# Digital Twin Mobility Profiling: A Spatio-Temporal Graph Learning Approach

Xin Chen1, Mingliang Hou1, Tao Tang2, Achhardeep Kaur2 and Feng Xia2(🖂)
1School of Software, Dalian University of Technology, Dalian 116620, China
2School of Engineering, IT and Physical Sciences, Federation University
Australia, Ballarat, VIC 3353, Australia
{xin.chen.jx; teemohold<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

With the arrival of the big data era, mobility profiling has become a viable
method of utilizing enormous amounts of mobility data to create an intelligent
transportation system. Mobility profiling can extract potential patterns in
urban traffic from mobility data and is critical for a variety of traffic-
related applications. However, due to the high level of complexity and the
huge amount of data, mobility profiling faces huge challenges. Digital Twin
(DT) technology paves the way for cost-effective and performance-optimised
management by digitally creating a virtual representation of the network to
simulate its behaviour. In order to capture the complex spatio-temporal
features in traffic scenario, we construct alignment diagrams to assist in
completing the spatio-temporal correlation representation and design dilated
alignment convolution network (DACN) to learn the fine-grained correlations,
i.e., spatio-temporal interactions. We propose a digital twin mobility
profiling (DTMP) framework to learn node profiles on a mobility network DT
model. Extensive experiments have been conducted upon three real-world
datasets. Experimental results demonstrate the effectiveness of DTMP.

###### Index Terms:

digital twin, mobility profiling, spatio-temporal graph learning,
transportation network, cyber-physical system, graph convolution network

##  I Introduction

With the emergence of industrial revolution Industry 4.0, Industrial Internet-
of-Things (IIoT) and smart city, Intelligent Transportation System (ITS) is
anticipated to satisfy various technical demands and diverse, realistic
requirements concerning accuracy, environment friendly and reliability [1, 2].
Nowadays, ITSs are an essential topic from both a technological and a business
standpoint. They mostly rely on Transportation Engineering models and
algorithms, which are profoundly affecting human travel habits. Furthermore,
the proliferation mobility data bring us to the era of transportation big
data. To efficiently leverage mobility data, the concept of mobility profiling
has been proposed as a promising way to design and virtualized ITS for urban
traffic management and planning. Mobility profiling offers an effective
simulation approach to meet the diverse use case requirements. It is a
critical enabler for realising ITS application with rigorous accuracy
requirements on a cloud computing platform.

Mobility profiles are a summary of mobility-specific characteristics such as
average speed and mobility flow [3]. The process of extracting an entity’s
mobility patterns from data, such as a user’s mobility interest and the crowd
flow of a bus stop, is referred to as mobility profiling. Effective mobility
profiling may aid our understanding of urban transportation, provide accurate
predictions or projections, and increase the efficiency of government or
policymakers’ decision-making.

However, due to the high complexity of the industrial process of mobility
data, directly employing mobility profiling in ITS suffers significant
challenges. Existing ITS applications, in particular, are unable to capture
the possible association retained in mobility data [4]. It’s essential to use
mobility profiles to find urban mobility trends based on data acquired by IoT
devices and to provide precise and reliable predictions. These predictions can
be utilized to provide reliable recommendations to human planners and to carry
out transportation management autonomously.

Figure 1: Virtual replica of physical transportation network using mobility
network DT model.

The recent advancement in Digital Twin (DT) technology opens up a lot of
possibilities for cyber-physical integration. Cyber-Physical System (CPS) is
transformational technology that manage integrated systems between their
physical assets and computing capability[5]. DT is a technology that creates
reflection of a physical net in the digital world. We agree that mobility
profiling can greatly benefit from a mobility network DT. Firstly, a mobility
DT creates a virtual replica of the physical transportation network, as shown
in Fig. 1 and can be used to do a variety of data analyses on various what-if
scenarios without disrupting the physical transportation network [6].
Secondly, a mobility network DT can generate accurate mobility profiles to
support downstream prediction tasks after any configuration changes. A DT of
the transportation network is crucial to achieve cost-efficient and
performance-optimal transportation management, as well as to continue
monitoring ITS performance under a set of operational circumstances without
modifying the physical network.

Artificial Intelligence (AI) techniques have recently advanced to the point
where they can now meet the requirements of mobility network development [4,
7]. When it comes to processing and analyzing large amounts of transportation
data, cloud computing stands out as the most efficient method available [8, 9,
10]. However, an intelligent DT system can only be developed by applying
advanced AI techniques to the collected data. The Internet of Things (IoT) is
used to collect massive mobility data from the physical world, and then use
the data as input into an AI model to construct a digital twin. As a last
phase, the developed DT can be used to optimize the performance of the ITS. On
the other hand, transportation networks are fundamentally represented in
graphs, many adaptive graph approaches [11, 12, 13] have emerged to learn the
spatio-temporal features based on Graph Neural Network (GNN). These approaches
have achieved outstanding performance by introducing graph learning into their
frameworks without a priory knowledge. In contrast to this, most existing
approaches employ a graph to depict the interaction between nodes, which may
not be suited to capture finely grained data-source specific patterns. As
shown in Fig. 2, node relationships include sequence similarity relationships
as well as temporal delay or lead relationships. Aside from that, current
approaches [14, 15, 16] always seek to learn spatial and temporal dependencies
individually, ignoring interactions between them. As a result, these models
are limited in their ability to provide reliable profiling results on graph-
structured data while capturing complicated spatio-temporal connections. This
constraint limits the creativity of a mobility network DT and its
implementation in ITS.

(a) Stations dependency under spatial and temporal.

(b) Flow from A, B and C stations. Stations A, B and C are on the same line
and station C has some similarity in flow to stations A and B, with B having a
smaller temporal shift and A having a larger temporal shift.

Figure 2: An illustrative example of spatio-temporal dependency.

To address the aforementioned limitations, we design a mobility network
profiling model (called DTMP) based on the Spatio-Temporal Graph Neural
Network to discover the complicated and fine-grained spatial and temporal
correlations in physical transportation networks and generate the accurate
node profiles to support downstream analytical tasks in cloud platform. In
particular, we develop a learning mechanism that is capable of learning
complex mobility patterns without relying on domain knowledge of network
structure. A supervised end-to-end learning process can be used to create the
adaptive station profiles from the data collected. Furthermore, we design an
unique alignment graph to help express fine-grained spatio-temporal
interactions and propose a dilated alignment convolution network (DACN) to
learn it. Finally, We employ gated Temporal Convolution Network (TCN) [11] to
assist the adaptive DACN to learn the temporal correlations.

In this paper, we present the spatio-temporal mobility network DT based on
mobility data collected by automatic fare collection system deployed on buses.
The mobility profiles refer to the learned virtual representations of bus
stations, which can be utilized to support downstream analytical tasks. A
summary of this paper’s major contribution is given below:

  1. 1.

We propose to use spatial and alignment graphs to represent fine-grained
spatio-temporal interactions and propose to alignment graph convolutional
network (AGCN) to capture spatio-temporal dependencies in urban mobility.

  2. 2.

We propose a novel framework digital twin mobility profiling (DTMP) to learn
node profiles on a mobility network DT. To develop accurate node profiles, the
DTMP employs DACN and parallelizes gated TCN.

  3. 3.

We undertake comprehensive experiments on three real-world data sets.
Experimental results demonstrate the effectiveness of the proposed framework.
We have released the code on GitHub111https://github.com/chenxino/DTMP.

The remainder of the paper is set out as follows. We briefly cover related
work in Section 2. The details of the proposed framework DTMP are given in
Section 3. Performance evaluation will be presented in Section 4 where
experimental results are analyzed. Finally, we conclude the paper in Section
5.

##  II Related Work

This work is closely related to digital twins, mobility profiling and spatio-
temporal graph learning. We briefly introduce previous work in this section.

###  II-A Digital Twins

Digital twins can be defined as machines or computer-based models that are
mirroring, emulating, simulating, or ‘twining’ the life of a physical
entity[17, 18]. Specifically, a Digital Twin can be any of the following three
types: 1) Digital Twin Prototype (DTP); 2) Digital Twin Instance (DTI); and 3)
Digital Twin Aggregate (DTA) [19]. A DTP is a completely digital model of a
not been created thing, whereas a DTI is a virtual twin of an existing object
that focuses on only one of its properties. A DTA is a collection of DTIs that
may be an exact digital replica of a previously existing physical object.

Various types of digital twins are developed in ITS, including DT for
automobile components, vehicles, vehicular networks, and road infrastructures.
For example, Wang et al. [20] proposed a framework for providing advisory
speed assistance to the driver by vehicle-to-cloud (V2C) communication network
DT. Cioroaica et al. [21] developed a digital twin, called hardware-in-the-
loop (vHiL) tested model to evaluate the trust-building capability of smart
systems within an ecosystem. Chen et al. [22] proposed a framework to allocate
complementary computation resources for mobile users in an mobile edge
computational environment by utilizing unmanned aerial vehicles (UAVs). They
developed a deep reinforcement learning (DRL) technique for task scheduling on
the UAV and decreasing response time from the UAV to mobile users. The DRL
model is trained by constructing a digital twin of the full mobile edge
computational system. Kumar et al. [23] created an entire virtual model via
digital twin to replicate the real-world scenario. Deep learning algorithms
were used to predict driver behaviour.

As per best of our knowledge, we are the first to develop a station-based
mobility network DT in order to extract precise bus station profiles to be
used in downstream analytical operations. The DT we developed in this research
can be classified as DTI because it only focuses on mobility-related aspects
(spatio-temporal relations) of bus stations.

###  II-B Mobility Profiling

Mobility profiling refers to the effort of extracting characteristics and
features for a specific entity (user, station, area, etc.)[24, 3]. Generally
speaking, mobility profiling can be categorized into two classes: (1) static
profiling and (2) dynamic profiling. Static profiling seeks to learn entity
representations relying on varied mobility data that varies depending on the
temporal perspective. For example, Ermal et al. [25] proposed a subway station
crowd flow prediction framework by integrating time-invariant station profiles
such as remoteness from the downtown area, and mean flow, etc. Xia et al. [26]
presented a framework to rank the station importance by considering the static
subway profiles such as centrality, connectivity, etc. Dynamic profiling, on
the other hand, relates to modeling the entity representations while taking
into account the temporal elements that the profiles may alter over time. Zhao
et al. [27] proposed a spatio-temporal latent ranking framework called
STELLAR, capturing temporal impact on successive PoI (Point of Interest)
recommendations. Wang et al. [28] proposed an adversarial substructured
representation learning method to extract mobility user profiles on mobile
activity graphs. Wang et al. [29] proposed a framework by incorporating
reinforcement learning into spatial knowledge graph representation learning to
generate the dynamic mobility user profiles.

###  II-C Spatio-temporal Graph Learning

The basic process of spatio-temporal learning is to model the spatial
dependency in a set of spatial graphs with different time slots and then learn
the temporal dependency of these spatial graphs by a temporal learning
architecture, e.g., Recurrent Neural Networks (RNN) [14]. Li et al. [15]
designed the diffusion convolutional layer to extract the spatial features and
adopted the Gated Recurrent Unit (GRU) to capture the temporal dependency. Ye
et al. [30] design the coupling mechanism to perform graph convolution in
different layers using different graphs, and then parasitize GCN to GRU to
capture spatio-temporal dependencies. Yu et al. [31] proposed stacked ST-Conv
blocks for learning the spatio-temporal correlations on the predefined spatio-
temporal graphs. The gated Convolutional Neural Networks (CNN) is utilized to
learn the temporal dependency. However, predefined spatio-temporal graph
learning could not capture the complex spatio-temporal patterns preserved in
mobility data. Wu et al. [11] proposed incorporating adaptive graph learning
mechanism into fixed predefined graph learning to learn the spatio-temporal
dependencies. Bai et al. [12] presented a full adaptive learning architecture,
including node adaptive parameter learning and data adaptive graph generation,
to learn the task-specific spatio-temporal features from mobility data. Li et
al. [32] constructed a spatio-temporal fusion graph to model the spatio-
temporal correlations explicitly. Then, it adopts gated dilated CNN to learn
the local and global spatio-temporal correlations on the fusion graph.
Oreshkin et al. [33] proposed a framework for learning spatio-temporal
dependencies without using predefined graphs and relying only on learn able
fully connected layers and gated graph architecture.

To learn station representations from mobility data, we use an adaptive graph
learning technique, which is different from prior spatio-temporal graph
learning research. Moreover, in order to learn the spatio-temporal
interactions, we create and design a convolutional alignment graph network.
Temporal dependency can be further learned with gated TCN in parallel with
DACN.

TABLE I: Frequently used notations Symbol | Description
---|---
$\mathcal{G}(V,A)$ | a traffic network
$V=\\{1,2,...,N\\}$ | set of nodes in the traffic network $\mathcal{G}$
$A\in\mathbb{R}^{N\times N}$ | predefined graph depends on prior knowledge
$W_{\theta}=\\{W,b,\Theta\\}$ | set of learnable parameters in the model
$X_{\mathcal{G}}\in\mathbb{R}^{T\times N\times C}$ |  $T$-step historical feature matrix
$Y_{\mathcal{G}}\in\mathbb{R}^{T^{\prime}\times N\times C}$ | next $T^{\prime}$-step traffic feature matrix
$X^{(i)}_{\mathcal{G}}\in\mathbb{R}^{N\times C}$ | feature matrix in the $\mathcal{G}$ at time step $i$
$H\in\mathbb{R}^{T\times N\times F}$ | output of middle layer
$A^{(S)}\in\mathbb{R}^{N\times N}$ | spatial adjacency matrix
$A^{(T)}\in\mathbb{R}^{N\times N}$ | alignment adjacency matrix
$A^{(S_{i})}\in\mathbb{R}^{N\times N}$ | spatial adjacency matrix with $i$ shift
$E_{1},E_{2}\in\mathbb{R}^{N\times e}$ | adaptive node embeddings

##  III Design of DTMP

###  III-A Spatio-temporal Prediction Problem Formulation

Figure 3: DTMP’s framework. The model consists of multiple spatio-temporal
modules and an output layer. Each spatial-temporal module transmits the result
to the output layer through skip-connected.

In the transportation system, we define a graph $\mathcal{G}(V,A)$ to
represent a traffic network, where $|V|=N$ is the set of nodes, $N$ denotes
the number of vertices. $A\in\mathbb{R}^{N\times N}$ is the matrix showing the
proximity between the nodes. At time step $i$, the graph $\mathcal{G}$ has a
feature matrix $X^{(i)}_{\mathcal{G}}\in\mathbb{R}^{N\times C}$, also called
graph signals. $X^{(i)}_{\mathcal{G}}$ records the $C$ types of traffic
features (such as flow, speed) of each nodes observed at time step $i$. We
describe the traffic prediction problem as having $T$-step historical data
$X_{\mathcal{G}}$, finding a mapping function $\mathcal{F}$ to predict the the
next $T^{\prime}$-step traffic condition $Y_{\mathcal{G}}$, formulated as:

$\displaystyle X_{\mathcal{G}}$
$\displaystyle=(X^{1}_{\mathcal{G}},X^{2}_{\mathcal{G}},...,X^{(T)}_{\mathcal{G}})$
(1) $\displaystyle Y_{\mathcal{G}}$
$\displaystyle=(X^{(T+1)}_{\mathcal{G}},X^{(T+2)}_{\mathcal{G}},...,X^{(T+T^{\prime})}_{\mathcal{G}})$
$\displaystyle Y_{\mathcal{G}}$
$\displaystyle=\mathcal{F}_{W_{\theta}}(X_{\mathcal{G}})$

where $W_{\theta}$ denotes all the learnable parameters in the model. Table I
lists frequently used notations in this paper.

###  III-B Framework of DTMP

We present the DTMP’s framework in Fig. 3. The model is mainly composed of
fully connected layers and stacked spatio-temporal modules. Each spatio-
temporal module is mainly included of parallel DACN and gated TCN.
Furthermore, coupling mapping transforms node embedding vectors between
spatio-temporal modules, resulting in each module having its own graph
structure. The node embedding is the station portrait learned by the model. By
using skip-connected, each module transfers the result to the output layer.
The spatio-temporal module’s output is combined in the output layer, which
translates it to the target dimension through two fully connected layers with
activation functions. Spontaneous spatial and temporal information are
extracted using DACN in the spatio-temporal module. Temporal dependencies are
further captured using the gated TCN.

###  III-C Alignment Convolution Module

The spatial correlation between nodes has always been regarded as the key
factor and research hotspot in the traffic prediction problem. With the
popularity of GNN, existing work mainly focuses on constructing an appropriate
spatial graph and using GNN to capture the spatial dependencies in the traffic
sequence[34]. Relatively rare is the temporal relationship between the nodes
taken into account. However, graph signals are time-series data. There should
be a temporal relationship between two time series. We propose two types of
graphs in the transportation network to better characterize the temporal and
spatial relationships, spatial graph $A^{(S)}$ and alignment graph $A^{(T)}$.
It’s a spatial graph that describes the similarity of temporal patterns
between nodes, similar to the prior work in this area as well, whereas, nodes
in the alignment graph are sorted according to how far ahead or behind they
are in the temporal pattern between them.

##### Alignment Graph

Figure 4: Alignment graph. Node b’s traffic flow status is $t$ time steps
behind node a.

To design a time alignment graph, we use the temporal shift between graph
signals. Nodes in the alignment graph can only be connected to their neighbors
in the spatial graph since the alignment graph is dependent on the spatial
graph. When two nodes are connected in a spatial network, we can conclude that
they have similar graph signals, but this is only true after temporal lags or
leads have been formed. Therefore, we build the alignment graph
$A^{(T)}\in\mathbb{R}^{N\times N}$ based on $A^{(S)}$ with each element
$a^{(T)}_{i,j}\geqslant 0$, as shown in Fig. 4. There is an alignment value
only for two points with edges in spatial graph. The $a^{(T)}_{i,j}$
represents the leading alignment of node $j$ relative to node $i$. The
alignment graph is directed, if $a^{(T)}_{i,j}>0$ is true, then
$a^{(T)}_{j,i}=0$.

##### AGCN

We present a novel GNN titled alignment graph convolution network (AGCN),
which implements graph convolution using the spatial graph and the alignment
graph. General graph convolution does not employ alignment operation on the
neighbor information, the spatial adjacency matrix to determine aggregated
neighbors and aggregation pattern, whereas AGCN does. Here’s how to calculate
it:

$\operatorname{shift}(H,d)=H>>d$ (2)
$H^{(l+1)}_{i}=\sigma(\sum_{j=1}^{n}(\tilde{A}^{(S)}_{i,j}\times\operatorname{shift}(H^{(l)}_{j},A^{(T)}_{i,j}))\times
W+b_{i}),$ (3)

where $H^{(l+1)}_{i}$ is the output of node $i$ through AGCN,
$\tilde{A}^{(S)}$ is the normalized spatial adjacency matrix, $W$ is learnable
parameters and $b_{i}$ is bias. Shift operation is for rolling the graph
signals along the time dimension. Elements that are shifted beyond the last
position are discarded. As the formula (2), where $>>$ is right-shift operator
and $d$ is the number of times the elements of $H$ have been shifted in time.

In traffic prediction, the convolution module can capture the temporal and
spatial dependency concurrently, rather than learning them separately. There
are, however, a couple of drawbacks to using AGCN on two graphs at the same
time. Such graph convolution requires high computational costs because only
one layer of the convolution network needs to complete $N\times N$ shift
operations. The overhead caused by the shift operation makes the calculation
speed extremely slow with limited computing resources. Another is the
generation method of the graph. Through the distance between nodes, we can
only get an inaccurate and incomplete alignment graph. It’s not possible for a
static, established network to include all the alignment information between
nodes. Meanwhile, due to the discontinuous nature of the alignment graph, the
adaptive generation method is likewise ineffective for producing the alignment
graph during training.

##### DACN

We propose an improved convolution network with a dilated aligned structure
(DACN). First, we decompose the spatial graph according to the temporal
alignment graph. Since the values of the alignment graph are discrete, the
spatial graph is decomposed into multiple spatial graphs with the same shift
value, which is expressed by the formula (4). Each sub-graph is initialized as
an all-zero matrix, and then the spatial graph is assigned to the sub-graphs
according to the alignment graph.

$A^{(S)}=\begin{cases}A^{(S_{0})},&\operatorname{If}\
A^{(T)}_{i,j}=0,A^{(S_{0})}_{i,j}=A^{(S)}_{i,j}\\\
A^{(S_{1})},&\operatorname{If}\
A^{(T)}_{i,j}=1,A^{(S_{1})}_{i,j}=A^{(S)}_{i,j}\\\ \vdots\\\
A^{(S_{(n-1)})},&\operatorname{If}\
A^{(T)}_{i,j}=n-1,A^{(S_{(n-1)})}_{i,j}=A^{(S)}_{i,j},\\\ \end{cases}$ (4)

The original AGCN can be replaced by several AGCNs after the spatial graph has
been decomposed. After decomposition, we construct a new DACN that uses
spatial graphs produced from decomposition, as shown in Fig. 5. With a certain
dilation rate $k$, we pick the spatial graph every $k$ step to apply AGCN to
reduce the complexity of the model. Then, we connect AGCN in series according
to the shift from small to large, and each AGCN is skip-connected to the
output layer. After merging the outcomes of each AGCN(using concatenation-
based fusion in the paper), the output layer converts each result to the
relevant dimension. For each AGCN in DACN, since each spatial graph contains
only one shift method, we can directly perform an overall shift on the input
data and then conduct message propagation. So the formula (3) can be
simplified as:

$H^{(l+1)}_{i}=\sigma(\tilde{A}^{(S_{d})}_{i}\times\operatorname{shift}(H^{(l)},d)\times
W^{(l)}+b^{(l)}_{i})$ (5)

where $\tilde{A}^{(S_{d})}$ is the normalized spatial adjacency matrix with
$d$ shift. $H^{(l)}$, $H^{(l+1)}$ represents the input and output of the $l$
layer respectively. $H^{(l+1)}_{i}$ is the output of node $i$.

Figure 5: Dilated alignment convolution network and adaptive graph generation.
With kernel size 3 and dilation rate 2, it picks spatial graphs every 2 step
and applies the AGCN.

Compared with the original alignment graph convolution, the number of shift
operations performed in DACN based on graph decomposition is reduced from the
$\mathcal{O}(N^{2})$ to $\mathcal{O}(1)$, significantly reducing the
computational overhead. On the other hand, time alignment graph information is
allocated to each spatial graph, making it possible to obtain alignment graph
information through learning. We overcome the problem of generating the
discrete alignment graph. The graph generation problem has changed from
constructing an alignment adjacency matrix and a spatial adjacency matrix to
generating a series of spatial adjacency matrices with different alignment
values.

##### Adaptive Graph Generation

There are three main methods to generate the spatial graph: predefined graph
by distance, predefined graph by time sequence similarity, adaptive generation
graph by learn able parameters. The method of the predefined graph depends on
prior knowledge, often unable to include all spatial information. Spatial
correlation in the predefined graph has a deviation. Therefore, a new trend
appears using an adaptive graph to describe spatial dependency information.
For large data sets with many nodes, however, utilizing a direct adaptive
method to generate numerous graphs can rapidly lead to an increase in
parameters, which makes it more difficult to train the network and execute
tests. So we use a coupled adaptive matrix generation method to generate a
series of spatial adjacency matrices.

For zero-aligned spatial adjacency matrice, we randomly initialize two
learnable node embedding vectors
$\mathbf{E}^{0}_{1},\mathbf{E}^{0}_{2}\in\mathbb{R}^{N\times e}$ for all node
(for the dataset with prior knowledge, $\mathbf{E}^{0}_{1}$ and
$\mathbf{E}^{0}_{2}$ can be obtained through the singular value decomposition
of a given adjacency matrix), where $e$ is the embedding dimension of the
node. The zero-aligned spatial graph can be calculated and normalized by the
following formula:

$A^{(S_{0})}=\operatorname{ReLU}\left(\mathbf{E}_{1}\mathbf{E}_{2}^{(T)}\right)\\\
$ (6) $\tilde{A}^{(S_{0})}=\operatorname{SoftMax}\left(A^{(S_{0})}\right)$ (7)

For the spatial graphs of other layers, we use the same generation method.
Instead of using randomly generated node embedding, we use the previous layer
of node embedding coupling mapping to obtain new node embedding. The coupled
mapping formula is as follows:

$\begin{split}\mathbf{E}^{l+1}_{1}&=\mathbf{E}^{l}_{1}\times W+b\\\
\mathbf{E}^{l+1}_{2}&=\mathbf{E}^{l}_{2}\times W+b\end{split}$ (8)

where $W\in\mathbb{R}^{e\times e}$ are learnable parameters and
$b\in\mathbb{R}^{e}$ are bias. The coupling mapping of
$\mathbf{E}_{1},\mathbf{E}_{2}$ shapes parameters. There are $O(N\times N)$
parameters in the randomly generated graph, while the coupled graph only adds
$O(2\times N\times e)$ parameters for each graph. In DACN, the adaptive graph
generation method is shown in the right part of Fig. 5

###  III-D Gated Temporal Convolution Module

According to the DACN module, a sequence of spatial graphs would be used to
capture temporal and spatial dependencies, as well as similarity and shift
information. However, adaptive learning is limited and the temporal dependence
of the nodes themselves is also critical to traffic prediction. We employ the
gated temporal convolution [11] to extract the temporal dependency. We set
DACN and gated TCN modules in the same layer to use the same dilation rate to
make the gated temporal convolution module and alignment convolution module
better cooperate. The gated temporal convolution module is defined as follows:

$H^{(l+1)}=\phi\left(\Theta_{1}\times
H^{(l)}+b_{1}\right)\odot\sigma\left(\Theta_{2}\times H^{(l)}+b_{2}\right)$
(9)

where $H^{(l)}\in\mathbb{R}^{T\times N\times C}$ is the output of upper layer
network, $\phi(\cdot)$, $\sigma(\cdot)$ are activation and sigmoid function
respectively. $\Theta_{1}$, $\Theta_{2}$, $b_{1}$, and $b_{2}$ are model
learnable parameters, $\odot$ is the Hadamard product.

###  III-E Loss Function

We trained DTMP for traffic prediction using the Mean Absolute Error (MAE).
Hence, the loss function of the model can be formulated as:

$L(W_{\theta})=\frac{\sum_{i=1}^{T^{\prime}}|\mathbf{\widehat{Y}}_{\mathcal{G}}^{(i)}-\mathbf{Y}_{\mathcal{G}}^{(i)}|}{T^{\prime}\times
N\times C},$ (10)

where $W_{\theta}$ denotes the set of learnable parameters in the model,
$\mathbf{\widehat{Y}}_{\mathcal{G}}^{(i)}$ and
$\mathbf{Y}_{\mathcal{G}}^{(i)}$ denote the predicted and true values in graph
$\mathcal{G}$ at time $i$, respectively.

Algorithm 1 illustrates the whole training procedure. $Linear(\cdot)$
represents a transformation on the feature dimension $C$ by fully connected
layer.

Algorithm 1 DTMP Training Process

1:The datasets D, the number of spatio-temporal modules $L$, batch size is B;

2:Well Trained DTMP;

3:Initialize embedding vectors $\mathbf{E}_{1},\mathbf{E}_{2}$.

4:repeat

5: sample a batch($X\in\mathbb{R}^{B\times T\times N\times C}$,
$Y\in\mathbb{R}^{B\times T^{\prime}\times N\times C}$) from D.

6: Expand the feature dimension of $X$ from $C$ to $F$ to get
$H\in\mathbb{R}^{B\times T^{\prime}\times N\times F}$.

7: Initialize temporary variable $S$ to be null.

8: for $i=1,2,\cdots,L$ do

9: Through formula (5) and (9), calculate
$Temp=DACN(H,E_{1},E_{2})+GatedTCN(H)$

10: $H=H+Linear(Temp)$

11: $S=S+Linear(Temp)$

12: Convert $\mathbf{E}_{1},\mathbf{E}_{2}$ through formula (8)

13: end for

14: Obtain the prediction result $\widehat{Y}$ by executing two fully
connected layers on $S$.

15: Optimize the parameters by minimizing the loss (10) between $Y$ and
$\widehat{Y}$.

16:until convergence

##  IV Experiments

###  IV-A Datasets

We conduct experiments on three real-world traffic datasets to evaluate the
performance of DTMP. PeMSD4 and PeMSD8 are collected from the Caltrans
Performance Measurement System (PeMS). They refer to the San Francisco Bay
Area and San Bernardino, California. After linear interpolation, the records
are grouped into 5-minute windows. The Huaian dataset is generated in the City
of Huaian, Jiangsu Province, China, by Panda Bus Company. We take the station
as the node to extract the Spatio-temporal traffic data from bus transaction
records and the corresponding bus arrival time records, then aggregated into
15-minute windows. Since the bus operating time is from 5 to 23 o’clock, a day
contains 72-time steps in traffic flow. The detailed information is shown in
Table II.

TABLE II: Dataset description and statistics. Data | nodes | time steps | period
---|---|---|---
PeMSD4 | 307 | 16692 | 1/Jan/2018 - 28/Feb/2018
PeMSD8 | 170 | 17856 | 1/Jul/2016 - 31/Aug/2016
Huaian | 721 | 2160 | 1/Apr/2020 - 30/Apr/2020

(a) MAE

(b) MAPE

(c) RMSE

Figure 6: Comparison of prediction performance for each horizon on the PeMSD4
dataset.

###  IV-B Experimental Setup

We divide the three datasets into training sets, validation sets, and test
sets with a ratio of 6:2:2. The prediction problem chooses to use the 12-time
steps historical data to predict the traffic conditions in the next 12- time
steps. We implemented the code for DTMP traffic prediction in Python 3.7 with
PyTorch 1.5.0. The model uses 6 layers of the spatio-temporal module, and the
corresponding dilation rate is 1, 2, 4, 1, 2, 4, respectively. DACN’s kernel
size is set to 2, indicating that each DACN contains two general convolutions.
Dropout with 0.3 is applied to the outputs of the DACN. Gated TCN’s kernel
size is set to 2. We initialize two sets of node embeddings randomly, and each
embedding size is set to 10. In the training process, we optimize the model by
Adam optimizer with a learning rate of 0.003. Mean Absolute Error (MAE), Root
Mean Square Error (RMSE), and Mean Absolute Percentage Error (MAPE) are used
to evaluate the performance of predictive models.

###  IV-C Baselines

We compare DTMP with the following models:

  1. 1.

HA: Historical Average model, which calculates the average of historical
values to predict the future values.

  2. 2.

ARIMA [35]: Auto-Regressive Integrated Moving Average model, which is a
strategy for predicting time series.

  3. 3.

GRU [36]: Gate Recurrent Unit model, a sequence-to-sequence model that
utilizes GRUs in encoder and decoder.

  4. 4.

DCRNN (ICLR 2018) [15]: Diffusion Convolution Recurrent Neural Network, which
incorporates diffusion graph convolutions with GRU.

  5. 5.

STGCN (IJCAI 2018)[31]: Spatio-Temporal Graph Convolutional Network, which
integrates graph convolutions with gated dilated convolutions.

  6. 6.

GraphWavenet (IJCAI 2019) [11]: Graph WaveNet, which uses adaptive adjacency
to conduct graph convolutions and combines diffusion graph convolutions with
gated dilated convolutions.

  7. 7.

AGCRN (NIPS 2020) [12]: Adaptive Graph Convolutional Recurrent Network, which
fusings node adaptive parameter learning and adaptive graph generation with
GRU.

  8. 8.

STFGNN (AAAI 2021) [32]: Spatial-Temporal Fusion Graph Neural Network, which
combines fusion graph module of various spatial and temporal graphs with a
novel gated convolution module.

###  IV-D Results and Analysis

####  IV-D1 Overall Comparison

Table III shows the performance of DTMP and baseline models on PeMSD4, PeMSD8,
and Huaian datasets. DTMP outperform in three datasets in terms of all
evaluation metrics. It includes classic temporal methods including HA, ARIMA,
and GRU with a large margin. Compared with other GCN-based spatio-temporal
methods, DTMP surpasses predefined graphs (STGCN, DCRNN) and further improves
the method of adaptive graphs (Graph WaveNet, AGCRN) with a significant
margin. In PeMSD4 and PeMSD8 datasets, compared with AGCRN and STFGNN, our
method achieves small promotion, but we obtain the bigger improvement in
Huaian datasets. We consider the Huaian dataset has more nodes than the PeMSD
datasets, and the nodes have more complex dependencies. DTMP has learned
multiple adaptive graphs and has the ability to capture this complex
relationship. In Fig. 6, we also display the evaluation measures at each
horizon in the PeMSD4 dataset. Although AGCRN balances long-term and short-
term predictions, our method’s forecast errors are smaller than those on the
12 horizons.

TABLE III: Performance on PEMSD4, PEMSD8 and Huaian datasets. model | dataset |  | PeMSD4 |  |  | PeMSD8 |  |  | Huaian |
---|---|---|---|---|---|---|---|---|---|---
metrics | MAE | RMSE | MAPE | MAE | RMSE | MAPE | MAE | RMSE | MAPE
HA | 38.30 | 56.76 | 39.63% | 31.98 | 47.49 | 21.46% | 127.4137 | 227.13 | 107.41%
ARIMA | 26.10 | 34.40 | 41.70% | 23.11 | 26.43 | 26.80% | 82.54 | 94.69 | 68.85%
GRU | 26.93 | 42.42 | 18.00% | 20.72 | 32.28 | 15.05% | 89.38 | 204.57 | 51.75%
DCRNN | 24.66 | 37.76 | 17.71% | 18.82 | 28.84 | 13.49% | 80.22 | 148.44 | 52.03%
STGCN | 24.80 | 38.69 | 17.42% | 19.86 | 30.72 | 13.93% | 72.96 | 134.92 | 43.50%
Graph Wavenet | 20.40 | 32.41 | 13.83% | 16.27 | 25.74 | 10.33% |  56\. 86 |  100.27 |  38.42%
AGCRN | 19.83 | 32.26 | 12.97% |  15.95 |  25.22 |  10.09% | 70.92 | 168.63 | 41.72%
STFGNN |  19.83 |  31.88 |  13.02% | 16.64 | 26.22 | 10.06% | 74.73 | 128.81 | 45.66%
DTMP | 19.07 | 30.93 | 12.59% | 15.11 | 24.15 | 9.63% | 54.83 | 95.68 | 35.48%

####  IV-D2 Effect of Station Profiling

We use station 0 in the Huaian dataset as the research object and exhibit the
relationships between it and other stations in 12 graphs to evaluate the
efficacy of station profiling in DACN. These graphs are calculated by the
station profiles (node embedding) and related model parameters after training.
As shown in the Fig. 7(a), the $i$-th row represents the correlation strength
with each station in the $i$-th graph. The X-axis represents station number. A
stronger correlation with station 0 is found at stations 28 and 342. Fig. 7(b)
shows the flow of stations 0, 28 and 342. We can see that compared with
station 0, the flow at stations 28 and 342 have the same trend, but station
342 has a leading shift different from station 28. It demonstrates how station
profiles capture the complex relationships between stations.

(a) Graph of station 0 for DACN.

(b) Flow of stations 0, 28, 342

Figure 7: The case of station 0. (a) The adaptively learned relationships
between nodes, with 28 and 342 being the two most relevant station. (b) The
traffic flow between these two stations and station 0.

####  IV-D3 Ablation Study

We perform DTMP ablation experiments on PeMSD4 and Huaian datasets by
eliminating many components. We design three variant models of DTMP as
follows:

  1. 1.

No coupling mapping: DTMP without the coupling mapping between the spatio-
temporal modules and between AGCNs. We utilize the same graph in different
spatio-temporal modules.

  2. 2.

No alignment: DTMP without DACN module. We replaced the DACN with a ordinary
GCN based on adaptive graph.

  3. 3.

No gated TCN: DTMP without gated TCN. We pass the outputs of DACN to fully
connected layers ignore gated TCN.

The results are shown in Table IV, and we could conclude that the complete
model possesses the best performance. Coupling mapping contributes to the
model capture more complex Spatio-temporal relationships. DACN, compared with
GCN, can capture temporal dependency. The introduction of DACN allows
information to flow directly between different nodes at different times. It
significantly improves performance on datasets with a large number of nodes.
Gated TCN assists DACN to capture the temporal dependency further.

TABLE IV: Ablation experiments on the PeMSD4 and Huaian datasets. dataset | model elements | MAE | RMSE | MAPE
---|---|---|---|---
PeMSD4 | No coupling mapping | 19.50 | 31.44 | 13.02%
No alignment | 19.95 | 32.00 | 13.31%
No gated TCN | 21.54 | 35.07 | 14.63%
All | 19.07 | 30.93 | 12.59%
Huaian | No coupling mapping | 59.74 | 106.62 | 39.79%
No alignment | 61.77 | 114.30 | 38.97%
No gated TCN | 57.83 | 103.79 | 36.40%
All | 54.83 | 95.68 | 35.48%

####  IV-D4 Parameter Sensitivity

To further investigate the model, we conduct parameter experiments on the
PeMSD4 dataset, as shown in the Table V. In DACN, kernel size is one key
parameter. It determines the number of adaptive graphs used in each DACN and
the number of convolution operations performed. The table shows the model
performance results when the kernel size is 1, 2, 3, and 4. The evaluation
metrics first fall as the kernel size grows larger, indicating that expanding
the kernel size can capture more full and intricate dependencies. When the
kernel size exceeds 2, therewith, the metrics begin to rise. The reason for
this is that the model has grown in complexity, making training more
difficult. The node embedding records the dependencies between nodes and its
dimension is another important parameter. The larger the dimension, the more
information can be stored, but the more difficult it is to train the model.
Therefore, the model performs best when the dimension is 10. In general, the
model maintains a high performance under different parameters and shows the
robustness of the model.

TABLE V: Parameter experiments on the PeMSD4 dataset. Hyperparameter | dataset | PeMSD4
---|---|---
metrics | MAE | RMSE | MAPE
Kernel size of DACN | 1 | 19.43 | 31.29 | 12.94%
2 | 19.07 | 30.93 | 12.59%
3 | 20.01 | 32.02 | 13.15%
Embedding dimension | 2 | 19.82 | 31.79 | 13.61%
5 | 19.65 | 31.54 | 13.16%
10 | 19.07 | 30.93 | 12.59%
15 | 19.36 | 31.15 | 12.65%
20 | 19.21 | 30.96 | 12.71%

##  V Conclusion

For learning interactive spatio-temporal relationships, this paper has
proposed a novel graph convolution network based on a spatio-temporal graph.
Furthermore, we have designed an adaptive approach to generating alignment and
spatial graphs based on mobility profiling and have proposed dilated alignment
convolution network to simplify the calculation. It is combined with gated TCN
and stacked to obtain the spatio-temporal graph learning framework DTMP for
traffic forecasting and mobility profiling, and digital twin technology is
used as a cost-effective tool. Extensive experiments and analysis have
confirmed the model’s improved forecasting performance compared to other
techniques, as well as its ability to extract station information. An entirely
new perspective on graph convolution networks was gained through the use of
alignment procedures in order to discover time-dependent correlations.

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# A Continuum Erdős-Beck Theorem

Paige Bright  Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02142-4307, USA<EMAIL_ADDRESS>and  Caleb Marshall  Department
of Mathematics
University of British Columbia, 1984 Mathematics Road
Vancouver, BC, Canada V6T 1Z2<EMAIL_ADDRESS>
###### Abstract.

We prove a version of the Erdős–Beck Theorem from discrete geometry for
fractal sets in all dimensions. More precisely, let $X\subset\mathbb{R}^{n}$
Borel and $k\in[0,n-1]$ be an integer. Let $\dim(X\setminus H)=\dim X$ for
every $k$-dimensional hyperplane $H\in\mathcal{A}(n,k)$, and let
$\mathcal{L}(X)$ be the set of lines that contain at least two distinct points
of $X$. Then, a recent result of Ren shows

$\dim\mathcal{L}(X)\geq\min\\{2\dim X,2k\\}.$

If we instead have that $X$ is not a subset of any $k$-plane, and

$0<\inf_{H\in\mathcal{A}(n,k)}\dim(X\setminus H)=t<\dim X,$

we instead obtain the bound

$\dim\mathcal{L}(X)\geq\dim X+t.$

We then strengthen this lower bound by introducing the notion of the “trapping
number” of a set, $T(X)$, and obtain

$\dim\mathcal{L}(X)\geq\max\\{\dim X+t,\min\\{2\dim X,2(T(X)-1)\\}\\},$

as consequence of our main result and of Ren’s result in $\mathbb{R}^{n}$.
Finally, we introduce a conjectured equality for the dimension of the line set
$\mathcal{L}(X)$, which would in particular imply our results if proven to be
true.

###### 2020 Mathematics Subject Classification:

28A75, 28A78

###### Contents

  1. 1 Introduction
    1. 1.1 Examples and a Conjecture
    2. 1.2 Outline of Paper
  2. 2 The Planar Theorem
    1. 2.1 The Dimension of Line Families
    2. 2.2 Furstenberg Set Estimates
    3. 2.3 Planar Erdős–Beck Theorems for Fractal Sets
  3. 3 The Higher Dimensional Theorem
    1. 3.1 Preliminaries for Higher Dimensions
    2. 3.2 Higher Dimensional Erdős-Beck Theorems for Fractal Sets

##  1\. Introduction

In discrete geometry, a classic result of Beck [Bec83] shows that given $N$
points in $\mathbb{R}^{2}$, either $\sim N$ of the points lie on a line, or
there are $\sim N^{2}$ lines spanned by the set, i.e. $\sim N^{2}$ lines which
contain at least two of the points.

######  Remark 1.

Here, $\sim$ denotes equality up to some positive multiplicative constant.
Similarly, $A\lesssim B$ denotes $A\leq CB$ for some multiplicative constant
$C$.

About two years ago, this result was generalized to the continuum setting by
Orponen–Shmerkin–Wang (OSW, [OSW22]), showing that given
$X\subset\mathbb{R}^{2}$ Borel, either

  1. 1)

there exists a line $\ell$ such that $\dim(X\setminus\ell)<\dim X$, or

  2. 2)

$\dim\mathcal{L}(X)\geq\min\\{2\dim X,2\\}$, where $\mathcal{L}(X)$ is the set
of all lines containing at least two points of $X$ contained in the metric
measure space $\mathcal{A}(2,1)$, i.e. the space of all lines in
$\mathbb{R}^{2}$.

######  Remark 2.

We define the dimension of a subset of the affine Grassmannian subordinate to
suitable metrics on $\mathcal{A}(n,1)$. For the sake of exposition, we
postpone any formal definition until Section 2.1.

The proof of the continuum version of Beck’s theorem from Orponen, Shmerkin,
and Wang used one of their main results on radial projections proven in the
plane, as well as a classical lowerbound on the dimension of a (dual)
Furstenberg set (see Section 2.2). Recently, Kevin Ren proved a higher
dimensional radial projection theorem strengthening OSW’s result, and as a
consequence, Ren was able to derive an analogous continuum version of Beck’s
theorem in $\mathbb{R}^{n}$ ([Ren23], Theorem 1.1 and Corollary 1.5).

Going back to discrete geometry: there is a more general version of Beck’s
theorem, called the Erdős–Beck theorem. This result shows the following: given
$N$ points in the plane, $P$, and a parameter $0\leq t\leq N$, suppose that

$\sup_{L\in\mathcal{A}(2,1)}|P\cap L|\geq N-t.$

Then $P$ spans $\gtrsim Nt$ many lines. This result was proven by Jozsef Beck
in [Bec83] and resolved a conjecture of Erdős–hence, this result is often
referred to in the literature as the Erdős-Beck Theorem.

Our main focus of this paper is finding a continuum analogue of the Erdős–Beck
theorem in $\mathbb{R}^{n}$. In particular, our main result is the following.
In what follows, $\mathcal{A}(n,k)$ is the affine Grassmannian of $k$ planes
in $\mathbb{R}^{n}$.

######  Theorem 3.

Let $X\subset\mathbb{R}^{n}$ be Borel, and let $k\in[1,n-1]$ be an integer.
Then,

  1. 1)

if $\dim(X\setminus H)=\dim X$ for every $H\in\mathcal{A}(n,k)$, then
$\dim\mathcal{L}(X)\geq\min\\{2\dim X,2k\\}$.

  2. 2)

if there exists an $H\in\mathcal{A}(n,k)$ such that $\dim(X\setminus H)<\dim
X$, we let $0<t\leq\dim X$ such that $\dim(X\setminus P)\geq t$ for all
$P\in\mathcal{A}(n,k)$. Then,

$\dim\mathcal{L}(X)\geq\dim X+t.$

Firstly, note that if we are in the second case of the above theorem, and the
only value of $t$ such that $\dim X\setminus P\geq t$ for all
$P\in\mathcal{A}(n,k)$ is $t=0$, then one can apply this same theorem with
$n=k$ and $X^{\prime}=X\cap H$. Furthermore, note that, if $\dim X>k$ for some
$k$, then Condition $(1)$ of Theorem 3 necessarily holds. Hence, we always
have the bound

(1) $\dim\mathcal{L}(X)\geq\max\\{2(\lfloor\dim X\rfloor-1),0\\}\quad\forall
X\subset\mathbb{R}^{n}.$

However, for sets satisfying Condition $(2)$, we may improve this bound by
examining the interaction between the sts $X_{1}:=X\cap H$ and
$X_{2}:=X\setminus H$, where $H\in\mathcal{A}(n,k)$ satisfies $\dim(X\setminus
H)<\dim X$. We will also further strengthen the lower bound of Theorem 3 by
defining the notion of a “trapping number” of a set–a concept related to the
estimate (1) appearing above.

######  Remark 4.

We view our result as a natural continuum analogoue of the Erdős-Beck Theorem,
with cardinality replaced by Hausdorff dimension. In particular, our estimate
gives a lower bound on the dimension of the line set $\mathcal{L}(X)$ as a
function of the dimensions of $X$ and its largest dimensional co-planar
subset.

Initially, the reader may be surprised that the Erdős-Beck bound of $Nt$
becomes a sum bound of $\dim X+t$ in our result. However, such “logarithmic”
phenomenon occur frequently in these kind of discrete-to-continuum Theorems.
Some examples in this vein include partial results towards an Erdős-Beck
Theorem in the Continuum as in [OSW22] and [Ren23], as well as further afield
analogues such as the Kakeya problem in finite fields, where the (conjectured)
dimensional estimate of $\dim K=n$ in $\mathbb{R}^{n}$ becomes a cardinality
estimate of $\\#K\gtrsim|\mathbb{F}_{q}|^{n}$ in the finite field
$\mathbb{F}_{q}$. See, for example, [Dvi09].

Let us now discuss how Theorem 3 can be strengthened by considering the
following definition.

######  Definition 5.

Let $X\subset\mathbb{R}^{n}$ be Borel with $\dim(X)>0$, and define the
trapping number of $X$ to be

$T(X):=\begin{cases}\min\\{k\in\mathbb{N}_{+}:\text{there exists an
}H\in\mathcal{A}(n,k)\text{ such that }\dim(X\setminus H)<\dim X\\}\\\
1~{}~{}\text{if no such $k$ exists}\end{cases}.$

One can intuitively think of the trapping number of a set $X$ as a way to
recognize if (a large proportion of) $X$ is contained in some lower
dimensional $k$-plane. Notice then that Ren’s continuum Beck’s theorem [Ren23]
proves the following:

######  Theorem 6 ([Ren23], Corollary 1.5).

Let $X\subset\mathbb{R}^{n}$ be Borel. Then,

$\dim\mathcal{L}(X)\geq\min\\{2\dim X,2(T(X)-1)\\}.$

######  Remark 7.

This was technically proven when $T(X)\geq 2$, though the result is trivial
when $T(X)=1$, so we include it for completeness.

Comparing this to our main result, Theorem 3, we obtain the following:

######  Corollary 8.

Let $X\subset\mathbb{R}^{n}$ be Borel and let $k\in[0,n-1]$ be an integer.
Then,

  1. 1)

if $\dim(X\setminus H)=\dim X$ for every $H\in\mathcal{A}(n,k)$, then
$\dim\mathcal{L}(X)\geq\min\\{2\dim X,2k\\}$.

  2. 2)

if there exists an $H\in\mathcal{A}(n,k)$ such that $\dim(X\setminus H)<\dim
X$, we let $0<t\leq\dim X$ such that $\dim(X\setminus P)\geq t$ for all
$P\in\mathcal{A}(n,k)$. Then,

$\dim\mathcal{L}(X)\geq\max\\{\dim X+t,\min\\{2\dim X,2(T(X)-1)\\}\\}.$

###  1.1. Examples and a Conjecture

We illustrate the relationship between the parameters $k$, $t$, $T(X)$, and
the dimension of the set $X$ through examples.

Firstly, notice that, since $\dim(X\setminus H)\leq\dim X$ for all $k$-planes
$H$, we always have that

$\dim X+\inf_{H\in\mathcal{A}(n,k)}\dim(X\setminus H)\leq 2\dim X,\quad\forall
1\leq k\leq n.$

Hence, the term $\dim X+t$ appearing in Corollary 8 dominates precisely when
$X\subset\mathbb{R}^{n}$ satisfies the inequality

(2) $2(T(X)-1)\leq\dim X+\inf_{H\in\mathcal{A}(n,k)}\dim(X\setminus H)\leq
2\dim X$

where $\dim X$ satisfies $\dim X\leq k$. The following example furnishes a set
$X\subset\mathbb{R}^{n}$ which satisfies the inequalities (2) (so Corollary 8
obtains a lower bound of $\dim X+t$).

######  Example 9.

Let $1\leq k\leq n-1$ be given and suppose that $X\subset\mathbb{R}^{n}$ Borel
can be written as $X=X_{1}\cup X_{2}$, where each $X_{j}$ is contained in some
distinct $k$-plane $H_{j}\in\mathcal{A}(n,k)$ and satisfies $k-1<\dim
X_{1}<\dim X_{2}=\dim X\leq k$. Then, since

$\dim(X\setminus H_{2})=\dim(X_{1})<\dim(X),$

we know that $T(X)\leq k$. However, for any $k-1$ plane
$H^{\prime}\in\mathcal{A}(n,k-1)$, we know that

$\dim(X_{2}\setminus H^{\prime})=\dim(X_{2})=\dim(X).$

Hence, $T(X)\geq k$. So, one knows that:

$\min\\{2\dim X,2(T(X)-1)\\}\leq 2k-2.$

However, since

$\inf_{H\in\mathcal{A}(n,k)}\dim(X\setminus H)=\dim(X\setminus
H_{2})=\dim(X_{1}),$

we have, setting $t=\dim X_{1}$,

$\dim X+t>(k-1)+(k-1)=2k-2$

Hence, the bound $\dim X+t$ of Corollary 8 dominates in this scenario.

We now provide some intuition for how the parameters $t$ and $T(X)$ come into
play for sets with varying geometric structure, and give an example of a set
where the term $\min\\{2\dim X,2(T(X)-1)\\}$ does, in fact, dominate $\dim
X+t$.

######  Example 10.

We work in $\mathbb{R}^{n}$ and suppose we are given some integer $1\leq k\leq
n-1$ and $0<\beta\leq 1$. In this example, we construct a set
$X\subset\mathbb{R}^{n}$ with dimension $\dim X=k-1+\beta$ and trapping
dimension $T(X)=k+1$. We then provide a lower bound for $\dim\mathcal{L}(X)$
from Corollary 8 and determine what happens as $\beta\rightarrow 1$.

To this end, suppose that $S\subset\mathbb{R}^{n}$ is an embedding of the
$k$-sphere $\mathbb{S}^{k}$ in $\mathbb{R}^{n}$ (which, of course, has
dimension $k$). We then take a subset $X\subset S$ which satisfies $\dim
X=k-1+\beta$. Now, we know that

$\dim(S\cap H)\leq k-1,\quad\forall H\in\mathcal{A}(n,k)\text{ intersecting
$S$}.$

Hence, for every $H\in\mathcal{A}(n,k)$, we have $\dim(X\setminus H)=\dim X$.
So, the trapping number of $X$ satisfies $T(X)\geq k+1$. However, the
$k$-sphere $S$ is itself contained in some $k+1$ hyperplane, which guarantees
that $T(X)\leq k+1$. So, $X$ has trapping number $T(X)=k+1$. We then have that

$T(X)-1=k\geq k-1+\beta=\dim X.$

This, in turn, implies that $\min\\{2\dim X,2(T(X)-1)\\}=2\dim X$, so that

$\dim\mathcal{L}(X)\geq\max\\{\dim X+t,2\dim X\\}=2\dim X.$

A similar result is, of course, given by Theorem 3, since we may take $t=\dim
X$. However, we note that

$\dim X=k-1+\beta\rightarrow 2k\textrm{ as }\beta\rightarrow 1.$

In particular, whenever $X$ is a full-dimensional subset of $S$, we see that

$\dim\mathcal{L}(X)\geq 2\dim X=2(T(X)-1)=2k.$

In the special case where $k=n-1$, this inequality is an equality, since one
necessarily has $\dim\mathcal{L}(X)\leq 2(n-1)$ for any
$X\subset\mathbb{R}^{n}$.

The last example demonstrates a set for which the lower bound of Corollary 8
is, in fact, an equality. However, we do not expect our bound to be sharp for
all Borel sets $X$. Instead, we conjecture the following to be true.

######  Conjecture 11.

Let $X\subset\mathbb{R}^{n}$ and let $T(X)=T$ be the trapping number of $X$.
Suppose that $H\in\mathcal{A}(n,T)$ is the unique hyperplane such that
$\dim(X\setminus H)<\dim X,$ and set $X_{1}=X\cap H$ and $X_{2}=X\setminus H$.
Then, one has

(3) $\dim\mathcal{L}(X)=\max\\{\dim(X_{1}\times X_{2}),\min\\{\dim(X_{1}\times
X_{1}),2(T-1)\\}\\}$

The spirit behind Conjecture 11 is the following. For $X=X_{1}\sqcup X_{2}$ in
$\mathbb{R}^{n}$ let $\mathcal{L}(X_{1},X_{2})$ denote the family of affine
lines connecting pairs of points between $X_{1}$ and $X_{2}$. Then, we can
write

$\mathcal{L}(X):=\mathcal{L}(X_{1})\cup\mathcal{L}(X_{1},X_{2})\cup\mathcal{L}(X_{2}).$

So, if one had the equalities

$\dim\mathcal{L}(X_{1},X_{2})=\dim(X_{1}\times
X_{2}),\quad\dim\mathcal{L}(X_{1})=\dim(X_{1}\times X_{1}),$

then this would prove Conjecture 11. In particular, since:

$\dim(X_{1}\times X_{2})\geq\dim X_{1}+\dim X_{2}=\dim
X+\inf_{H\in\mathcal{A}(n,T)}\dim(X\setminus H),$ $\dim(X_{1}\times X_{1})\geq
2\dim X_{1}=2\dim X,$

we see that Conjecture 11 in fact implies the lower bound of our main result,
which is Theorem 3.

######  Remark 12.

We note that bilinear line set estimates of the form $\mathcal{L}(X,Y)$ can be
studied in a similar manner, which is a direction we hope to pursue in future
work.

To help motivate Conjecture 11 and demonstrate that it supersedes Theorem 3
for certain sets, we recall the following result concerning the dimension of
product sets (see [Mat99, Theorem 8.10] for a reference).

######  Theorem 13.

Let $A\subset\mathbb{R}^{n}$ and $B\subset\mathbb{R}^{m}$ be non-empty Borel
sets. Then

(4) $\dim(A\times B)\geq\dim A+\dim B.$

Many product sets $K=A\times B$ attain (4) with equality–for example, when $A$
has equal Hausdorff and packing dimensions. Yet, the inequality (4) can also
be strict. Such product sets are precisely those which motivate Conjecture 11.
We give an example in $\mathbb{R}^{2}$ below, where point-line duality
arguments allow us to easily calculate the dimension of
$\mathcal{L}(X_{i},X_{j})$ in terms of the dimension of the product set
$X_{i}\times X_{j}$ for $i,j=1,2$.

We now construct a set $E\subset\mathbb{R}^{2}$ with $\dim(E)=0$ such that
$\dim\mathcal{L}(E)=1$. Moreover, our set $E$ has the property that
$E=E_{1}\sqcup E_{2}$ where each $E_{i}$ is contained in some line $\ell_{i}$
and $\mathcal{L}(E)=\dim(E_{1}\times E_{2})=1$. In particular, then, this
example demonstrates a set $E$ which simultaneously sharpens the conjectured
inequality (3) while also departing from the lower bound of Corollary 8.

######  Example 14.

We adapt an example from [BP16, p.90] on $\mathbb{R}$ to lines in
$\mathbb{R}^{2}$. For any $S\subset\mathbb{N}$, let

$A_{S}:=\\{x\in[0,1]:\sum\limits_{k=1}^{\infty}x_{k}2^{-k}\\},\textrm{ where
}x_{k}\in\begin{cases}\\{0,1\\},\,k\in S\\\ \\{0\\},\quad k\in
S^{C}\end{cases}.$

Note that $[0,1]\subset A_{S}+A_{S^{C}}$ for any choice of
$S\subset\mathbb{N}$. It is shown in [BP16] that

$\overline{\dim_{\mathscr{M}}}A_{S}=\lim_{N\rightarrow\infty}\frac{\\#(S\cap\\{1,...,N\\})}{N},$

where $\overline{\dim_{\mathscr{M}}}A_{S}$ denotes the upper Minkowski
dimension of $A_{S}$. Now, choose $S$ so that

$S:=\mathbb{N}\cap\bigg{(}\bigcup_{j=1}^{\infty}[(2j-1)!,2j!)\bigg{)}\quad\quad
S^{C}:=\mathbb{N}\cap\bigg{(}\bigcup_{j=1}^{\infty}[2j!,(2j+1)!)\bigg{)},$

and let $A=A_{S}$ and $B=A_{S^{C}}$. We then have that

$\lim_{N\rightarrow\infty}\frac{\\#(S\cap\\{1,...,N\\})}{N}=\lim_{N\rightarrow\infty}\frac{\\#(S^{C}\cap\\{1,...,N\\})}{N}=0,$

so that $\dim A=\dim B=0$.

Now, let $E\subset\mathbb{R}^{2}$ be defined as $E=E_{1}\sqcup E_{2}$ where

$E_{1}:=\\{(a,0):a\in A\\}\quad\quad\quad E_{2}:=\\{(0,b):b\in B\\}.$

Then $\dim(E)=0$ and each $E_{i}$ is a subset of the two coordinate axes
$\ell_{x}$ and $\ell_{y}$. We know, then, that

$\dim E=T(E)=\dim E+\inf_{H\in\mathcal{A}(2,k)}\dim(E\setminus
H)=0\quad\quad\forall k=0,1,2,$

so that Corollary 8 gives the trivial lower bound of $\dim\mathcal{L}(E)\geq
0$.

Nevertheless, one has $\dim\mathcal{L}(E)=1.$ To verify this, first write

$\mathcal{L}(E)=\mathcal{L}(E_{1})\sqcup\mathcal{L}(E_{2})\sqcup\mathcal{L}(E_{1},E_{2}),$

and notice that $\dim\mathcal{L}(E_{1})=\dim\mathcal{L}(E_{2})=0$. Now, since
$\ell_{x}$ and $\ell_{y}$ are perpendicular, no lines in
$\mathcal{L}(E_{1},E_{2})$ are vertical. So, we may use the point-line duality
dimension bound (see, for example, [OS23] for a discussion) to obtain

$\dim\mathcal{L}(E_{1},E_{2})=\dim(E_{1}\times E_{2})=\dim(A\times B)=1.$

In particular, this shows the equality–not inequality–conjectured in (3).

######  Remark 15.

Example 14 demonstrates the improvement of Conjecture 11 over Corollary 8 by
first finding two sets $A,B\subset\mathbb{R}$ such that $\dim(A\times B)>\dim
A+\dim B$ and then embedding these sets in perpendicular lines in
$\mathbb{R}^{2}$. It is likely that a similar method could give further
examples relevant to Conjecture 11 in $\mathbb{R}^{n}$ for $n\geq 3$. That is,
taking sets $A\subset\mathbb{R}^{m}$ and $B\subset\mathbb{R}^{n}$ such that
$\dim(A\times B)>\dim A+\dim B$. Such sets are constructed in [Hat71], with
parameters $\alpha=\dim A$, $\beta=\dim B$ and $\gamma=\dim(A\times B)$
satisfying $\alpha\leq m$ and

$\alpha+\beta<\gamma<\min\\{\alpha+m,\beta+n\\}.$

We plan to study such generalizations at a later time.

###  1.2. Outline of Paper

We begin by proving Theorem 3, and thus Corollary 8, in the plane in Section
2. This is done to outline the approach we will take in higher dimensions. We
also take this time to develop useful lemmata regarding the affine
Grassmannian and the relation to radial projections. Then, in Section 3, we
discuss and prove the higher dimensional results. In this section, we
generalize the planar lemmata and utilize results of Ren [Ren23] and the first
author, Fu, and Ren [BFR24].

Acknowledgements. A number of ideas for this paper were conceived while the
authors were working on the Study Guide Writing Workshop 2023 at UPenn. We
would like to thank our collaborators on the study guide: Ryan Bushling and
Alex Ortiz. We would also like to thank Hong Wang for words of insight and
encouragement when first attempting this problem at the Study Guide Workshop.
Finally, we also must thank Josh Zahl for insightful discussions, especially
for helping us formalize Corollary 8 and Conjecture 11.

##  2\. The Planar Theorem

###  2.1. The Dimension of Line Families

For each $n\geq 2$, we work with $\mathcal{G}(n,1)$, the set of linear
subspaces of $\mathbb{R}^{n}$ and $\mathcal{A}(n,1)$, the family of affine
lines in $\mathbb{R}^{n}$.

Before beginning our work on line families, we make a small note of the
following notation. Whenever $x\in\mathbb{R}^{n}$, the map
$\pi_{x}:\mathbb{R}^{n}\setminus\\{x\\}\rightarrow\mathbb{S}^{n-1}$ will
denote radial projection onto the sphere centered at $x$; however, if whenever
$L\in\mathcal{A}(n,1)$, we let $\pi_{L}:\mathbb{R}^{n}\rightarrow L$ denote
the orthogonal projection map onto $L$.

We will view both $\mathcal{G}(n,1)$ and $\mathcal{A}(n,1)$ as a topological
metric spaces. For $\mathcal{G}(n,1)$, we need only use the standard metric,

$d_{\mathcal{G}(n,1)}(L,L^{\prime}):=\lVert\pi_{L}-\pi_{L^{\prime}}\rVert.$

where $\pi_{L}$ denotes orthogonal projection onto $L$.

This metric induces a natural notion of measure and dimension on
$\mathcal{G}(n,1)$, which is defined to the(normalized) $\mathcal{H}^{s}$
measures on $\mathbb{S}^{n-1}$ and radial projection about the origin. Since
we often employ radial projection through the origin, we define it below.

######  Definition 16.

For $x\in\mathbb{R}^{n}\setminus\\{0\\}$, we let $\pi_{O}(x)$ denote the
radial projection of $x$ onto $\mathbb{S}^{n-1}$ centered at the origin. That
is

$\pi_{O}(x):=\frac{x}{|x|},\quad\quad\forall
x\in\mathbb{R}^{n}\setminus\\{0\\}.$

In particular, if $L\in\mathcal{G}(n,1)$ is any line through the origin,
$\pi_{O}(L)$ consists of two antipodal points. Moreover, whenever
$\mathcal{L}\subset\mathcal{G}(n,1)$ is (viewed as) a collection of lines in
$\mathbb{R}^{n}$, we let

$\Theta_{\mathcal{L}}:=\pi_{O}(\bigcup_{L\in\mathcal{L}}L)=\bigcup_{L\in\mathcal{L}}(L\cap\mathbb{S}^{n-1}),$

In words, $\Theta_{\mathcal{L}}$ is the set of directions determined by
$\mathcal{L}$. We then use a notion of measure and dimension on
$\mathcal{G}(n,1)$, which is discussed, for example, in [Mat99].

######  Definition 17.

For each $0\leq s\leq n-1$, the s-Hausdorff measure $\gamma_{s}$ on
$\mathcal{G}(n,1)$ is defined as,

$\gamma_{s}(\mathcal{L}):=\mathcal{H}^{s}\big{|}_{\mathbb{S}^{n-1}}\big{(}\Theta_{\mathcal{L}}\big{)},\quad\forall\,\mathcal{L}\subset\mathcal{G}(n,1),$

The Hausdorff dimension of $\mathcal{L}$ is then defined in the usual way but
relative to the measures $\gamma_{s}$.

For the affine Grassmanian $\mathcal{A}(n,1)$, we use a family of metrics
parametrized by $w\in\mathbb{R}^{n}$.

######  Definition 18.

For each $w\in\mathbb{R}^{n}$, we define a metric
$d^{w}:=d_{\mathcal{A}(n,1)}^{w}(\cdot,\cdot)$ on $\mathcal{A}(n,1)$ by
setting,

$d_{\mathcal{A}(n,1)}^{w}(\ell,\ell^{\prime})=\lVert\pi_{L_{w}}-\pi_{L_{w}^{\prime}}\rVert+|a_{w}-a_{w}^{\prime}|,\quad\forall\ell,\ell^{\prime}\in\mathcal{A}(n,1).$

where $\pi_{L_{w}}$ denotes projection onto the line $L_{w}$, which is
parallel to $\ell$ and passes through $w$, and $a_{w}$ is the point of
intersection of the lines $\ell$ and the $n-1$-dimensional hyperplane through
$w$ which is perpendicular to $L_{w}$.

This family of metrics $d_{\mathcal{A}(n,1)}^{w}$ allows us to perform the
following important “change-of-coordinates” for a family of lines passing
through a common point. Indeed, notice that if $\mathcal{L}_{w}$ is a family
of lines passing through $w\in\mathbb{R}^{n}$, then $a_{w}=a_{w}^{\prime}=w$
for each $\ell,\ell^{\prime}\in\mathcal{L}_{w}$. Consequently

$\sup_{\ell,\ell^{\prime}\in\mathcal{L}_{w}}d^{w}(\ell,\ell^{\prime})=\sup_{L_{w},L_{w}^{\prime}}\lVert\pi_{L_{w}}-\pi_{L_{w}^{\prime}}\rVert=\sup_{L,L^{\prime}}\lVert\pi_{L}-\pi_{L^{\prime}}\rVert,$

where $L\sim L_{w}$ and $L^{\prime}\sim L_{w}^{\prime}$ are the linear
subspaces parallel to $L_{w}$ and $L_{w}^{\prime}$. So, relative to the metric
$d^{w}$, the diameter of a collection of lines $\mathcal{L}_{w}$ passing
through $w\in\mathbb{R}^{n}$ is computable in terms of
$d_{\mathcal{G}(n,1)}$–the standard metric on the Grassmanian of linear
subspaces.

For us, the most important property of the metrics $d^{w}$s are their
equivalence.

######  Proposition 19.

Suppose that $w_{1}$ and $w_{2}$ are points in $\mathbb{R}^{n}$. Then, the
associated metrics $d^{w_{1}}$ and $d^{w_{2}}$ are bilipschitz equivalent.
That is, there exist constants $C_{1},C_{2}$–which are allowed to depend on
$w_{1}$ and $w_{2}$–such that:

(5) $C_{1}d^{w_{1}}(\ell,\ell^{{}^{\prime}})\leq
d^{w_{2}}(\ell,\ell^{\prime})\leq C_{2}d^{w_{1}}(\ell,\ell^{\prime})$

for all lines $\ell,\ell^{\prime}\in\mathcal{A}(n,1)$.

###### Proof.

The ideas behind this Proposition are discussed in [Mat99]. However, we give a
proof of the following restricted special case, as it is essentially the
situation we encounter in the following key result (Lemma 20). Specifically:
we will be studying line families that intersect in a common point–so, the
following special case considers the transformation from the origin to this
point of intersection.

Suppose $\ell,\ell^{\prime}\in\mathcal{A}(n,1)$ are non-parallel lines, and
set $w:=\ell\cap\ell^{\prime}$. Let us show that there exist constants $C_{1}$
and $C_{2}$ such that

$C_{1}d^{w}(\ell,\ell^{\prime})\leq d^{O}(\ell,\ell^{\prime})\leq
C_{2}d^{w}(\ell,\ell^{\prime})$

where $O$ is the origin. Importantly, our constants $C_{1}$ and $C_{2}$ will
depend upon the distance $W:=|O-w|=|w|$, which is valid since our choice of
$w$ fixes the metric, but not the lines in question.

Figure 1. We work with the change-of-coordinates which takes us from the
origin to the point of intersection of the lines $\ell$ and $\ell^{\prime}$.
This change-of-coordinates is highly important for what comes later.

It is clear that the projection operators satisfy,

$\lVert\pi_{L_{w}}-\pi_{L_{w}^{\prime}}\rVert=\lVert\pi_{L}-\pi_{L^{\prime}}\rVert.$

which follows directly from the definition of the operator norm. So, call this
value $\delta>0$. Then, we have that

$d^{w}(\ell,\ell^{\prime})=\delta$
$d^{O}(\ell,\ell^{\prime})=\delta+|a-a^{\prime}|$

It is obvious that the left-hand inequality of (5) holds with constant
$C_{1}=1$. It remains to show the right-hand side of the inequality is valid.

Consider the two triangles $T$ and $T^{\prime}$ formed by connecting the
points $w$, $a$ and $O$ and $w$, $a^{\prime}$ and $O$, respectively. Notice
that, the assumption that $a\in L^{\perp}$ and $a^{\prime}\in L^{\prime\perp}$
implies that both $T$ and $T^{{}^{\prime}}$ are right-triangles, and both have
hypotenuse along the segment connecting $w$ and $O$. Moreover, if $\theta$ and
$\theta^{\prime}$ denote the the angles of triangles $T$ and $T^{\prime}$
(resp.) at vertex $w$, then we have

$\theta+\theta^{\prime}\lesssim\delta.$

This is just another way of asserting that both $\theta$ and $\theta^{\prime}$
are proportionally bounded above by the separation-in-direction of $\ell$ and
$\ell^{\prime}$.

(a) The right triangle formed by the origin, $a$ and $w$.

(b) The right-triangle formed by the origin, $a^{\prime}$ and $w$.

Figure 2. Both of these right-triangles are contained in two separate two-
dimensional hyperplanes. Hence, we can apply elementary trigonometry and the
triangle inequality to estimate their dimensions.

Combining our previous observations, we have

$|a-a^{\prime}|\leq|a-O|+|a^{\prime}-O|=\sin\theta|W|+\sin\theta^{\prime}|W|\lesssim(\theta+\theta^{\prime})|W|.$

Since $\theta+\theta^{\prime}\lesssim\delta$, this gives:

$d^{O}(\ell,\ell^{\prime})=\delta+|a-a^{\prime}|\lesssim\delta|W|=|W|\cdot
d^{w}(\ell,\ell^{\prime}).$

Hence, it suffices to take $C_{2}\gtrsim|W|$. The proof is finished for this
special case. ∎

Proposition 19 guarantees that any $s$-dimensional measures we generate from
the metrics $d^{w}$ are guaranteed to be absolutely continuous. In particular,
then, we can define our notion of $s$-dimensional Hausdorff measure on
$\mathcal{A}(n,1)$ relative to the metric $d_{\mathcal{A}(n,1)}^{O}$ (so $w$
is chosen to be the origin). This is the metric given in [Mat99]. This brings
us to the following Lemma–which we think of as an identity for calculating the
dimension of line families which happen to have common intersection at a
point.

######  Lemma 20.

Let $\mathcal{L}_{x}$ be a family of lines in $\mathbb{R}^{n}$, and suppose
there exists a common point $x\in\ell$ for each $\ell\in\mathcal{L}_{x}$.
Then,

$\dim\mathcal{L}_{x}=\dim\pi_{x}\bigg{(}\bigcup_{\ell\in\mathcal{L}}\ell\setminus\\{x\\}\bigg{)}.$

###### Proof.

The proof is a summation of what we have already discussed. Namely, since our
notion of dimension is stable under translation, we can freely choose $x=O$ to
be the origin. This is critical, because in this special situation,
$\mathcal{L}_{O}$ can be taken as a subset of $\mathcal{G}(n,1)$. In
particular, we can define the dimension of our line family $\mathcal{L}_{x}$
relative to the measure $\gamma_{s}$ defined in Definition 17. The
details–again, freely assuming now that $x=O$ due to Proposition 19–are as
follows:

$\displaystyle\dim\mathcal{L}_{x}$ $\displaystyle:=$
$\displaystyle\sup\\{s:\gamma_{s}(\mathcal{L}_{O})=0\\}$ $\displaystyle=$
$\displaystyle\sup\\{s:\mathcal{H}^{s}(\Theta_{\mathcal{L}_{O}})=0\\}$
$\displaystyle=$
$\displaystyle\sup\big{\\{}s:\mathcal{H}^{s}\big{(}\bigcup_{\ell\in\mathcal{L}_{O}}(\ell\cap\mathbb{S}^{n-1})\big{)}=0\big{\\}}$
$\displaystyle=$
$\displaystyle\dim\pi_{O}\bigg{(}\bigcup_{\ell\in\mathcal{L}}(\ell\setminus\\{O\\})\bigg{)}$

∎

######  Remark 21.

In the special case of $\mathcal{A}(2,1)$, one may leverage the duality of
points and lines to greatly simplify the notion of dimension (at least in
certain special cases). A discussion of this idea is given in [OS23] and
[OSW22]. However, when working in $\mathcal{A}(n,1)$ for $n\geq 3$, this
duality is not available to us. For this reason, we think of Lemma 20 as a
substitute for this duality–at least in the special case where our line family
has common intersection in a point.

These lemmata on dimension of line sets, as well as classic lower bounds on
the dimension of Furstenberg sets, are utilized in the proof of Theorem 3.

###  2.2. Furstenberg Set Estimates

We now go into some background regarding Furstenberg set estimates, which has
been a rapidly developing topic over the past few years and one that we will
utilize in this paper. An $(s,t)$-Furstenberg set in $\mathbb{R}^{n}$ is a set
of points $F$ such that there exists a (non-empty) family of lines
$\mathcal{L}\subset\mathcal{A}(n,1)$ such that $\dim\mathcal{L}\geq t>0$, and
such that $\dim(F\cap\ell)\geq s>0$ for all $\ell\in L.$

The Furstenberg set problem strives to find a lower bound on the dimension of
all $(s,t)$-Furstenberg sets–a problem that was recently solved by Kevin Ren
and Hong Wang in the plane [RW23]. Before this result was proven, there were a
number of partial results towards the full conjecture which prove useful for
this paper. One such result is the the following:

$\dim F\geq s+\min\\{s,t\\}$

where $F$ is an $(s,t)$-Furstenberg set. The case when $t\leq s$ was proven by
Lutz and Stull using information theory [LS20] and the case $s\leq t$ is
essentially due to Wolff [Wol99]. Both cases were also proven by
Héra–Shmerkin–Yavicoli in [[HSY20], Theorem A.1].

However, studying (generalized) dual Furstenberg sets has been proving
fruitful in the past few years. A (generalized) dual $(s,t)$-Furstenberg set
of lines in $\mathbb{R}^{n}$ is a set of lines $\mathcal{L}$ such that there
exists a non-empty set of pins $F^{\ast}\subset\mathbb{R}^{n}$ such that $\dim
F^{\ast}\geq t>0$, and for all $x\in F^{\ast}$, there exists a set of lines
$\mathcal{L}_{x}\subset\mathcal{L}$ through $x$ such that
$\dim\mathcal{L}_{x}\geq s\geq 0$. The (generalized) dual Furstenberg set
problem then asks for a lower bound on $\dim\mathcal{L}$. As it happens, in
two dimensions, the notion of a dual Furstenberg set is precisely dual to a
Furstenberg set using point-line duality (see [OSW22] Section 3.1 for more
details). In particular, via point-line duality in two dimensions, one obtains

$\dim\mathcal{L}\geq s+\min\\{s,t\\}$

where $\mathcal{L}$ is a dual $(s,t)$-Furstenberg set. This lower bound is the
one we shall utilize in the proof of the continuum Erdős–Beck Theorem.

######  Remark 22.

While in higher dimensions, generalized dual Furstenberg sets are not dual to
a Furstenberg set, the notion still generalizes the lower dimensional case,
and are thusly referred to as generalized dual Furstenberg sets. See Section
3.1 for an analogous lower bound.

###  2.3. Planar Erdős–Beck Theorems for Fractal Sets

We now prove our main result in the plane, which we restate here for
convenience.

######  Theorem 23.

Let $X\subset\mathbb{R}^{2}$ be Borel. Then,

  1. 1)

if $\dim(X\setminus\ell)=\dim X$ for every $\ell\in\mathcal{A}(2,1)$, then
$\dim\mathcal{L}(X)\geq\min\\{2\dim X,2\\}$.

  2. 2)

if there exists an $\ell\in\mathcal{A}(2,1)$ such that
$\dim(X\setminus\ell)<\dim X$, we let $0<t\leq\dim X$ such that
$\dim(X\setminus\ell)\geq t$ for all $\ell\in\mathcal{A}(2,1)$. Then,

$\dim\mathcal{L}(X)\geq\dim X+t.$

###### Proof.

If it is the case that for all lines $\ell$ we have $\dim(X\setminus\ell)=\dim
X$, then we may apply the continuum version of Beck’s theorem and obtain

$\dim\mathcal{L}(X)\geq\min\\{2\dim X,2\\}.$

Otherwise, there exists an $\ell\in\mathcal{A}(2,1)$ such that $\dim
X\setminus\ell<\dim X$. Fix $0<t\leq\dim X$ such that
$\dim(X\setminus\ell^{\prime})\geq t$ for all lines $\ell^{\prime}$. Given
there exists a line $\ell$ such that $\dim(X\setminus\ell)<\dim X$, we have
that $\dim(X\cap\ell)=\dim X:=s$. Therefore, by Lemma 25, we have that

$\dim\pi_{x}(X\setminus\\{x\\})\geq s,\quad\forall x\in X\setminus\ell.$

Hence, by Lemma 20, the set of lines through $x$,
$\mathcal{L}_{x}\subset\mathcal{L}(X)$, satisfies

$\dim\mathcal{L}_{x}=\dim\pi_{x}(X\setminus\\{x\\})\geq s$

for all $x\in X\setminus\ell$.

Hence, we have that

$\mathcal{L}(X)\supset\bigcup_{x\in X\setminus\ell}\mathcal{L}_{x},$

where $\dim(X\setminus\ell)\geq t$ (by assumption) and
$\dim\mathcal{L}_{x}\geq s$ for all $x\in X\setminus\ell$. This is an dual
$(s,t)$-Furstenberg set (note that here we used $X\setminus\ell$ as our non-
empty set of pins with dimension at least $t>0$ by assumption). Thus, using
point-line duality, we see that

$\dim\left(\bigcup_{x\in X\setminus\ell}\mathcal{L}_{x}\right)\geq\dim F,$

where $F$ is an $(s,t)$-Furstenberg set. Note that by assumption, $0<t\leq
s=\dim X$. Thus, the result of Lutz–Stull gives that $\dim F\geq s+t$. In
total, we have that

$\dim\mathcal{L}(X)\geq\dim\left(\bigcup_{x\in
X\setminus\ell}\mathcal{L}_{x}\right)\geq\dim F\geq s+t.$

This gives the desired result. ∎

######  Corollary 24.

Let $X\subset\mathbb{R}^{2}$ be Borel. Then,

  1. 1)

if $\dim(X\setminus\ell)=\dim X$ for every $\ell\in\mathcal{A}(2,1)$, then
$\dim\mathcal{L}(X)\geq\min\\{2\dim X,2\\}$.

  2. 2)

if there exists an $\ell\in\mathcal{A}(2,1)$ such that
$\dim(X\setminus\ell)<\dim X$, we let $0<t\leq\dim X$ such that
$\dim(X\setminus\ell)\geq t$ for all $\ell\in\mathcal{A}(2,1)$. Then,

$\dim\mathcal{L}(X)\geq\max\\{\dim X+t,\min\\{2\dim X,2(T(X)-1)\\}.$

###### Proof.

Compare Theorem 23 with Theorem 6 (due to Ren) to obtain the result. ∎

##  3\. The Higher Dimensional Theorem

###  3.1. Preliminaries for Higher Dimensions

In the course of proving Theorem 3, we will use the following two lemmata.

######  Lemma 25.

Let $X\subset\mathbb{R}^{n}$ with $\dim X=s\in[k,k+1)$ for some integer $1\leq
k\leq n-1$. Assume that there exists a $k$-plane $H\in\mathcal{A}(n,k)$ so
that $\dim(X\cap H)=s$. Then,

(6) $\dim\pi_{x}(X\setminus\\{x\\})\geq s,\quad\forall x\in X\setminus H.$

###### Proof of Lemma 25.

Firstly, note that if $X\setminus H=\emptyset$, this is vacously true, so we
may assume that $X\setminus H\neq\emptyset.$

Call $X_{1}:=X\cap H$ and $X_{2}:=X\setminus H$. We show that,

$\dim\pi_{x_{2}}(X_{1})=s,\quad\forall x_{2}\in X_{2}.$

which clearly implies (6). Indeed, this follows because the mapping
$\pi_{x_{2}}:H\rightarrow\mathbb{S}^{n-1}$ is locally bi-Lipschitz for each
fixed $x_{2}\in X_{2}$. However, since locally bi-Lipschitz mappings preserve
dimension, we have

$\dim\pi_{x_{2}}(X_{1})=\dim X_{1}=s.$

Hence, to finish the proof, we show that the mapping
$\pi_{x_{2}}:X_{1}\rightarrow\mathbb{S}^{n-1}$ is locally bi-Lipschitz.

We will assume (without loss of generality) that

$H:=\\{x\in\mathbb{R}^{n}:x_{k+1}=\cdots=x_{n-1}=0;x_{n}=1\\}$

and that $x_{2}=0$. Note that, for an arbitrary $k$-dimensional hyperplane
$H\subset\mathbb{R}^{n}$ and point $x_{2}\in\mathbb{R}^{n}$, we can always
perform an affine transformation to arrive at this special case. Since affine
transformations preserve Hausdorff dimension, we are fine making this
reduction. In this special case, notice that

$\pi_{x_{2}}(H)=\\{\theta\in\mathbb{S}^{n-1}:\theta_{k+1}=\cdots=\theta_{n-1}=0\\}=\Theta.$

So, the mapping $\pi_{x_{2}}|_{H}$ admits an inverse function
$\psi:\Theta\rightarrow H$, defined via:

$\psi(\theta_{1},...,\theta_{k},\theta_{k+1},...,\theta_{n}):=\left(\frac{\theta_{1}}{\theta_{n}},...,\frac{\theta_{k}}{\theta_{n}},0,...,0,1\right).$

Now, for each $n=1,2,3,...$, let

$X_{n}:=\\{x\in X_{1}:|x-e_{n}|\leq n\\},$

where $e_{n}$ is the $n$-th standard normal vector in $\mathbb{R}^{n}$.
Similarly, let $\Theta_{n}:=\pi_{x_{2}}(X_{n})$, which is a subset of
$\Theta$. The mapping $\pi_{x_{2}}|_{X_{n}}$ is Lipschitz with constant $\sim
1$. Moreover, this mapping admits an inverse $\psi_{n}:\Theta_{n}\rightarrow
X_{n}$, which is also Lipschitz with constant $\sim n$. Therefore,

$\dim\pi_{x_{2}}(X_{n}):=\dim\Theta_{n}=\dim X_{n}.$

In particular,

$\dim\pi_{x_{2}}(X_{1})\geq\sup_{1\leq n<\infty}(\dim X_{n})=\dim X.$

Since $X_{1}\subset X$, this concludes the proof. ∎

In addition to the previous two lemmata, we will require some recent results—
the first of which is a continuum higher dimensional Beck’s theorem by Ren,
and the second of which is a dual Furstenberg set estimate from the first
author, Fu, and Ren—both of which we state without proof.

######  Theorem 26 ([Ren23], Corollary 1.5).

Let $X\subset\mathbb{R}^{n}$ be a Borel set with $\dim(X\setminus H)=\dim X$
for all $k$-planes $H$. Then, the line set $\mathcal{L}(X)$ spanned by pairs
of distinct points in $X$ satisfies

$\dim\mathcal{L}(X)\geq\min\\{2\dim X,2k\\}.$

######  Theorem 27 ([BFR24], Theorem 4).

Let $0\leq s\leq n-1$ and let $0<t\leq n$. Given $\mathcal{L}$ is a dual
$(s,t)$-Furstenberg set of lines in $\mathbb{R}^{n}$, we have,

$\dim\mathcal{L}\geq s+\min\\{s,t\\}.$

###  3.2. Higher Dimensional Erdős-Beck Theorems for Fractal Sets

We now prove the higher-dimensional Erdős-Beck Theorem, Theorem 3, restated
here.

######  Theorem 28.

Let $X\subset\mathbb{R}^{n}$ be Borel, and let $k\in[1,n-1]$ be an integer.
Then,

  1. 1)

if $\dim(X\setminus H)=\dim X$ for every $H\in\mathcal{A}(n,k)$, then
$\dim\mathcal{L}(X)\geq\min\\{2\dim X,2k\\}$.

  2. 2)

if there exists an $H\in\mathcal{A}(n,k)$ such that $\dim(X\setminus H)<\dim
X$, we let $0<t\leq\dim X$ such that $\dim(X\setminus P)\geq t$ for all
$P\in\mathcal{A}(n,k)$. Then,

$\dim\mathcal{L}(X)\geq\dim X+t.$

###### Proof.

Fix an integer $k\in[1,n-1]$. If it is the case that for all $k$-planes
$H\in\mathcal{A}(n,k)$, we have $\dim(X\setminus H)=\dim X$, then by Theorem
26 from [Ren23], we have that

$\dim\mathcal{L}(X)\geq\min\\{2\dim X,2k\\}.$

Otherwise, there exists an $H$ such that $\dim(X\setminus H)<\dim X$. Then,
fix $0<t\leq\dim X$ such that $\dim(X\setminus P)\geq t$ for all $k$-planes
$P\in\mathcal{A}(n,k)$. We now go into case work. Given there exists a
$k$-plane $H\in\mathcal{A}(n,k)$ such that $\dim(X\setminus H)<\dim X$, we
have that $\dim(X\cap H)=\dim X:=s$. Therefore, by Lemma 25, we have that

$\dim\pi_{x}(X\setminus\\{x\\})\geq s,\quad\forall x\in X\setminus H.$

Hence, by Lemma 20, for all $x\in X\setminus H$, the set of lines through $x$,
$\mathcal{L}_{x}\subset\mathcal{L}(X)$, satisfies

$\dim\mathcal{L}_{x}=\dim\pi_{x}\left(X\setminus\\{x\\}\right)\geq s$

for all $x\in X\setminus H$.

In total, we have that

$\mathcal{L}(X)\supset\bigcup_{x\in X\setminus H}\mathcal{L}_{x},$

where $X\setminus H$ is non-empty, $\dim X\setminus H\geq t$, and
$\dim\mathcal{L}_{x}\geq s$ for all $x\in X\setminus H$. By Theorem 27, this
implies that

$\dim\mathcal{L}(X)\geq\dim\left(\bigcup_{x\in X\setminus
H}\mathcal{L}_{x}\right)\geq s+\min\\{t,s\\}=s+t:=\dim X+t.$

∎

######  Corollary 29.

Let $X\subset\mathbb{R}^{n}$ be Borel and let $k\in[1,n-1]$ be an integer.
Then,

  1. 1)

if $\dim(X\setminus H)=\dim X$ for every $H\in\mathcal{A}(n,k)$, then
$\dim\mathcal{L}(X)\geq\min\\{2\dim X,2k\\}$.

  2. 2)

if there exists an $H\in\mathcal{A}(n,k)$ such that $\dim(X\setminus H)<\dim
X$, we let $0<t\leq\dim X$ such that $\dim(X\setminus P)\geq t$ for all
$P\in\mathcal{A}(n,k)$. Then,

$\dim\mathcal{L}(X)\geq\max\\{\dim X+t,\min\\{2\dim X,2(T(X)-1)\\}\\}.$

###### Proof.

Compare Theorem 28 with Theorem 6 (due to Ren) to obtain the result. ∎

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