}_{i-1}(t_{i})}{\overline{p}_{i}(t_{i})}$
(157)

This guarantees that

$\displaystyle\dfrac{\overline{p}_{i-1}(t_{i})}{\overline{p}_{i}(t_{i})}$
$\displaystyle<\dfrac{e^{-\beta_{max}H^-(i-1)}}{e^{-\beta_{max}H^-(i)}}$ (158)

Plugging in these definitions establishes both that

$\displaystyle\dfrac{\int\mathrm{d}P^\alpha
p^{eq}_{H^+,{\beta^\alpha}}(i-1)}{\int\mathrm{d}P^\alpha
p^{eq}_{H^+,{\beta^\alpha}}(i)}$
$\displaystyle<\dfrac{\overline{p}_{i-1}(t_{i})}{\overline{p}_{i}(t_{i})}$
(159)

and

$\displaystyle\dfrac{\int\mathrm{d}P^\alpha
p^{eq}_{H^-,{\beta^\alpha}}(i-1)}{\int\mathrm{d}P^\alpha
p^{eq}_{H^-,{\beta^\alpha}}(i)}$
$\displaystyle>\dfrac{\overline{p}_{i-1}(t_{i})}{\overline{p}_{i}(t_{i})}$
(160)

$\int\mathrm{d}P^\alpha p^{eq}_{H^*,{\beta^\alpha}}(i)$ is a differentiable
function of $H^*(i)$, and $H^+(j)=H^-(j)=H^*(j)$ for all $j\neq i$. Therefore
applying the intermediate value theorem with Eq. 159 and Eq. 160 means that
there is some value $H^*(i)$ such that Eq. 156 holds. The second step of the
iterative procedure finishes by setting $H^*(i)$ to this intermediate value.
At that point we increment $i$ by $1$.

At the end of the second step for any particular $i$, we have established that
Eq. 156 holds for that $i$ in both of the possible situations
$\overline{p}_{i}(t_{i})=\overline{p}_{i-1}(t_{i})$ and
$\overline{p}_{i}(t_{i})<\overline{p}_{i-1}(t_{i})$. Therefore by the end of
the iterative procedure we have established that Eq. 156 holds for all $i$.

This completes the proof that there is a solution for $H^*(x)$, as claimed.
Uniqueness of this solution follows from the fact that the derivative of
$\int\mathrm{d}P^\alpha p^{eq}_{H^*,{\beta^\alpha}}(i)$ with respect to
$H^*(i)$ is non-negative, and so the intermediate value arising in the second
step of the iterative procedure is unique.

##  Appendix D Proof of non-increasing value of information for no-uncertainty
rate matrices

Consider the case where the rate matrix $K_{xx^{\prime}}(t)$ during the
interval $t\in[t_{i},\tau)$ has no uncertainty. We can see how the
thermodynamic value of information of $\alpha$ depends on $\tau$ in this case,
by taking the derivative of the LHS of $I_{\tau}$ with respect to $\tau$:

$\displaystyle\dfrac{dI_{\tau}}{d\tau}$
$\displaystyle=\dfrac{\mathrm{d}S\left(\overline{p}_{\tau}(X)\right)}{\mathrm{d}\tau}-\int\mathrm{d}P^\alpha\dfrac{\mathrm{d}S\left(p_{\tau}^\alpha\right)}{\mathrm{d}\tau}$
$\displaystyle=\int\mathrm{d}P^\alpha\,\sum_{x,x^{\prime}}K_{xx^{\prime}}(\tau)p^\alpha_{\tau}(x^{\prime})\left[\ln\dfrac{\int\mathrm{d}P^{\alpha^{\prime}}K_{xx^{\prime}}(\tau)p_{\tau}^{\alpha^{\prime}}(x^{\prime})}{\int\mathrm{d}P^{\alpha^{\prime}}K_{x^{\prime}x}(\tau)p_{\tau}^{\alpha^{\prime}}(x)}-\ln\dfrac{K_{xx^{\prime}}(\tau)p_{\tau}^\alpha(x^{\prime})}{K_{x^{\prime}x}(\tau)p_{\tau}^\alpha(x)}\right]$
(161)

Therefore the time-derivative of the value of information reduces to a
difference of EP rates, distinguished from each other by whether we know
$\alpha$ or not. In addition, if we multiply and divide by $P_{\alpha}$ inside
the rightmost logarithm in Eq. 161 (i.e., change the two conditional
probability distributions into joint probability densities), and consider
$\mathrm{d}P^\alpha=\mathrm{d}\alpha P^\alpha$ we get

$\displaystyle\dfrac{dI_{\tau}}{d\tau}$
$\displaystyle=\sum_{x,x^{\prime}}\int\mathrm{d}\alpha
P^\alpha\,K_{xx^{\prime}}(\tau)p^\alpha_{\tau}(x^{\prime})\left[\ln\dfrac{\int\mathrm{d}\alpha
P^{\alpha^{\prime}}K_{xx^{\prime}}(\tau)p_{\tau}^{\alpha^{\prime}}(x^{\prime})}{\int\mathrm{d}\alpha
P^{\alpha^{\prime}}K_{x^{\prime}x}(\tau)p_{\tau}^{\alpha^{\prime}}(x)}-\ln\dfrac{P^\alpha
K_{xx^{\prime}}(\tau)p_{\tau}^\alpha(x^{\prime})}{P^\alpha
K_{x^{\prime}x}(\tau)p_{\tau}^\alpha(x)}\right]$ (162)

In particular, if $P_{\alpha}$ is a probability mass function, $p(\alpha)$,
then Eq. 162 can be written in terms of the joint probability
$p_{\tau}(x,\alpha)=p(\alpha)p^\alpha_{\tau}(x)$:

$\displaystyle\sum_{x,x^{\prime}}\sum_{\alpha}K_{xx^{\prime}}(\tau)p_{\tau}(x^{\prime},\alpha)\left[\ln\dfrac{\sum_{\alpha^{\prime}}K_{xx^{\prime}}(\tau)p_{\tau}(x^{\prime},\alpha^{\prime})}{\sum_{\alpha^{\prime}}K_{x^{\prime}x}(\tau)p_{\tau}(x,\alpha^{\prime})}-\ln\dfrac{K_{xx^{\prime}}(\tau)p_{\tau}(x^{\prime},\alpha)}{K_{x^{\prime}x}(\tau)p_{\tau}(x,\alpha)}\right]$
(163)

Each $(x,x^{\prime})$ pair in the outer sum in Eq. 163 in which $x=x^{\prime}$
gives a value of zero, since both of the logarithms equal zero if
$x^{\prime}=x$. In addition, $K^{xx^{\prime}}(\tau)$ is non-negative for all
$x^{\prime}\neq x$. So we can apply the log-sum inequality (assuming $a_{i}$
and $b_{i}$ non-negative):

$\sum_{i}a_{i}\log\frac{a_{i}}{b_{i}}\geq\left(\sum_{i}a_{i}\right)\log\frac{\sum_{i}a_{i}}{\sum_{i}b_{i}}$
(164)

separately for each $(x,x^{\prime}\neq x)$ pair. This means that
$\frac{\mathrm{d}I_{\tau}}{\mathrm{d}\tau}\leq 0$, as claimed.