# Envisioning New Futures of Positive Social Technology: Beyond Paradigms of
Fixing, Protecting, and Preventing

JaeWon Kim  0000-0003-4302-3221 University of WashingtonSeattleWAUSA
<EMAIL_ADDRESS>,  Lindsay Popowski  0000-0002-5649-0286 Stanford
UniveristyStanfordCAUSA<EMAIL_ADDRESS>,  Anna Fang
0000-0003-0055-3011 Carnegie Mellon UniversityPittsburghPAUSA
<EMAIL_ADDRESS>,  Cassidy Pyle  0000-0003-4578-0226 University of
MichiganAnn ArborMIUSA<EMAIL_ADDRESS>,  Guo Freeman  0000-0001-5107-7794
Clemson UniversityClemsonSCUSA<EMAIL_ADDRESS>,  Ryan M. Kelly
0000-0002-8773-6656 RMIT UniversityMelbourneVICAustralia
<EMAIL_ADDRESS>,  Angela Y. Lee  0000-0002-9527-5730 Stanford
UniversityStanfordCAUSA<EMAIL_ADDRESS>,  Fannie Liu
0000-0002-5656-3406 JPMorgan Chase & Co.New YorkNYUSA<EMAIL_ADDRESS>,  Angela D. R. Smith  0000-0001-5546-5452 The University of Texas at
AustinAustinTXUSA<EMAIL_ADDRESS>,  Alexandra To  0000-0001-7298-7607
Northeastern UniversityBostonMAUSA<EMAIL_ADDRESS>and  Amy X. Zhang
0000-0001-9462-9835 University of WashingtonSeattleWAUSA<EMAIL_ADDRESS>
(2024)

###### Abstract.

Social technology research today largely focuses on mitigating the negative
impacts of technology and, therefore, often misses the potential of technology
to enhance human connections and well-being. However, we see a potential to
shift towards a holistic view of social technology’s impact on human
flourishing. We introduce Positive Social Technology (Positech), a framework
that shifts emphasis toward leveraging social technologies to support and
augment human flourishing. This workshop is organized around three themes
relevant to Positech: 1) “Exploring Relevant and Adjacent Research” to define
and widen the Positech scope with insights from related fields, 2) “Projecting
the Landscape of Positech” for participants to outline the domain’s key
aspects and 3) “Envisioning the Future of Positech,” anchored around strategic
planning towards a sustainable research community. Ultimately, this workshop
will serve as a platform to shift the narrative of social technology research
towards a more positive, human-centric approach. It will foster research that
goes beyond fixing technologies to protect humans from harm, to also pursue
enriching human experiences and connections through technology.

positive social technology; social computing

††journalyear: 2024††copyright: rightsretained††conference: Companion of the
2024 Computer-Supported Cooperative Work and Social Computing; November 9–13,
2024; San Jose, Costa Rica.††booktitle: Companion of the 2024 Computer-
Supported Cooperative Work and Social Computing (CSCW Companion ’24), November
9–13, 2024, San Jose, Costa Rica††isbn: 979-8-4007-1114-5/24/11††doi:
10.1145/3678884.3681833††ccs: Human-centered computing Collaborative and
social computing

##  1\. Introduction

Much of social computing research today primarily focuses on addressing the
flaws of current platforms, particularly in social media. However, this often
limits exploring new, potentially more beneficial socio-technical systems.
Indeed, today’s research interventions are often confined within the bounds of
platforms like Facebook and Instagram (González-Bailón et al., 2023; Guess et
al., 2023a, b; Nyhan et al., 2023), as opposed to pioneering new models, such
as in research prototypes from two decades ago (Miller et al., 2003).
Furthermore, legislative approaches focus reactively on addressing problems on
specific, existing platforms rather than proactively considering new
possibilities (pre, [n.d.]; Schatz and [d Hi], 2023; Stewart and [r Ut-2],
2023; bil, [n.d.]) for the future of social technologies, furthering the view
of social media as a fixed entity.

While prevention is important, focusing on positive and healthy potentials is
equally important. We draw inspiration from positive psychology (Seligman and
Csikszentmihalyi, 2000), which arose from the understanding that people seek
more than just relief from suffering; they aspire to live meaningful,
fulfilling lives, nurturing their best qualities and enriching their
experiences of love, work, and play (Seligman, 2019). Decades of research in
positive psychology have demonstrated that enhancing positive affect or life
satisfaction is not merely about removing negative affect (Seligman, 2002).

Similar opportunities for positive impact are emerging in social computing
research. To illustrate this, we consider social media as a representative
example. Much research has understandably focused on addressing issues such as
problematic social media use (Nesi, 2020; Odgers and Jensen, 2020; Ito et al.,
2020; Valkenburg et al., 2022; Van Der Wal et al., [n.d.]; Sala et al., 2024;
Davis et al., 2023; Tran et al., 2019), the adverse effects of social media on
mental health (Choukas-Bradley et al., 2023; Hamilton et al., 2022; Jensen et
al., 2019; Keles et al., 2020; Nesi, 2020; Odgers and Jensen, 2020; Ito et
al., 2020; Valkenburg et al., 2022; Sala et al., 2024), the effect of
misinformation/disinformation on political dynamics (Suarez-Lledo et al.,
2021; Wang et al., 2019), and the implications of privacy violations (Zhao et
al., 2022; Hargittai and Marwick, 2016; Ali et al., 2022; Agha et al., 2023;
Wisniewski et al., 2022; Razi et al., 2020). However, an additional,
complementary approach allows us to more comprehensively address social
media’s core potential: fostering and amplifying human flourishing, or “living
within an optimal range of human functioning, one that connotes goodness,
generativity, growth, and resilience,” (Fredrickson and Losada, 2005) through
the cultivation of positive emotions, engagement, relationships, meaning,
purpose, and accomplishments, following the PERMA model of well-being
(Seligman, 2011). To truly harness social media as technology centered on
human flourishing, our research, design, and development must prioritize and
be sensitive to both inter- and intra-personal growth and needs. We see great
promise in integrating scientific knowledge and theories to make life more
fulfilling into research on social technologies and in designing and creating
new sociotechnical environments that better support human and societal
flourishing through social interaction.

###  1.1. Positive Social Technology (Positech)

Given the historical evolution of social computing research and the rise of
positive psychology, we propose Positech as the theme of our workshop.
Positech is a framework that advocates for a shift in how we engage with
social technology, emphasizing its potential to support and enhance human
flourishing. It is characterized by focusing on the potential and
opportunities for innovation and positive change that center on human
flourishing rather than merely rectifying issues in humans or technology.
Positech is both a framework and a set of efforts encompassing the development
of technologies, legislative actions, industry initiatives, theoretical
research, and meta-scientific research.

Figure 1. Venn Diagram of Positech and Related Concepts

The image depicts a Venn diagram with three overlapping circles within a
larger rectangular frame titled “Positech.” Each circle represents a different
field: “Positive Technology,” “Positive Computing,” and “Asset-based
Approach.” The overlapping sections of the circles are labeled with letters
(A), (B), (C), (D), and (E) to indicate areas of intersection between these
fields. Area (A) is the portion of the “Positive Technology” circle that does
not overlap with the other fields. Area (B) is the portion of the “Positive
Computing” circle that does not overlap with the other fields. Area (C) is the
portion of the “Asset-based Approach” circle that does not overlap with the
other fields and is outside of the “Positech” rectangle. Area (D) is the
portion within Positech that is outside the all other circles, indicating the
unique portion of the “Positech” rectangle. Area (E) is the central
overlapping area where all three circles intersect, indicating a shared space
between “Positive Technology,” “Positive Computing,” and “Asset-based
Approach.”

Figure 1 shows the relationships between Positech and its adjacent fields.
Here, we lay out the distinct guiding questions of each field for each labeled
area in the figure.

  1. (A)

Positive Technology: How can we use technology to increase emotional quality,
engagement, and connectedness in personal experiences (e.g., therapy)? (Riva
et al., 2012; Botella et al., 2012; Gaggioli et al., 2017)

  2. (B)

Positive Computing: How can we account for well-being determinants such as
autonomy, competence, relatedness, compassion, engagement, and meaning at
every stage of interaction design? (Calvo and Peters, 2014)

  3. (C)

Asset-based Approach: How can we build upon users’ strengths and self-
determination in using and engaging with technology? (Wong-Villacres et al.,
2020)

  4. (D)

Positive Social Technology (Positech): How are we balancing the needs for
fixing and innovating social technologies in supporting human flourishing in
areas such as positive affect and meaning?

An example outcome at the intersection of these fields (labeled (E) in Figure
1) could include research on a proof-of-concept for a new social media
platform designed to explore new possibilities for increasing human
connectedness. This platform would leverage people’s intrinsic motivations to
empathize with others to enhance the effectiveness of peer support.

Positech is distinct from positive technology or computing, which focuses on
designing technology to improve personal well-being and quality of life. While
principles of positive technology and computing guide systems creation, they
represent only one aspect of Positech. Positech involves not just creating
technology but also driving social technology development through broader
ethical and societal dialogue about technology’s role. Unlike the asset-based
approach, Positech does not define users’ roles in technology use. It aims to
maximize technology’s potential in supporting universally valued aspects of
human life, such as fostering connections and self-actualization, without
making value judgments on individual capacities, which can vary across
cultures and contexts. Positech also puts a particular emphasis on social
technology.

It is important to note that within the Positech framework, we prioritize
enriching interpersonal and personal experiences, viewing technology primarily
as a supportive tool. While social technologies hold transformative potential
for facilitating human connections, they should not replace them by default.
Our goal is for technology to amplify direct human interactions and, where
necessary, create new avenues for meaningful engagement. This ensures that
technology enhances rather than substitutes for meaningful and authentic human
connections, maintaining our focus on supporting the core elements of human
flourishing.

###  1.2. Relevance to the CSCW Community

We recognize that within the CSCW community, there are already numerous
research initiatives that embody the principles of Positech, focusing on
proactively enhancing human flourishing and well-being through technology (To
et al., 2023; Wohn et al., 2018; Freeman and Wohn, 2017; Maloney and Freeman,
2020; Freeman and Acena, 2021; Freeman et al., 2016; Zytko et al., 2015;
Musick et al., 2021; Li et al., 2021, 2023, 2024; Liu et al., 2021; Zhang et
al., 2022). Our vision is not to present these ideas as new but to acknowledge
and unify these scattered efforts under the Positech framework. By doing so,
we aim to consolidate and amplify the impact of work that aligns with our
goals:

  1. (1)

Beyond fixing people, toward the goals of positive psychology such as human
flourishing and well-being. This direction is in contrast to work that
exclusively documents harms or seeks to effect behavior change without the
intention of the user.

  2. (2)

Beyond fixing technology, toward new paradigms and design patterns that
prioritize positive psychology. This direction contrasts with work that aims
to mitigate harms that exist on current platforms.

These directions are not entirely separate but depict two different
orientations toward our goals that best represent different types of research
contributions. By focusing on both immediate and broader possibilities, we
envision a future where CSCW is instrumental in unlocking the full potential
of social technologies to contribute to human flourishing. However, it is
crucial to acknowledge that social technology research has played a
significant role in bringing us to our present understanding, and it is not
the intention to diminish the value of these contributions. Rather, we call
for a broader perspective that includes seeking new possibilities that would
complement rather than replace current research momentum.

##  2\. Organizers

JaeWon Kim is a PhD candidate at the University of Washington Information
School. Her research focuses on understanding how social technologies support
or undermine relationship building and designing and building social
technologies that center on trusting relationship building, especially for the
youth.

Lindsay Popowski is a PhD candidate in the Stanford University Computer
Science Department. Her research is in the field of social computing systems,
with an emphasis on the design of social media. She works to design and build
online spaces that facilitate unique social goals, such as strong
interpersonal relationships, effort-sharing, and preserving conversational
context.

Anna Fang is a PhD student at Carnegie Mellon University in the Human-Computer
Interaction Institute, School of Computer Science. Her work is at the
intersection of computational social science and mental well-being. She is
predominantly interested in proactive approaches — those that are self-
sustaining, self-correcting, or promote positive behaviors — for emotional and
mental health, rather than retrospective handling of harm after it has
occurred.

Cassidy Pyle is a Ph.D. Candidate at the University of Michigan School of
Information. Her work examines how marginalized and stigmatized communities’
interactions with and on socio-technical systems (e.g., social media,
algorithms) may simultaneously afford and constrain self-expression, emotional
well-being, and access to college and career opportunities.

Guo Freeman is an Associate Professor of Human-Centered Computing at Clemson
University. Her work focuses on how interactive technologies such as digital
games, live streaming, social VR, and AI shape interpersonal relationships and
group behavior; and how to design safe, inclusive, and supportive social VR
spaces to mitigate emergent harassment risks.

Ryan M. Kelly is an Associate Professor in the School of Computing
Technologies at RMIT University. His research focuses on the design and
evaluation of communication technologies for fostering meaningful connections
and alleviating social isolation. This includes designing for sensitive
settings and vulnerable user groups, such as people with chronic health
conditions or older adults living in long-term institutional care.

Angela Y. Lee is a Ph.D. Candidate in the Department of Communication at
Stanford University. Her research investigates how communication technologies
affect well-being.

Fannie Liu is a VP Applied Research Lead on the Global Tech Applied Research
AR/VR team at JPMorgan Chase & Co. Her research involves the design of novel
social experiences that promote social connection, collaboration, and well-
being.

Angela D. R. Smith is an Assistant Professor in the School of Information at
the University of Texas at Austin, where she co-leads the Research on Equity,
Access, and incLusIon in Technology and societY (REALITY) Lab. Her research
explores the design of socially responsible technology experiences by
examining racial and cultural inequities.

Alexandra To is an Assistant Professor jointly appointed in the Art+Design
department and the Khoury College of Computer Sciences at Northeastern
University. She works in HCI, game design, critical race theory, and identity.

Amy X. Zhang is an Assistant Professor in the University of Washington Allen
School of Computer Science and Engineering, where she leads the Social Futures
Lab dedicated to reimagining social and collaborative systems to empower
people and improve society. Her research areas broadly include social
computing, CSCW, human-AI interaction, and HCI.

##  3\. Agenda

Our workshop at CSCW 2024 will bring together 15-30 researchers/
practitioners/designers to discuss the area of Positech: aiming to envision
new futures of positive social technology and strategize how to support each
other’s efforts. While we acknowledge that the absence of a remote
participation option may present equity concerns for some potential
participants, we will be holding this workshop in person only due to budgetary
and logistical limitations.

We aim to explore the overlapping fields of research that contribute to
Positech, project a current landscape and future agenda, and plan how to build
community and support our peers. This workshop puts a specific focus on
community-building and collaborative agenda-setting; we, therefore, dedicate
most of our time to small-group brainstorms and discussions. We also plan to
collaboratively develop a course syllabus through the workshop to advance and
project the Positech research agenda.

Table 1. Workshop Agenda Time |  Activity
---|---
9:00-10:00am |  Welcome and Attendee Introduction
10:00-10:20am |  Coffee Break
10:20-11:20am |  Panel: Positech Histories and Influences
11:20-12:00pm |  Discussion: Positech Boundaries and Overlaps
12:00-1:30pm |  Lunch Break
1:30-2:50pm |  Brainstorm, Share, Synthesize: Positech Challenges and Promising Directions
2:50-3:10pm |  Coffee Break
3:10-3:50pm |  Brainstorm, Share: How to Support Positech Research
3:50-4:40pm |  Brainstorm, Share: Positech Community Futures
4:40-5:00pm |  Closing
6:30pm |  (Optional) Dinner Social Event

###  3.1. Themes

Our workshop activities are motivated by particular goals, which we realize
via three themes.

####  3.1.1. Theme 1: Exploring Relevant and Adjacent Research

We explore this theme via the “Positech Histories and Influences” Panel. We
will invite panelists across fields that overlap with Positech to explore the
following questions:

  * •

Historical Lessons: What can we learn from prior research on social
technologies? How can that inform future research directions? How have
previous platforms/research projects influenced current social technologies?

  * •

Defining Positech: What do we think are important distinctions to draw between
Positech and related fields or frameworks? How can we synthesize insights from
related fields such as positive computing, positive technologies, strength-
based technology, and asset-based technology with lessons from adjacent
disciplines like psychology to inform the development of social technologies
aimed at human flourishing?

####  3.1.2. Theme 2: Projecting the Landscape of Positech

We address this theme via two Discussions: “Positech Boundaries and Overlaps”
and “Positech Challenges and Promising Directions.”

We aim to map the current landscape of Positech. Key questions for discussion
include: What constitutes Positech? What are its challenges and opportunities?
To facilitate this exploration, participants will collaborate in groups to
design a course syllabus regarding themes around Positech: origins,
overlapping fields and frameworks, and different ongoing research directions.

####  3.1.3. Theme 3: Envisioning the Future of Positech

This theme is addressed through the “How to Support Positech Research” and
"Positech Community Futures” Discussions.

For the final activity of our workshop, we will engage in a brainstorming
session to outline our post-workshop plans. This will be conceptualized as a
research question: How can we maintain an asynchronous, geographically
dispersed, cross-generational, and (potentially) culturally diverse research
community? Addressing community continuity is a complex challenge that has
garnered much attention in social computing (Bali et al., 2023; Burke et al.,
2009; Kraut et al., 2011), and social technologies are perhaps our best tool
for sustaining such a community across geographical distances. 111One example
idea that may come out of this discussion is a Slack (Slack, [n.d.]) bot
designed to facilitate interaction among participants. This bot would pair
members for virtual coffee chats, using conversation starters linked to their
research interests to foster sustained dialogue and collaboration.

We plan to implement strategies developed during the session to ensure the
momentum generated by our workshop continues to build. Relevant questions
include:

  * •

Ideal scenarios. What does an ideal future look like with Positech fully
integrated into social technology research and society in general?

  * •

Sustaining Positech. How can we keep the momentum alive post-workshop? How
should this community function and interact? How can we assist others in the
Positech community with research?

  * •

Unintended consequences. How can we anticipate unintended consequences of new
technologies or communities proactively and prevent them at an early stage?
How can we ensure that Positech actually prioritizes and supports the well-
being of individuals and communities?

###  3.2. Deliverables

We aim to come away from the workshop with two main takeaways:

  1. (1)

Positech Syllabus: a course syllabus on Positech. This syllabus will outline
the course’s overview and motivation, feature 8-15 weekly themes covering
various Positech aspects, include recommended weekly readings, and detail
assignments such as projects and reading responses. We will share the result
on our website as a resource for the wider community, along with a broader
list of exemplary papers. (Developed during first two discussions, including
insights from the panel.)

  2. (2)

Positech Community Plan: a set of steps to develop and maintain a community of
researchers engaged in the Positech vision. This will include ideas for
maintaining communication between attendees, facilitating discovery for
newcomers to the area, and ways to connect members with support within the
community. (Developed during last two discussions.)

##  4\. Workshop Logistics

###  4.1. Website

The link to our website is https://positech-cscw-2024.github.io/. We have
prepared the website with an overview of our workshop, a call for
participation, and key information about the workshop, including the workshop
agenda and organizer information. We also plan to publicize any outcome of our
workshop, such as the hypothetical Positech course syllabi and workshop
submissions, if given the authors’ consent.

###  4.2. Pre-workshop Plan

Call for participation. We will invite 25-30 individuals interested in the
goal of envisioning new futures of positive social technology to participate
in our in-person-format workshop. Submissions are welcomed in several forms:

  * •

Position papers or drafts (2-6 pages, ACM single-column format, excluding
references) discussing themes of the workshop. Alternate forms, such as design
fiction, are encouraged.

  * •

“Encore” submissions of relevant conference or journal papers.

  * •

Statements of research interest (up to 2 pages, ACM single-column format,
excluding references).

Selecting participants. Submissions will be reviewed by the organizers based
on: 1) relevance to the workshop’s theme, 2) quality of the submission, and 3)
diversity of perspectives. This approach will ensure a rich exchange of ideas
among participants from varied backgrounds and specialties.

Workshop preparation. Prior to the workshop, we will survey participants to
understand their accessibility needs, constraints, and expectations. This
feedback will inform the logistics and schedule, with experienced organizers
ensuring an engaging experience for both in-person and remote attendees.

###  4.3. Accessibility

We will ask all attendees to supply an accessable PDF of their submissions.
For online workshop attendees, we will enable (auto-generated) live closed
captioning, which participants will be able to turn on as desired. Any special
accessibility requirements can be communicated to us, and we will do our best
to provide an accommodation in partnership with the Accessibility Chair.

###  4.4. Post-workshop plan

Our workshop is designed to generate (a) a syllabus, and (b) a plan for
community building after the workshop. We plan to make the syllabus widely
available and publicize it after the workshop. We also plan to follow the
community-building plan to keep our ties and build new ones after the
workshop.

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# Minkowski dimension of the boundaries of the lakes of Wada

Zhangchi Chen

> Abstract
>
> The lakes of Wada are three disjoint simply connected domains in $S^{2}$
> with the counterintuitive property that they all have the same boundary. The
> common boundary is a indecomposable continuum. In this article we calculated
> the Minkowski dimension of such boundaries. The lakes constructed in the
> standard Cantor way has $\ln(6)/\ln(3)\approx 1.6309$-dimensional boundary,
> while in general, for any number in $[1,2]$ we can construct lakes with such
> dimensional boundaries.

Keywords: Minkowski dimension, Lakes of Wada.

##  1 Introduction

A famous counterintuitive example in planar topology, called the lakes of
Wada, is that there exists three disjoint simply connected domains
$U_{1},U_{2},U_{3}$ in $S^{2}$ such that $\partial U_{1}=\partial
U_{2}=\partial U_{3}$. The classical constructions in [8, P60] and in [4]
explain the infinite procedure, while [6, Sect. 8] presents it with an
interesting scenario of the perils of philanthropy.

To simplify computations later, we introduce the lakes of Wada on the unit
square constructed in the standard Cantor way:

Day 1: we dig a simply connected blue lake $B_{1}$ by digging
$[0,\tfrac{2}{3}]\times[\tfrac{1}{3},\tfrac{2}{3}]$ on the island so that
every point on the island is at most $\tfrac{\sqrt{2}}{3}$ far away from the
blue lake.

Day 2: we dig a simply connected red lake $R_{2}$ on the island with width
$\tfrac{1}{9}$ so that every point on the island is at most
$\tfrac{\sqrt{2}}{9}$ far away from the red lake.

Day 3: we dig a simply connected green lake $G_{3}$ on the island with width
$\tfrac{1}{27}$ so that every point on the island is at most
$\tfrac{\sqrt{2}}{27}$ far away from the green lake.

Figure 1: Day 1 Figure 2: Day 2 Figure 3: Day 3

Day 4: we extend our blue lake as $B_{4}$ by digging a continuous canal of
width $\tfrac{1}{81}$ on the island so that every point on the island is at
most $\tfrac{\sqrt{2}}{81}$ far away from the blue lake. The canal starts from
the peak point of $B_{1}$: $(\tfrac{2}{3},\tfrac{1}{2})$, separate into two
directions: up and down, and terminates at two peak points in violet colour in
the lower left corner of the figure.

Day 5: we extend our rad lake as $R_{5}$ by digging a continuous canal of
width $\tfrac{1}{243}$ on the island so that every point on the island is at
most $\tfrac{\sqrt{2}}{243}$ far away from the red lake. The canal starts from
the peak point of $R_{2}$, in dark red colour in the upper left corner of the
figure, separate into two directions: up and down, and terminates at two peak
points, in violet colour near $(\tfrac{2}{3},\tfrac{1}{2})$.

Figure 4: Day 4 Figure 5: Day 5

On the following days, we extend the lake with colour green, red, blue in
turns, by digging a continuous canal of width $1/3$ of the day before and make
sure that every point on the island is closer to our lakes. The canal starts
from one of the peak points (at most two), separate into two directions and
terminates at two peak points.

Thus we get

$\displaystyle B_{1}$ $\displaystyle\subset B_{4}\subset B_{7}\subset\dots$
$\displaystyle R_{2}$ $\displaystyle\subset R_{5}\subset R_{8}\subset\dots$
$\displaystyle G_{3}$ $\displaystyle\subset G_{6}\subset G_{9}\subset\dots$

three increasing series of domains. The union

$U_{B}:=\bigcup_{k=0}^{\infty}B_{3k+1},\ \ \
U_{R}:=\bigcup_{k=0}^{\infty}R_{3k+2},\ \ \
U_{G}:=\bigcup_{k=0}^{\infty}G_{3k+3}$

are simply connected domains. Moreover their boundaries in the unit square
$(0,1)^{2}$ are the same, noted by $W_{\sf cantor}:=(0,1)^{2}\cap\partial
U_{B}=(0,1)^{2}\cap\partial U_{R}=(0,1)^{2}\cap\partial U_{G}$.

To obtain simply connected domaines on $S^{2}$ with the same boundary, one can
paste the unit square with its reflection, and deform the two squares into two
half-balls covering $S^{2}$. The common edge $V_{A}V_{B}$ and
$V_{A}^{\prime}V_{B}^{\prime}$ cover one meridian, while other edges
$V_{A}V_{D}V_{C}$ and $V_{A}^{\prime}V_{D}^{\prime}V_{C}^{\prime}$ cover the
opposite meridian.

Figure 6: Gluing two squares on $S^{2}$

The common boundary $W_{\sf cantor}$ is a nowhere dense closed continuous
fractal set of infinite length and of 0 area. The Minkowski-Bouligand (also
called ‘box-counting’) dimension and Hausdorff-Besicovitch dimension describe
how ‘rough’ this fractal set is. We abbreviate them as Minkowski dimension and
Hausdorff dimension.

######  Definition 1.1 (Minkowski dimension).

For any bounded set $S\subset\mathbb{R}^{n}$ and for any $\epsilon>0$, let
$N(\epsilon)$ be the number of boxes (squares) required to cover $S$. We
define the Minkowski dimension of $S$

$\mathcal{H}^{d}(S):=\lim\limits_{r\to
0+}\inf\big{\\{}\sum\limits_{i}r_{i}^{d}:\text{there is a countable cover of
$S$ by balls with radius $r_{i}\in(0,r)$}\big{\\}},$

and we define the upper Minkowski dimension of $S$ as

$\overline{\dim_{\sf box}}(S):=\limsup\limits_{\epsilon\to
0+}\frac{\ln\big{(}N(\epsilon)\big{)}}{\ln(1/\epsilon)}$

and the lower Minkowski dimension of $S$ as

$\underline{\dim_{\sf box}}(S):=\liminf\limits_{\epsilon\to
0+}\frac{\ln\big{(}N(\epsilon)\big{)}}{\ln(1/\epsilon)}.$

When the two numbers agree, we define the Minkowski dimension of $S$ as

$\dim_{\sf box}(S):=\lim\limits_{\epsilon\to
0+}\frac{\ln\big{(}N(\epsilon)\big{)}}{\ln(1/\epsilon)}.$

######  Definition 1.2 (Hausdorff dimension).

For any bounded set $S\subset\mathbb{R}^{n}$ and for any $d>0$ we define the
$d$-dimensional Hausdorff outer measure of $S$

$\mathcal{H}^{d}(S):=\lim\limits_{r\to
0+}\inf\big{\\{}\sum\limits_{i}r_{i}^{d}:\text{there is a countable cover of
$S$ by balls with radius $r_{i}\in(0,r)$}\big{\\}},$

and we define the Hausdorff dimension of $S$

$\dim_{H}(S):=\inf\\{d\geqslant 0~{}|~{}\mathcal{H}^{d}(S)=0\\}.$

The Minkowski dimension is relative easy to calculate, while the Hausdorff
dimension deals with countable unions better. There are several basic
properties of these dimensions. Assuming $S,S^{\prime}$ have Minkowski
dimensions.

  1. 1.

If $S$ is a dimension $d\in\mathbb{Z}$ manifold, then $\dim_{H}(S)=\dim_{\sf
box}(S)=d$;

  2. 2.

If $S^{\prime}\subset S$ then $\dim_{H}(S^{\prime})\leqslant\dim_{H}(S)$ and
$\dim_{\sf box}(S^{\prime})\leqslant\dim_{\sf box}(S)$;

  3. 3.

If $S=\bigcup\limits_{i=1}^{n}S_{i}$ then $\dim_{\sf box}(S)=\max\dim_{\sf
box}(S_{i})$;

  4. 4.

If $S=\bigcup\limits_{i=1}^{\infty}S_{i}$ then
$\dim_{H}(S)=\sup\dim_{H}(S_{i})$;

  5. 5.

In general $\dim_{H}(S)\leqslant\dim_{\sf box}(S)$ [2]. The equality holds
when $S$ is strictly self-similar.

From the properties we have $1\leqslant\dim_{H}(W_{\sf
cantor})\leqslant\dim_{\sf box}(W_{\sf cantor})\leqslant 2$. In Section 2 we
prove

######  Theorem 1.3.

The boundary of the lakes of Wada construced in the standard Cantor way
$W_{\sf cantor}$ has

$\dim_{\sf box}(W_{\sf cantor})=\frac{\ln(6)}{\ln(3)}\approx 1.6309.$

By changing parameters we can construct the lakes of Wada in the modified
Cantor way. Take a series of real numbers
$\mathcal{A}:=\\{a_{i}>2\\}_{i=1}^{\infty}$.

Day 1: we dig a simply connected red lake $B_{1}$ by digging a continuous
canal of width $w_{1}:=1-\frac{2}{a_{1}}$ from the left on the island so that
every point on the island is at most $\tfrac{\sqrt{2}}{a_{1}}$ far away from
the blue lake.

Day 2: we dig a simply connected red lake $R_{2}$ on the island with width
$w_{2}:=\frac{1}{a_{1}}(1-\frac{2}{a_{2}})$ so that every point on the island
is at most $\tfrac{\sqrt{2}}{a_{1}a_{2}}$ far away from the red lake.

Day 3: we dig a simply connected green lake $G_{3}$ on the island with width
$w_{3}:=\tfrac{1}{a_{1}a_{2}}(1-\frac{2}{a_{3}})$ so that every point on the
island is at most $\tfrac{\sqrt{2}}{a_{1}a_{2}a_{3}}$ far away from the green
lake.

In the future days, write

$w_{n}:=\prod\limits_{i=1}^{n-1}a_{i}^{-1}(1-\tfrac{2}{a_{n}}),\ \ \
t_{n}:=\prod\limits_{i=1}^{n}a_{i}^{-1},\ \ \ (n\geqslant 2)$

for the width of canals and $\frac{\sqrt{2}}{2}\times$(the maximal distances
between points on the islands and the extended canals on Day $n$).

Then for $k=1,2,\dots$, on day $3k+1$, we extend our blue lake as $B_{3k+1}$
by digging a continuous canal of width $w_{3k+1}$ on the island so that every
point on the island is at most $\sqrt{2}\,f_{3k+1}$ far away from the blue
lake. On day $3k+2$ we extend the red while keeping it simply connected, while
on day $3k+3$ we work on the green. Finally we get the boundary of the lakes
of Wada $W_{\mathcal{A}}$ depends on the series $\mathcal{A}$.

We write $W_{c}$ for $W_{\mathcal{A}}$ when $\mathcal{A}$ is a constant
sequence $c,c,c,\dots$ with $c>2$. In particular $W_{3}=W_{\sf cantor}$. In
Section 3 we prove

######  Theorem 1.4.

Using the notations above

$\dim_{\sf box}(W_{c})=\frac{\ln(2c)}{\ln(c)}\rightarrow 1,\ \ \ \
(c\rightarrow+\infty)$

In general

$\dim_{\sf
box}(W_{\mathcal{A}})=1+\lim\limits_{n\rightarrow+\infty}\frac{n\ln(2)}{\sum\limits_{i=1}^{n}\ln(a_{i})}$

######  Corollary 1.5.

For any real number $d\in[1,2]$ there exists a series
$\mathcal{A}:=\\{a_{i}>2\\}_{i=1}^{\infty}$ such that

$\dim_{\sf box}(W_{\mathcal{A}})=d.$

Higher dimensional lakes of Wada can be obtained by taking products of
2-dimensional lakes of Wada with $\mathbb{R}^{n-2}$.

######  Corollary 1.6.

In $\mathbb{R}^{n}$ with $n\geqslant 2$, for any real number $d\in[n-1,n]$
there exists three disjoint domains $U_{1},U_{2},U_{3}$ sharing the same
boundary

$W=\partial U_{1}=\partial U_{2}=\partial U_{3}$

and

$\dim_{\sf box}(W)=d.$

The Hausdorff dimension will be calculated in the next version of the paper.

Acknowledgement: The author thanks the organisers of meeting GDR (Groupe de
Recherche) GDM (Géométrie Différentielle et Mécanique) at La Rochelle 07-09
July 2021, especially Boris Kolev for introducing the lakes of Wada during a
chat in a bar after the meeting.

##  2 Minkowski dimension of the boundary in the standard Cantor case

From the construction

$W_{\sf cantor}=\bigcup\limits_{k=0}^{\infty}\partial
B_{3k+1}=\bigcup\limits_{k=0}^{\infty}\partial
R_{3k+2}=\bigcup\limits_{k=0}^{\infty}\partial G_{3k+3}.$

To simplify notations, we write $W_{3k+1},$ $W_{3k+2}$ and $W_{3k+3}$ for
$\partial B_{3k+1}$, $\partial R_{3k+2}$ and $\partial G_{3k+3}$ respectively.
Thus

$W_{\sf cantor}=\bigcup\limits_{n=0}^{\infty}W_{n}.$

First we calculate its dimension by box-counting.

Day 1: we cover the blue boundary $\partial B_{1}$ by exactly 1 square of
length 1. So $N(1)=1$. This square also covers $W_{\sf cantor}$.

Day 2: we cover the red boundary $\partial R_{2}$ by 7 squares of length
$3^{-1}$. So $N(3^{-1})=7$. These squares also covers $W_{\sf cantor}$.

Day 3: we cover the green boundary $\partial G_{3}$ by 43 squares of length
$3^{-2}$. So $N(3^{-2})=43$. These squares also covers $W_{\sf cantor}$.

Figure 7: Day 2: 7 squares of length $3^{-1}$

On day $n$, all the $N(3^{1-n})$ squares covering $W_{n}$ hence covering
$W_{\sf cantor}$, up to a rotation, are classified into 4 cases:

  * •

Case 1 (Terminal): The new canal ends in this square, like the square 1 on Day
2.

  * •

Case 2 (Straight): The new canal passes this square straightly, like squares
2, 4, 6, 7 on Day 2.

  * •

Case 3 (Turning): The new canal passes this square with a left or right turn,
like squares 3, 5 on Day 2.

  * •

Case 4 (Separation): The new canal enters this square from the left and
separates into two branches up and down. It appears exactly once everyday from
Day 4.

Figure 8: Terminal, Straight and Turning Figure 9: Separation comes from
extending a Terminal, and will split into two Terminals

In the first three days, there is exactly one Terminal and no Separation. From
Day 4, there will be exactly two Terminals and one Separation.

On day $n+1$, all the $N(3^{-n})$ squares required to cover $W_{n+1}$ are sub-
squares of the Terminal, the Straights and the Turnings used one day before.
In fact

  * •

each Terminal is covered by and requires 7 sub-squares;

  * •

each Straight and each Turning is covered by and requires 6 sub-squares.

  * •

each Separation is covered by and requires 5 sub-squares.

thus

$\left\\{\begin{aligned} N(1)&=1,\\\ N(3^{-n})&=6N(3^{1-n})+1,\ \ \
n=1,2,3,\dots\end{aligned}\right.$

We concludes that $N(3^{-n})=O(6^{n})$ so

$\lim\limits_{n\rightarrow+\infty}\frac{\ln\big{(}N(3^{-n})\big{)}}{\ln(3^{n})}=\frac{\ln(6)}{\ln(3)}.$

For any $\epsilon\in[3^{-n-1},3^{-n}]$ clearly we have

$N(3^{-n})\leqslant N(\epsilon)\leqslant N(3^{-n-1}).$

Thus

$\displaystyle\frac{\ln\big{(}N(3^{-n})\big{)}}{\ln(3^{n+1})}$
$\displaystyle\leqslant\frac{\ln\big{(}N(\epsilon)\big{)}}{\ln(1/\epsilon)}\leqslant\frac{\ln\big{(}N(3^{-n-1})\big{)}}{\ln(3^{n})}$
$\displaystyle\frac{\ln\big{(}N(3^{-n})\big{)}/\ln(3^{n})}{1+\ln(3)/\ln(3^{n})}$
$\displaystyle\leqslant\frac{\ln\big{(}N(\epsilon)\big{)}}{\ln(1/\epsilon)}\leqslant\frac{\ln\big{(}N(3^{-n-1})\big{)}/\ln(3^{n+1})}{1-\ln(3)/\ln(3^{n+1})}$

We concludes that the Minkowski dimension exists and

$\dim_{\sf box}(W_{\sf cantor})=\frac{\ln(6)}{\ln(3)}\approx 1.6309.$

##  3 Minkowski dimension of the boundary in the modified Cantor case

First we treat the case $\mathcal{A}:=\\{a_{i}\in\mathbb{Z}_{\geqslant
3}\\}_{i=1}^{\infty}$ of integer series.

Figure 10: $W_{4}$, the lakes of Wada constructed by a constante sequence
$4,4,4,\dots$

Recall the notations

$w_{n}=\prod\limits_{i=1}^{n-1}a_{i}^{-1}(1-\tfrac{2}{a_{n}}),\ \ \
t_{n}=\prod\limits_{i=1}^{n}a_{i}^{-1},\ \ \ (n\geqslant 2)$

for the width of canals and $\frac{\sqrt{2}}{2}\times$(the maximal distances
between points on the islands and the extended canals on Day $n$).

On Day $n$ we use $N(t_{n})$ squares of length
$t_{n}=\prod\limits_{i=1}^{n}a_{i}^{-1}$. They are also classified into one
Terminal (two Terminals and one Separation from Day 4) and finitely many
Straights and Turnings.

On Day $n+1$ those squares splits into sub-squares of length
$t_{n+1}=\prod\limits_{i=1}^{n+1}a_{i}^{-1}$, while

  * •

each Terminal is covered by and requires $3a_{n+1}-2$ sub-squares;

  * •

each Straight and each Turning is covered by and requires $2a_{n+1}$ sub-
squares;

  * •

each Separation is covered by and requires $a_{n+1}+2$ sub-squares.

Thus we get recurrence formulas

$\left\\{\begin{aligned} N(t_{0})&=N(1)=1,\\\
N(t_{n+1})&=2a_{n+1}N({t_{n}})+a_{n+1}-2,\ \ \
n=1,2,3,\dots\end{aligned}\right.$

We conclude that $N(t_{n+1})=O\big{(}\prod\limits_{i=1}^{n}2a_{i}\big{)}$.

If the sequence $\mathcal{A}$ satisfies

$\displaystyle\lim\limits_{n\rightarrow+\infty}\frac{\ln(a_{n+1})}{\sum\limits_{i=1}^{n}\ln(a_{i})}=0$
(1)

for example, if $\mathcal{A}$ is bounded or $a_{n}=O(n^{k})$ for some
$k\in\mathbb{Z}$, then by the same argument as before, we conclude that

$\dim_{\sf
box}(W_{\mathcal{A}})=\lim\limits_{n\rightarrow+\infty}\frac{\ln\big{(}N(t_{n})\big{)}}{\ln(t_{n})}=\lim\limits_{n\rightarrow+\infty}\frac{\ln\big{(}\prod\limits_{i=1}^{n}2a_{i}\big{)}}{\ln\big{(}\prod\limits_{i=1}^{n}a_{i}\big{)}}=1+\lim\limits_{n\rightarrow+\infty}\frac{n\ln(2)}{\sum\limits_{i=1}^{n}a_{i}}.$

But if $\mathcal{A}$ does not satisfy the condition (1), for example if
$a_{n}=2^{2^{n}}$, then we need to analyse more delicately.

For any $\epsilon\in(0,1)$ there is a unique $n$ such that
$t_{n+1}\leqslant\epsilon<t_{n}$, i.e.

$\prod\limits_{i=1}^{n}a_{i}<1/\epsilon\leqslant
a_{n+1}\prod\limits_{i=1}^{n}a_{i}.$

Recall that $a_{n+1}\in\mathbb{Z}_{\geqslant 3}$. Thus there is a unique
integer $p\in\\{1,2,\dots,a_{n+1}-1\\}$ such that

$p\prod\limits_{i=1}^{n}a_{i}<1/\epsilon\leqslant(p+1)\prod\limits_{i=1}^{n}a_{i}.$

On Day $n+1$, for any $q\in\\{1,2,\dots,a_{n+1}\\}$, if we use sub-squares of
length $q^{-1}t_{n}$ instead of $a_{n+1}^{-1}t_{n}$, then

  * •

each Terminal is covered by and requires $3q-2$ sub-squares;

  * •

each Straight and each Turning is covered by and requires $2q$ sub-squares;

  * •

each Separation is covered by and requires $q+2$ sub-squares.

Thus

$N(q^{-1}\,t_{n+1})=2qN({t_{n}})+q-2,\ \ \ q=1,2,\dots,a_{n+1}.$

In particular

$\displaystyle N(p^{-1}t_{n})$ $\displaystyle\leqslant N(\epsilon)\leqslant
N\big{(}(p+1)^{-1}t_{n}\big{)}$ $\displaystyle 2pN(t_{n})$
$\displaystyle\leqslant N(\epsilon)\leqslant
2(p+1)N\big{(}t_{n}\big{)}+p-2<(2p+3)N\big{(}t_{n}\big{)}$
$\displaystyle\frac{\ln(2p)+\ln\big{(}N(t_{n})\big{)}}{\ln(p+1)+\ln\big{(}\prod\limits_{i=1}^{n}a_{i}\big{)}}$
$\displaystyle\leqslant\frac{\ln\big{(}N(\epsilon)\big{)}}{\ln(\epsilon)}\leqslant\frac{\ln(2p+3)+\ln\big{(}N(t_{n})\big{)}}{\ln(p)+\ln\big{(}\prod\limits_{i=1}^{n}a_{i}\big{)}}$

Both sides converges to
$1+\lim\limits_{n\rightarrow+\infty}\frac{n\ln(2)}{\ln(\prod\limits_{i=1}^{n}a_{i})}$
The Minkowski dimension is then

$\dim_{\sf
box}(W_{\mathcal{A}})=1+\lim\limits_{n\rightarrow+\infty}\frac{n\ln(2)}{\ln(\prod\limits_{i=1}^{n}a_{i})}$

In particular, when $\mathcal{A}$ is a constant sequence $c,c,c,\dots$ we get

$\dim_{\sf box}(W_{c})=1+\frac{\ln(2)}{\ln(c)}$

which tends to 1 when $c\rightarrow+\infty$.

In the general case when $a_{n}>2$ may not be integers, we may not be able to
decompose squares of length $t_{n}=\prod\limits_{i=1}^{n}a_{i}^{-1}$ into
squares of length $t_{n+1}=\prod\limits_{i=1}^{n+1}a_{i}^{-1}$. Instead, we
can keep square Terminal and Turnings, and rectangular Straights, like the
part 1, 2, 3 in the following figures.

Figure 11: $W_{\mathcal{A}}$ with $\mathcal{A}=3.8,4,4\dots$ Figure 12:
$W_{\mathcal{A}}$ with $\mathcal{A}=16/7,7/3,12/5\dots$

The number of rectangles are bounded by the sum of the number of Turnings, of
Terminals and of Separations, since there are at most one rectangle between
two Turnings, between one Turning and one Terminal, etc.

######  Lemma 3.1.

Let $T(n)$ be the number of Turnings after the $n$-th digging. Then

$T(n)=\frac{9}{8}\cdot 2^{n}-4,\ \ \ n\geqslant 5$

###### Proof.

We calculate $T(n)$ by recurrence. Start with $T(1)=0,T(2)=2,T(3)=6,T(4)=14$.
From Day $4$,

  * •

each Straight produces no Turnings;

  * •

each Turning produces two Turnings;

  * •

one Terminal where we start our canal, produces one Separation and two
Turnings;

  * •

the other Terminal produces two Turnings;

  * •

the Separation produces two Terminals and no Turnings.

Thus we get the recurrence formula

$T(n+1)=2T(n)+4,\ \ \ n\geqslant 4$

which implies that

$\displaystyle\frac{T(n+1)}{2^{n+1}}$
$\displaystyle=\frac{T(n)}{2^{n}}+2^{1-n}$
$\displaystyle=\frac{T(4)}{2^{4}}+(2^{1-n}+2^{2-n}+\dots+2^{1-4})$
$\displaystyle=\frac{14}{2^{4}}+2^{-2}-2^{1-n},\ \ \ n\geqslant 4$
$\displaystyle T(n+1)$ $\displaystyle=\frac{9}{8}\cdot 2^{n+1}-4,\ \ \
n\geqslant 4$ $\displaystyle T(n)$ $\displaystyle=\frac{9}{8}\cdot 2^{n}-4,\ \
\ n\geqslant 5$

∎

Thus there are at most $T(n)+3=\frac{9}{8}\cdot 2^{n}-1$ rectangles on Day $n$
with $n\geqslant 5$. We can cover each rectangle by sub-squares with at most
two overlaps.

So at each rectangle, the area that we covered twice are bounded by two sub-
squares of length $t_{n}$. In total, we over-covered at most
$2\big{(}T(n)+3\big{)}\,t_{n}^{2}$ area. Write $A(n)$ as the area of the
island after $n$-th digging. Then

$\displaystyle A(n)\leqslant\underbrace{N(t_{n})\,t_{n}^{2}}_{\text{total area
covered by squares}}\leqslant A(n)+\underbrace{2(T(n)+3)t_{n}^{2}}_{\text{area
of overlaps}}=A(n)+\big{(}\tfrac{9}{4}\cdot
2^{n}-\tfrac{9}{4}\big{)}\,t_{n}^{2}$ (2)

######  Lemma 3.2.

The are of the island after $n$-th digging satisfies

$2^{n}t_{n}<A(n)<2^{n+1}t_{n},\ \ \ n\geqslant 1$

###### Proof.

We calculate $A(n)$ by recurrence. Start with $A(0)=1$. After $n$-th digging,

Figure 13: The change of the area from $A(n-1)$ to $A(n)$

  * •

the area of each Straight, no matter a square or a rectangle, is shrunk by
$2/a_{n}$;

  * •

the area of each of the $T(n)$ Turnings is changed from $t_{n-1}^{2}$ to
$t_{n-1}^{2}\cdot\frac{2}{a_{n}}$, i.e. shrunk by $2/a_{n}$;

  * •

the area of each of the two Terminals is changed from $t_{n-1}^{2}$ to
$t_{n-1}^{2}\cdot\frac{1}{a_{n}^{2}}\cdot(3a_{n}-2)$;

  * •

the area of the Separation is changed from $t_{n-1}^{2}$ to
$t_{n-1}^{2}\cdot\frac{1}{a_{n}^{2}}\cdot(a_{n}+2)$;

Thus

$\displaystyle A(n)$ $\displaystyle=\left\\{\begin{aligned}
\Big{(}&A(n-1)-t_{n-1}^{2}\Big{)}\frac{2}{a_{n}}+t_{n-1}^{2}\cdot\frac{1}{a_{n}^{2}}\cdot(3a_{n}-2),\
\ \ n=1,2,3,\\\
\Big{(}&A(n-1)-3t_{n-1}^{2}\Big{)}\frac{2}{a_{n}}+2t_{n-1}^{2}\cdot\frac{1}{a_{n}^{2}}\cdot(3a_{n}-2)+t_{n-1}^{2}\cdot\frac{1}{a_{n}^{2}}\cdot(a_{n}+2),\
\ \ n\geqslant 4\end{aligned}\right.$
$\displaystyle=\frac{2}{a_{n}}A(n-1)+\frac{t_{n-1}^{2}}{a_{n}^{2}}(a_{n}-2),\
\ \ n\geqslant 1$ $\displaystyle=\frac{2}{a_{n}}A(n-1)+t_{n}^{2}\,(a_{n}-2),\
\ \ n\geqslant 1$ $\displaystyle\frac{A(n)}{2^{n}\,t_{n}}$
$\displaystyle=\frac{A(n-1)}{2^{n-1}\,t_{n-1}}+\frac{t_{n}\,(a_{n}-2)}{2^{n}},\
\ \ n\geqslant 1$

The last term is bounded by

$0<\frac{t_{n}\,(a_{n}-2)}{2^{n}}<\frac{t_{n}\,a_{n}}{2^{n}}=\frac{t_{n-1}}{2^{n}}$

Recall that
$t_{n}=\prod\limits_{i=1}^{n}a_{i}^{-1}<\prod\limits_{i=1}^{n}2^{-1}=2^{-n}$,
so

$0<\frac{t_{n}\,(a_{n}-2)}{2^{n}}<2^{1-2n}.$

Thus for $n\geqslant 1$

$\displaystyle\frac{A(n-1)}{2^{n-1}\,t_{n-1}}<\frac{A(n)}{2^{n}\,t_{n}}$
$\displaystyle<\frac{A(n-1)}{2^{n-1}\,t_{n-1}}+2^{1-2n},$ $\displaystyle
A(0)<\frac{A(n)}{2^{n}\,t_{n}}$ $\displaystyle<A(0)+2^{-1}+2^{-3}+\dots$
$\displaystyle 1<\frac{A(n)}{2^{n}\,t_{n}}$ $\displaystyle<1+1=2,$
$\displaystyle 2^{n}t_{n}<A(n)$ $\displaystyle<2^{n+1}t_{n}.$

∎

######  Definition 3.3.

Let $F(n),G(n)$ be two positive functions of $n\in\mathbb{Z}_{\geqslant 1}$.
We write

$F(n)\simeq G(n),$

if there exist some constant $c>1$ such that

$c^{-1}\,G(n)<F(n)<c\,G(n).$

So we can write $A(n)\simeq 2^{n}\,t_{n}$. In the equation (2)

$\displaystyle A(n)\leqslant N(t_{n})\,t_{n}^{2}$ $\displaystyle\leqslant
A(n)+\big{(}\tfrac{9}{4}\cdot 2^{n}-\tfrac{9}{4}\big{)}\,t_{n}^{2}$
$\displaystyle<2^{n+1}t_{n}+\tfrac{9}{4}\,2^{n}\,t_{n}^{2}$
$\displaystyle<2^{n+1}t_{n}+\tfrac{9}{4}\,2^{n}\,t_{n},\ \ \ (t_{n+1}<1)$
$\displaystyle<(2+9/4)\,A(n).$

Thus $N(t_{n})t_{n}^{2}\simeq A(n)\simeq 2^{n}\,t_{n}$, implies
$N(t_{n})\simeq\frac{2^{n}}{t_{n}}$ and

$\displaystyle\dim_{\sf box}(W_{\mathcal{A}})$
$\displaystyle=\lim\limits_{n\rightarrow+\infty}\frac{\ln\big{(}N(t_{n})\big{)}}{\ln(1/t_{n})}=\lim\limits_{n\rightarrow+\infty}\frac{\ln(2^{n}/t_{n})}{\ln(1/t_{n})}=\lim\limits_{n\rightarrow+\infty}\frac{n\ln(2)-\ln(t_{n})}{-\ln(t_{n})}$
$\displaystyle=1+\lim\limits_{n\rightarrow+\infty}\frac{n\ln(2)}{-\ln(t_{n})}=1+\lim\limits_{n\rightarrow+\infty}\frac{n\ln(2)}{\sum\limits_{i=1}^{n}\ln(a_{i})}.$

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  * [4] Hocking, J.G.; Young, G.S.: Topology, Second edition. Dover Publications, Inc., New York, (1988). x+374 pp. ISBN: 0-486-65676-4
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  * [6] Hubbard, J.H.; Oberste-Vorth, R.W.: Hénon mappings in the complex domain. II. Projective and inductive limits of polynomials, Real and complex dynamical systems (Hillerød, 1993), 89–132, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, (1995).
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Zhangchi Chen, Université Paris-Saclay, CNRS, Laboratoire de mathématiques
d’Orsay, 91405, Orsay, France
<EMAIL_ADDRESS>https://www.imo.universite-paris-
saclay.fr/$\sim$chen/