Calculus7

The Kolakoski-Cantor set

A 0-1 sequence can be interpreted as a point in the interval [0,1]. But this makes the long-term behavior of the sequence practically invisible due to limited resolution of our screens (and eyes). To make it visible, we can also plot the points obtained by shifting the binary sequence to the left (Bernoulli shift, which also goes by many other names). The resulting orbit is often dense in the interval, which doesn’t really help us visualize any patterns. But sometimes we get an interesting complex structure.

kol_set_small — The Kolakoski-Cantor set, KC

The vertical axis here is the time parameter, the number of dyadic shifts. The 0-1 sequence being visualized is the Kolakoski sequence in its binary form, with 0 and 1 instead of 1 and 2. By definition, the n-th run of equal digits in this sequence has length ${x_n+1}$ . In particular, 000 and 111 never occur, which contributes to the blank spots near 0 and 1.

Although the sequence is not periodic, the set is quite stable in time; it does not make a visible difference whether one plots the first 10,000 shifts, or 10,000,000. The apparent symmetry about 1/2 is related to the open problem of whether the Kolakoski sequence is mirror invariant, meaning that together with any finite word (such as 0010) it also contains its complement (that would be 1101).

There are infinitely many forbidden words apart from 000 and 111 (and the words containing those). For example, 01010 cannot occur because it has 3 consecutive runs of length 1, which implies having 000 elsewhere in the sequence. For the same reason, 001100 is forbidden. This goes on forever: 00100100 is forbidden because it implies having 10101, etc.

The number of distinct words of length n in the Kolakoski sequence is bounded by a power of n (see F. M. Dekking, What is the long range order in the Kolakoski sequence?). Hence, the set pictured above is covered by ${O(n^p)}$ intervals of length ${2^{-n}}$ , which implies it (and even its closure) is zero-dimensional in any fractal sense (has Minkowski dimension 0).

The set KC apparently does not have any isolated points; this is also an open problem, of recurrence (whether every word that appears in the sequence has to appear infinitely many times). Assuming this is so, the closure of the orbit is a totally disconnected compact set without isolated points, i.e., a Cantor set. It is not self-similar (not surprising, given it’s zero-dimensional), but its relation to the Bernoulli shift implies a structure resembling self-similarity:

kolhalf — KC is covered by two copies scaled by 1/2

Applying the transformations ${x\mapsto x/2}$ and ${x\mapsto (1+x)/2}$ yields two disjoint smaller copies that cover the original set, but with some spare parts left. The leftover bits exist because not every word in the sequence can be preceded by both 0 and 1.

kolx2 — KC covered by two copies scaled by 2

Applying the transformations ${x\mapsto 2x}$ and ${x\mapsto 2x-1}$ yields two larger copies that cover the original set. There are no extra parts within the interval [0,1] but there is an overlap between the two copies.

The number ${c = \inf KC\approx 0.146778684766479}$ appears several times in the structure of the set: for instance, the central gap is ${((1-c)/2, (1+c)/2)}$ , the second-largest gap on the left has the left endpoint ${(1-c)/4}$ , etc. The Inverse Symbolic Calculator has not found anything about this number. Its binary expansion begins with 0.001 001 011 001 001 101 001 001 101 100… which one can recognize as the smallest binary number that can be written without doing anything three times in a row. (Can’t have 000; also can’t have 001 three times in a row; and 001 010 is not allowed because it contains 01010, three runs of length 1. Hence, the number begins with 001 001 011.) This number is obviously irrational, but other than that…

In conclusion, the Python code used to plot KC.

import numpy as np
import matplotlib.pyplot as plt
n = 1000000
a = np.zeros(n, dtype=int)
j = 0                  
same = False  
for i in range(1, n):
    if same:
        a[i] = a[i-1]    
        same = False
    else:
        a[i] = 1 - a[i-1]
        j += 1            
        same = bool(a[j])
v = np.array([1/2**k for k in range(60, 0, -1)])
b = np.convolve(a, v, mode='valid')
plt.plot(b, np.arange(np.size(b)), '.', ms=2)
plt.show()

Pisot constant beyond 0.843

In a 1946 paper Charles Pisot proved a theorem involving a curious constant ${\gamma_0= 0.843\dots}$ . It can be defined as follows:

${\gamma_0= \sup\{r \colon \exists }$ monic polynomial ${p}$ such that ${|p(e^z)| \le 1}$ whenever ${|z|\le r \}}$

Equivalently, ${\gamma_0}$ is determined by the requirement that the set ${\{e^z\colon |z|\le \gamma_0\}}$ have logarithmic capacity 1; this won’t be used here. The theorem is stated below, although this post is really about the constant.

Theorem: If an entire function takes integer values at nonnegative integers and is ${O(e^{\gamma |z|})}$ for some ${\gamma < \gamma_0}$ , then it is a finite linear combination of terms of the form ${z^n \alpha^z}$ , where each ${\alpha }$ is an algebraic integer.

The value of ${\gamma_0}$ is best possible; thus, in some sense Pisot’s theorem completed a line of investigation that began with a 1915 theorem by Pólya which had ${\log 2}$ in place of ${\gamma_0}$ , and where the conclusion was that ${f}$ is a polynomial. (Informally speaking, Pólya proved that ${2^z}$ is the “smallest” entire-function that is integer-valued on nonnegative integers.)

Although the constant ${\gamma_0}$ was mentioned in later literature (here, here, and here), no further digits of it have been stated anywhere, as far as I know. So, let it be known that the decimal expansion of ${\gamma_0}$ begins with 0.84383.

A lower bound on ${\gamma_0}$ can be obtained by constructing a monic polynomial that is bounded by 1 on the set ${E(r) = \{e^z \colon |z|\le r \}}$ . Here is E(0.843):

It looks pretty round, except for that flat part on the left. In fact, E(0.82) is covered by a disk of unit radius centered at 1.3, which means that the choice ${p(z) = z-1.3}$ shows ${\gamma_0 > 0.82}$ .

082disk — p(z) = z-1.3 gives lower bound 0.82

How to get an upper bound on ${\gamma_0}$ ? Turns out, it suffices to exhibit a monic polynomial ${q}$ that has all zeros in ${E(r)}$ and satisfies ${|q|>1}$ on the boundary of ${E(r)}$ . The existence of such ${q}$ shows ${\gamma_0 < r}$ . Indeed, suppose that ${p}$ is monic and ${|p|\le 1}$ on ${E(r)}$ . Consider the function $u(z) = \frac{\log|p(z)|}{\deg p} - \frac{\log|q(z)|}{\deg q}$ . By construction ${u<0}$ on the boundary of ${E(r)}$ . Also, ${u}$ is subharmonic in its complement, including ${\infty}$ , where the singularities of both logarithms cancel out, leaving ${u(\infty)=0}$ . This contradicts the maximum principle for subharmonic functions, according to which ${u(\infty)}$ cannot exceed the maximum of ${u}$ on the boundary.

The choice of ${q(z) = z-1.42}$ works for ${r=0.89}$ .

So we have ${\gamma_0}$ boxed between 0.82 and 0.89; how to get more precise bounds? I don’t know how Pisot achieved the precision of 0.843… it’s possible that he strategically picked some linear and quadratic factors, raised them to variable integer powers and optimized the latter. Today it is too tempting to throw some optimization routine on the problem and let it run for a while.

But what to optimize? The straightforward approach is to minimize the maximum of ${|p(e^z)|}$ on the circle ${|z|=r}$ , approximated by sampling the function at a sufficiently fine uniform grid ${\{z_k\}}$ and picking the maximal value. This works… unspectacularly. One problem is that the objective function is non-differentiable. Another is that taking maximum throws out a lot of information: we are not using the values at other sample points to better direct the search. After running optimization for days, trying different optimization methods, tolerance options, degrees of the polynomial, and starting values, I was not happy with the results…

Turns out, the optimization is much more effective if one minimizes the variance of the set ${\{|p(\exp(z_k))|^2\}}$ . Now we are minimizing a polynomial function of ${p(\exp(z_k)}$ , which pushes them toward having the same absolute value — the behavior that we want the polynomial to have. It took from seconds to minutes to produce the polynomials shown below, using BFGS method as implemented in SciPy.

As the arguments for optimization function I took the real and imaginary parts of the zeros of the polynomial. The symmetry about the real axis was enforced automatically: the polynomial was the product of quadratic terms ${(z-x_k-iy_k) (z-x_k+iy_k)}$ . This eliminated the potentially useful option of having real zeros of odd order, but I did not feel like special-casing those.

Three digits

Real part: 0.916, 1.186, 1.54, 1.783
Imaginary part: 0.399, 0.572, 0.502, 0.199

Here and below, only the zeros with positive imaginary part are listed (in the left-to-right order), the others being their conjugates.

Real part: 0.878, 1.0673, 1.3626, 1.6514, 1.8277
Imaginary part: 0.3661, 0.5602, 0.6005, 0.4584, 0.171

Four digits

Real part: 0.8398, 0.9358, 1.1231, 1.357, 1.5899, 1.776, 1.8788
Imaginary part: 0.3135, 0.4999 ,0.6163, 0.637, 0.553, 0.3751, 0.1326

Real part: 0.8397, 0.9358, 1.1231, 1.3571, 1.5901, 1.7762, 1.879
Imaginary part: 0.3136, 0.5, 0.6164, 0.6372, 0.5531, 0.3751, 0.1326

No, I didn’t post the same picture twice. The polynomials are just that similar. But as the list of zeros shows, there are tiny differences…

Five digits

Real part: 0.81527, 0.8553, 0.96028, 1.1082, 1.28274, 1.46689, 1.63723, 1.76302, 1.82066, 1.86273
Imaginary part: 0.2686, 0.42952, 0.556, 0.63835, 0.66857, 0.63906, 0.54572, 0.39701, 0.23637, 0.08842

Real part: 0.81798, 0.85803, 0.95788, 1.09239, 1.25897, 1.44255, 1.61962, 1.76883, 1.86547, 1.89069
Imaginary part: 0.26631, 0.4234, 0.54324, 0.62676, 0.66903, 0.65366, 0.57719, 0.44358, 0.26486, 0.07896

Again, nearly the same polynomial works for upper and lower bounds. The fact that the absolute value of each of these polynomials is below 1 (for lower bounds) or greater than 1 (for upper bounds) can be ascertained by sampling them and using an upper estimate on the derivative; there is enough margin to trust computations with double precision.

Finally, the Python script I used. The function “obj” is getting minimized while function “values” returns the actual values of interest: the minimum and maximum of polynomial. The degree of polynomial is 2n, and the radius under consideration is r. The sample points are collected in array s. To begin with, the roots are chosen randomly. After minimization runs (inevitably, ending in a local minimum of which there are myriads), the new starting point is obtained by randomly perturbing the local minimum found. (The perturbation is smaller if minimization was particularly successful.)

import numpy as np
from scipy.optimize import minimize

def obj(r):
    rc = np.concatenate((r[:n]+1j*r[n:], r[:n]-1j*r[n:])).reshape(-1,1)
    p = np.prod(np.abs(s-rc)**2, axis=0)
    return np.var(p)

def values(r):
    rc = np.concatenate((r[:n]+1j*r[n:], r[:n]-1j*r[n:])).reshape(-1,1)
    p = np.prod(np.abs(s-rc), axis=0)
    return [np.min(p), np.max(p)]

r = 0.84384
n = 10
record = 2 
s = np.exp(r * np.exp(1j*np.arange(0, np.pi, 0.01)))
xr = np.random.uniform(0.8, 1.8, size=(n,))
xi = np.random.uniform(0, 0.7, size=(n,))
x0 = np.concatenate((xr, xi))

while True:
    res = minimize(obj, x0, method = 'BFGS')
    if res['fun'] < record:
        record = res['fun']
        print(repr(res['x']))
        print(values(res['x']))
        x0 = res['x'] + np.random.uniform(-0.001, 0.001, size=x0.shape)
    else:
        x0 = res['x'] + np.random.uniform(-0.05, 0.05, size=x0.shape)

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Top tag road-bike: Bikes designed for road use only. Could be any road-only bike, but typically means bikes optimized for speed / racing / club rides with drop handlebars, narrow tires and a crouched-forward rider position.

Puzzling

For those who create, solve, and study puzzles.

Top tag riddle: A riddle gives indirect clues about an unnamed object or concept to be identified. It is often presented in the form of a poem.

Network Engineering

For network engineers.

Top tag cisco: Cisco is a major provider of networking equipment. Cisco devices often run IOS or NX-OS. This is a generic tag to be used when no more specific tags are available. See the partial list of tags in the full Tag Wiki.

Christianity

For committed Christians, experts in Christianity and those interested in learning more.

Top tag catholicism: The Catholic Church and its teachings and views on specific subjects.

Aviation

For aircraft pilots, mechanics, and enthusiasts.

Top tag aircraft-design: Aircraft Design describes the different choices that aircraft designers (typically aerospace engineers) make to create an aircraft.

Quantitative Finance

For finance professionals and academics.

Top tag options: A contract that gives the owner the right, but not the obligation, to buy or sell a security at a fixed price in the future.

Theoretical Computer Science

For theoretical computer scientists and researchers in related fields.

Top tag cc.complexity-theory: P versus NP and other resource-bounded computation.

German Language

For speakers of German wanting to discuss the finer points of the language and translation.

Top tag meaning: Bedeutung – Questions on definitions and nuances of words, phrases or sentences.

Philosophy

For those interested in the study of the fundamental nature of knowledge, reality, and existence.

Top tag logic: Logic is the study of formal systems of reasoning, especially of the deductive variety. It is one of the few fundamental philosophical subdisciplines, along with metaphysics, ontology and aesthetics. Logic has taken on considerable importance in recent mathematical developments, and one of the concerns of [tag:philosophy-of-mathematics] involves the role, extent and conceptual architecture of various kinds of logical or formal axiomatic systems.

Board & Card Games

For people who like playing board games, designing board games or modifying the rules of existing board games.

Top tag magic-the-gathering: Magic: The Gathering is a collectible card game for 2 or more players, set in a variety of fantastical realms. The cards represent the player’s resources and available spells and their interactions lead to an often high paced battle of wits.

Sound Design

For sound engineers, producers, editors, and enthusiasts.

Top tag pro-tools: AVID produced DAW. Use this tag for general questions regarding Pro Tools.

Anime & Manga

For anime and manga fans.

Top tag naruto: Naruto is a shounen manga/anime by Masashi Kishimoto about the young ninja Uzumaki Naruto who strives to become the “Hokage”, the leader of his village.

Programming Puzzles & Code Golf

For programming puzzle enthusiasts and code golfers.

Top tag code-golf: Code-golf is a competition to solve a particular problem in the fewest bytes of source code. If you want to score by characters instead of bytes, please state this explicitly in the challenge. If source code length is not the primary scoring criterion, consider using another tag instead.

Skeptics

For scientific skepticism.

Top tag medical-science: Use this tag for questions about the science of medicine and its practices. Use [medications] for questions about the actual cures that people take, and use [alternative-medicine] for claims about cures and practices which are claimed to be alternative to official medical science

Gardening & Landscaping

For gardeners and landscapers.

Top tag identification: Use this tag for questions that ask “what is this thing?” Pictures are helpful, as are descriptive titles. When you get an answer, consider adding a further tag (for the flower, tree, beetle etc that has been identified) so that your question is cataloged correctly. If the question is about determining the cause of a plant problem (e.g. disease), the diagnosis tag is more appropriate.

Craft CMS

For administrators, end users, developers and designers for Craft CMS.

Top tag plugin-development: Questions having to do with constructing plugins.

Islam

For Muslims, experts in Islam, and those interested in learning more about Islam.

Top tag sharia: Sharia (Islamic Law) based on the teachings of the Qur’an and Sunnah.

History

For historians and history buffs.

Top tag united-states: The United States of America is a nation-state stretching across North America between Canada and Mexico, plus Alaska in the continent’s northwest, the mid-Pacific archipelago of Hawaii and several territories in the Pacific & Caribbean. Founded by European émigrés rebelling from British control in the 1700s, the US spread across the continent by conquest and land-purchase to become the most powerful country on the planet after the Cold War in the 1900s.

Physical Fitness

For physical fitness professionals, athletes, trainers, and those providing health-related needs.

Top tag running: Moving rapidly on foot. Questions are about proper running form, race preparation, measuring the benefits and avoiding the pitfalls of running.

Computational Science

For scientists using computers to solve scientific problems.

Top tag linear-algebra: Questions on the algorithmic/computational aspects of linear algebra, including the solution of linear systems, least squares problems, eigenproblems, and other such matters.

CiviCRM

For administrators and users of the CiviCRM Constituent Relationship Management software.

Top tag wordpress: Questions specific to CiviCRM installed on WordPress.

Ethereum

For users of Ethereum, the decentralized application platform and smart contract enabled blockchain.

Top tag go-ethereum: Go Ethereum (short: Geth) is a Golang implementation of the Ethereum protocol.

Law

For legal professionals, students, and others with experience or interest in law.

Top tag united-states: For questions specific to the United States as a whole, or that span multiple state jurisdictions.

Software Quality Assurance & Testing

For software quality control experts, automation engineers, and software testers.

Top tag automated-testing: Use for questions involving problems with automated tests. Relevant for designing test automation, debugging test automation, automation tooling questions, and questions about when it is appropriate to automate. Questions regarding specific tools should tag those tools as well.

French Language

For students, teachers, and linguists wanting to discuss the finer points of the French language.

Top tag grammaire: La grammaire est l’étude systématique des éléments constitutifs d’une langue.

Space Exploration

For spacecraft operators, scientists, engineers, and enthusiasts.

Top tag launch: Questions regarding the takeoff or the liftoff phase of the flight of a rocket and the set of activities required for preparation of the launch vehicle leading to it.

Data Science

For Data science professionals, Machine Learning specialists, and those interested in learning more about the field.

Top tag machine-learning: Methods and principles of building “computer systems that automatically improve with experience.”

Tridion

For Tridion developers and administrators.

Top tag 2011: This tag is for questions specific to SDL Tridion 2011. Version tags should be used **only** when necessary to provide context to the question, and not to simply state that “you are using SDL Tridion 2011” unless that information is needed. A Service Pack and Hotfix Rollup was issued for this version in mid 2012.

Writers

For authors, editors, reviewers, professional writers, and aspiring writers.

Top tag fiction: Fiction is a form of prose writing that deals with at least partly artificial or imagined events and characters. This tag should be used for any questions relating to fiction, including fiction formatting and technique, fiction critiques, and the publishing of fiction.

Linguistics

For professional linguists and others with an interest in linguistic research and theory.

Top tag syntax: The study of the internal structure of expressions, especially between words and phrases, and the principles and processes that determine it. This includes words order, but also the grammatical relations that hold between words, as well as structural ambiguity, binding, reference, and similar issues. Common approaches are numerous phrase structure grammars (GPSG, HPSG, LFG, G&B, X-bar, Minimalism, …) and, on the other hand, dependency grammars.

Hinduism

For followers of the Hindu religion and those interested in learning more about Hinduism.

Top tag mythology: For questions about stories that are part of Hindu religious beliefs. Hindu mythology can be found throughout Hindu scriptures like the Vedas, Puranas, Ramayana, and Mahabharata.

Parenting

For parents, grandparents, nannies and others with a parenting role.

Top tag toddler: Age specific questions from about 1 year to about 3 years. Younger: infant. Older: pre-schooler.

Video Production

For engineers, producers, editors, and enthusiasts spanning the fields of video, and media creation.

Top tag video: Video is the technology of electronically manipulating still images that represent motion.

Joomla

For Joomla! administrators, users, developers and designers.

Top tag joomla-3.x: For question regarding Joomla 3.x of the Content Management System. This includes versions 3.0 up to 3.6

Economics

For professional and academic economists and analysts.

Top tag macroeconomics: Macroeconomics is a branch of economics dealing with the aggregate economy as a whole, rather than individual markets.

Homebrewing

For dedicated home brewers and serious enthusiasts.

Top tag fermentation: The anaerobic process by which yeast convert sugars into alcohol and carbon dioxide.

Astronomy

For astronomers and astrophysicists.

Top tag star: Questions regarding large spheres of plasma undergoing fusion.

Cognitive Sciences

For practitioners, researchers, and students in cognitive science, psychology, neuroscience, and psychiatry.

Top tag cognitive-psychology: For questions focusing on the interaction of many internal mental processes. If your question involves only one of memory, attention, language, decision-making, or perception then use the associated specialized tag instead of cognitive-psychology.

elementary OS

For developers and users of elementary OS and applications.

Top tag release-loki: Questions specific to elementary OS 0.4, codenamed “Loki”. Loki is the current stable version of elementary OS.

Chinese Language

For students, teachers, and linguists wanting to discuss the finer points of the Chinese language.

Top tag translation: Questions on specific points regarding translating Chinese to/from another language. Do your own research first!

Politics

For people interested in governments, policies, and political processes.

Top tag united-states: The United States of America is a Constitutional Republic located primarily between the Pacific and Atlantic Oceans in the middle of North America. The United States is a federal representative democracy consisting of 50 individual states, each with their own semi-autonomous government. At the national level, the government consists of a bicameral legislature (the Senate and the House of Representatives), a President, and independent Supreme Court.

Biblical Hermeneutics

For professors, theologians, and those interested in exegetical analysis of biblical texts.

Top tag greek: Koiné (from κοινή, “common”) Greek was the form of post-classical Greek spoken and written in Hellenistic and Roman antiquity. It is the language of the Septuagint (LXX), Christian New Testament, and most early Christian theological writings.

Vi and Vim

For people using the vi and Vim families of text editors.

Top tag vimrc: Vim reads initialization commands from a file called vimrc on startup. This can be used to set settings, define functions, execute autocommands, and more.

Spanish Language

For students, teachers, and linguists wanting to discuss the finer points of the Spanish language.

Top tag traducción: Preguntas sobre traducciones y adaptaciones de frases, palabras, términos y conceptos de otras idiomas al idioma español. Questions about translations or adaptations of sentences, words, terms and concepts from other languages into Spanish.

Engineering

For professionals and students of engineering.

Top tag mechanical-engineering: Questions within the problem domain of mechanical engineering. Mechanical Engineering can be a broad field; consider choosing more specific tags if they apply.

Reverse Engineering

For researchers and developers who explore the principles of a system through analysis of its structure, function, and operation.

Top tag ida: Interactive Disassembler Professional (IDA Pro), a proprietary multi-platform disassembler by Hex-Rays.

Tor

For researchers, developers, and users of Tor.

Top tag tor-browser-bundle: The Tor Browser Bundle lets you use Tor without needing to install any software by pairing Firefox and Vidalia.

Health

For medical specialists, students, dietitians, and anyone with health-related questions.

Top tag nutrition: Questions relating to the intake, safety, effects of nutrients, and their biochemical interactions with our body.

Pets

For pet owners, caretakers, breeders, veterinarians, and trainers.

Top tag cats: Small, furry (usually), domesticated members of the feline family.

Project Management

For project managers.

Top tag scrum: For questions about using or implementing the Scrum framework.

Buddhism

For people practicing or interested in Buddhist philosophy, teaching, and practice.

Top tag personal-practice: Questions containing or motivated by some aspect of personal experience. The experience can be during formal Buddhist practice or during everyday activity however the experience will be related to Buddhist practice or doctrine. The questions will necessarily have a subjective quality however the question or answers will contain references to Buddhist scripture, doctrine or an established teacher.

The Great Outdoors

For people who love outdoor activities, excursions, and outdoorsmanship.

Top tag gear: Whenever applicable use a more specific tag like backpack, shoes, stoves, tents, … Only questions that do not fit into such a tag and are related to any physical gear for engaging in those activities discussed in The Great Outdoors should use the equipment tag.

Open Data

For developers and researchers interested in open data.

Top tag data-request: Indicates a request for relevant open data sources. Please read https://opendata.meta.stackexchange.com/questions/284/how-a-good-data-request-question-should-look-like for writing a good request.

Sports

For participants in team and individual sport activities.

Top tag rules: Questions about the rules of sports, including regulations for particular competitions or formats. Questions must also be tagged for the specific sport or competition.

Chess

For serious players and enthusiasts of chess.

Top tag opening: Questions relating to the first few moves in a game

Windows Phone

For enthusiasts and power users of Windows Phone OS.

Top tag 8.1: Questions relating to features introduced in Windows Phone 8.1

Robotics

For professional robotic engineers, hobbyists, researchers and students.

Top tag arduino: Arduino is an open-source electronics prototyping platform based on flexible, easy-to-use hardware and software. It’s intended for artists, designers, hobbyists, and anyone interested in creating interactive objects or environments.

Expatriates

For people living abroad on a long-term basis.

Top tag usa: Questions regarding emigrating from, immigrating to and living in the United States of America

Ask Patents

For people interested in improving and participating in the patent system.

Top tag patentability: For questions relating to the legal requirements necessary for applications to be granted a patent.
Please include a tag relating to the relevant type of intellectual property protection you are asking about. There are invention patents, design patents and utility models. For an explanation of the differences, read the full tag description.

Startups

For entrepreneurs faced with delivering a new product or service under conditions of significant uncertainty.

Top tag legal: Questions about legality related to the startup.

Earth Science

For those interested in the geology, meteorology, oceanography, and environmental sciences.

Top tag meteorology: Meteorology is the study of how the earth’s atmosphere works, including weather forecasting. Use this tag for questions about the earth’s weather. When asking questions specifically about the atmosphere, also include the [atmosphere] tag.

Russian Language

For students, teachers, and linguists wanting to discuss the finer points of the Russian language.

Top tag перевод: All questions about translation of words, phrases, idioms or collocations from English to Russian. Read the FAQ section about translations.

Personal Productivity

For people wanting to improve their personal productivity.

Top tag time-management: Questions regarding how to handle the use of time in one’s life.

Stack Apps

For apps, scripts, and development with the Stack Exchange API.

Top tag support: You need help with the use of the Stack Exchange API, apps built on the API, or assistance building scripts that work on Stack Exchange websites.

Sitecore

For developers and end users of the Sitecore CMS and multichannel marketing software.

Top tag xdb: For questions relating to the Sitecore Experience Database (xDB). If you’re using Sitecore 7.2 or earlier, use the `dms` tag instead.

Bricks

For LEGO® and building block enthusiasts.

Top tag ev3: EV3 is the current generation of LEGO MINDSTORMS, released in 2013.

Genealogy & Family History

For expert genealogists and people interested in genealogy or family history.

Top tag united-states: For questions about ancestors and records in the USA. This includes previous political entities within the geographical boundaries of the present-day USA, such as the relevant European colonies. Also tag the state if the question is specific to that state.

Lifehacks

For people looking to bypass life’s everyday problems with simple tricks.

Top tag cleaning: Tips and tricks relating to cleaning clothing, objects etc. quickly and easily. Please also use another applicable tag (e.g., [clothing], [kitchen], etc.), as THIS TAG IS AMBIGUOUS.

Mathematics Educators

For those involved in the field of teaching mathematics.

Top tag undergraduate-education: For questions about teaching students at the undergraduate (university) level.

Italian Language

For students, teachers, and linguists wanting to discuss the finer points of the Italian language.

Top tag word-usage: Questions about correctly using a word within a particular phrase or context

Woodworking

For professional and amateur woodworkers.

Top tag finishing: Finishing a woodworking project refers to the final steps to beautify and preserve it.

Amateur Radio

For amateur radio enthusiasts.

Top tag antenna: Questions about the design, selection, building, testing, mounting, or properties of antennas. See also specific tags: [antenna-theory], [antenna-construction], [dipole], etc.

Hardware Recommendations

For people seeking specific hardware recommendations.

Top tag laptop: Use this tag when requesting information about laptops or notebook computers. These devices have an integrated hardware keyboard and screen.

Monero

For developers and users of the secure, private and untraceable cryptocurrency Monero.

Top tag monero-wallet-cli: Name of official command line wallet available for Monero on Linux, Window and MacOS

History of Science and Mathematics

For people interested in the history and origins of science and mathematics.

Top tag mathematics: For questions about the quantitative study of topics such as numbers, structure, space, and change, carried out by investigating patterns.

Music Fans

For music historians, critics, and fans.

Top tag identify-this-song: For questions looking to identify a song. Be sure to include enough information (lyric snippets, a good quality sound clip, etc.) for someone to be able to identify it.

Open Source

For people organizing, marketing or licensing open source development projects.

Top tag licensing: Licensing refers to applying a license to an area of software. Only use this tag if your question concerns the application of a license to an area of interest. If your question concerns a specific license, use the tag that corresponds to your license. For more general questions, use this tag.

Freelancing

For self-employed and freelance workers.

Top tag contracts: Questions concerning contracts between a freelancer, agent and a client.

Latin Language

For linguists, teachers, and students wanting to discuss the finer points of the Latin language.

Top tag classical-latin: Questions concerning Latin of the classical era, approximately 75 BC to AD 300

Computer Graphics

For computer graphics researchers and programmers.

Top tag opengl: For questions involving use of the OpenGL graphics library.

Poker

For serious players and enthusiasts of poker.

Top tag texas-hold-em: Texas Hold’Em is a poker variant in which each player receives two cards (hole cards) that are dealt face down and then five community cards are dealt face-up by the dealer. All players may use their hole cards and the five community cards to make their best five-card poker high hand.

Martial Arts

For students and teachers of all martial arts.

Top tag training: The practice of martial arts. How, what, where, when, why.

Portuguese Language

For linguists, teachers and learners wanting to discuss the finer points of the Portuguese language.

Top tag significado: Para questões sobre o significado de uma palavra ou expressão, para saber o que quer dizer. For questions about the meaning of a word or expression, to know what things mean.

Sustainable Living

For folks dedicated to a lifestyle that can be maintained indefinitely without depleting available resources.

Top tag recycling: using materials that would otherwise be waste to manufacture new materials. For questions specifically on [upcycling] use that tag, for questions on [reuse] use that tag.

Ebooks

For ebook publishers and readers.

Top tag kindle: A brand of e-book readers and tablet devices produced by Amazon.

Esperanto Language

For teachers and students of the Esperanto language.

Top tag single-word-requests: Use this tag when you are searching for a word in order to express a specific meaning.

3D Printing

For 3D printing enthusiasts.

Top tag filament: For questions related to different filaments used as the print material.

Literature

For scholars and enthusiasts of literature.

Top tag symbolism: For questions concerning symbolic features in a work of literature.

Mythology

For enthusiasts and scholars of mythology.

Top tag greek: For questions about Greek mythology and legends, from ancient Greece to the present day.

Retrocomputing

For vintage-computer hobbyists interested in restoring, preserving, and using the classic computer and gaming systems of yesteryear.

Top tag hardware: For questions about the components of retro devices (computers, etc.)

Coffee

For people interested in all aspects of producing and consuming coffee.

Top tag brewing-process: Questions on the process of turning coffee beans into a beverage.

Artificial Intelligence

For people interested in conceptual questions about life and challenges in a world where “cognitive” functions can be mimicked in purely digital environment.

Top tag neural-networks: For questions about an artificial neural network (ANN), a network inspired by biological networks, which are used to estimate or approximate functions.

Beer, Wine & Spirits

For alcoholic beverage aficionados and those interested in beer, wine, or spirits.

Top tag taste: Questions about flavor and how it is influenced by various factors such as temperature and glassware.

Arts & Crafts

For artists and crafters.

Top tag drawing: For questions about the use of pens, pencils, chalk, etc. to create images or patterns on a 2-D surface.

Korean Language

For linguists, teachers and students of the Korean language.

Top tag grammar: Questions about the rules that govern and structure the language, and the composition of clauses, phrases and sentences. Also pertains to the syntax and morphology of the Korean language.

Ukrainian Language

For linguists, teachers and students of the Ukrainian language.

Top tag переклад: Для запитань про українські еквіваленти іншомовних слів // For questions about the Ukrainian equivalents of foreign-language words

Language Learning

For students, teachers, polyglots, and anyone interested in the techniques of second-language acquisition.

Top tag learning-methods: For questions about methods, techniques, and/or practices that assist in learning another language

Community Building

For community managers, administrators, and moderators.

Top tag user-behavior: Questions about user behavior, reactions to moderator decisions and other actions they take within the community.

Internet of Things

For users of everyday objects embedded with electronics to be sensed, monitored, and controlled remotely.

Top tag smart-home: For questions about any smart devices and services that help automate or remote control one’s home. Prominent examples are lights, home appliances, roller shutters and radiators.

DevOps

For software engineers working on automated testing, continuous delivery, service integration and monitoring, and building SDLC infrastructure.

Top tag jenkins: For questions about Jenkins, an open source automation server, and using Jenkins for topics such as building, testing, and deploying software, etc. For questions specifically about Jenkins Plugins use the jenkins-plugins tag.

Vegetarianism

For those committed to a vegan or vegetarian lifestyle and anyone interested in learning more.

Top tag veganism: In addition to being vegetarian (do not eat meat, poultry, or fish), vegans do not consume or use any animal products (such as eggs, milk, leather, honey, etc.)

Multipliers preserving series convergence

The Comparison Test shows that if ${\sum a_n}$ is an absolutely convergent series, and ${\{b_n\}}$ is a bounded sequence, then ${\sum a_nb_n}$ converges absolutely. Indeed, ${|a_nb_n|\le M|a_n|}$ where ${M}$ is such that ${|b_n|\le M}$ for all ${n}$ .

With a bit more effort one can prove that this property of preserving absolute convergence is equivalent to being a bounded sequence. Indeed, if ${\{b_n\}}$ is unbounded, then for every ${k}$ there is ${n_k}$ such that ${|b_{n_k}|\ge 2^k}$ . We can ensure ${n_k > n_{k-1}}$ since there are infinitely many candidates for ${n_k}$ . Define ${a_n=2^{-k}}$ if ${n = n_k}$ for some ${k}$ , and ${a_n=0}$ otherwise. Then ${\sum a_n}$ converges but ${\sum a_nb_n}$ diverges because its terms do not approach zero.

What if we drop “absolutely”? Let’s say that a sequence ${\{b_n\}}$ preserves convergence of series if for every convergent series ${\sum a_n}$ , the series ${\sum a_n b_n}$ also converges. Being bounded doesn’t imply this property: for example, ${b_n=(-1)^n}$ does not preserve convergence of the series ${\sum (-1)^n/n}$ .

Theorem. A sequence ${\{b_n\}}$ preserves convergence of series if and only if it has bounded variation, meaning ${\sum |b_n-b_{n+1}| }$ converges.

For brevity, let’s say that ${\{b_n\}}$ is BV. Every bounded monotone sequence is BV because the sum ${\sum |b_n-b_{n+1}| }$ telescopes. On the other hand, ${(-1)^n}$ is not BV, and neither is ${(-1)^n/n}$ . But ${(-1)^n/n^p}$ is for ${p>1}$ . The following lemma describes the structure of BV sequences.

Lemma 1. A sequence ${\{b_n\}}$ is BV if and only if there are two increasing bounded sequences ${\{c_n\}}$ and ${\{d_n\}}$ such that ${b_n=c_n-d_n}$ for all ${n}$ .

Proof. If such ${c_n,d_n}$ exist, then by the triangle inequality $\displaystyle \sum_{n=1}^N |b_n-b_{n+1}| = \sum_{n=1}^N (|c_n-c_{n +1}| + |d_{n+1}-d_n|) = \sum_{n=1}^N (c_{n+1}-c_n) + \sum_{n=1}^N (d_{n+1}-d_n)$ and the latter sums telescope to ${c_{N+1}-c_1 + d_{N+1}-d_1}$ which has a limit as ${N\rightarrow\infty}$ since bounded monotone sequences converge.

Conversely, suppose ${\{b_n\}}$ is BV. Let ${c_n = \sum_{k=1}^{n-1}|b_k-b_{k+1}|}$ , understanding that ${c_1=0}$ . By construction, the sequence ${\{c_n\}}$ is increasing and bounded. Also let ${d_n=c_n-b_n}$ ; as a difference of bounded sequences, this is bounded too. Finally,

$\displaystyle d_{n+1}-d_n = c_{n+1} -c_n + b_n - b_{n+1} = |b_n-b_{n+1}|+ b_n - b_{n+1} \ge 0$

which shows that ${\{d_n\}}$ is increasing.

To construct a suitable example where ${\sum a_nb_n}$ diverges, we need another lemma.

Lemma 2. If a series of nonnegative terms ${\sum A_n}$ diverges, then there is a sequence ${c_n\rightarrow 0}$ such that the series ${\sum c_n A_n}$ still diverges.

Proof. Let ${s_n = A_1+\dots+A_n}$ (partial sums); then ${A_n=s_n-s_{n-1}}$ . The sequence ${\sqrt{s_n}}$ tends to infinity, but slower than ${s_n}$ itself. Let ${c_n=1/(\sqrt{s_n}+\sqrt{s_{n-1}})}$ , so that ${c_nA_n = \sqrt{s_n}-\sqrt{s_{n-1}}}$ , and we are done: the partial sums of ${\sum c_nA_n}$ telescope to ${\sqrt{s_n}}$ .

Proof of the theorem, Sufficiency part. Suppose ${\{b_n\}}$ is BV. Using Lemma 1, write ${b_n=c_n-d_n}$ . Since ${a_nb_n = a_nc_n - a_n d_n}$ , it suffices to prove that ${\sum a_nc_n}$ and ${\sum a_nd_n}$ converge. Consider the first one; the proof for the other is the same. Let ${L=\lim c_n}$ and write ${a_nc_n = La_n - a_n(L-c_n)}$ . Here ${\sum La_n}$ converges as a constant multiple of ${\sum a_n}$ . Also, ${\sum a_n(L-c_n)}$ converges by the Dirichlet test: the partial sums of ${\sum a_n}$ are bounded, and ${L-c_n}$ decreases to zero.

Proof of the theorem, Necessity part. Suppose ${\{b_n\}}$ is not BV. The goal is to find a convergent series ${\sum a_n}$ such that ${\sum a_nb_n}$ diverges. If ${\{b_n\}}$ is not bounded, then we can proceed as in the case of absolute convergence, considered above. So let’s assume ${\{b_n\}}$ is bounded.

Since ${\sum_{n=1}^\infty |b_n-b_{n+1}|}$ diverges, by Lemma 2 there exists ${\{c_n\} }$ such that ${c_n\rightarrow 0}$ and ${\sum_{n=1}^\infty c_n|b_n-b_{n+1}|}$ diverges. Let ${d_n}$ be such that ${d_n(b_n-b_{n+1}) = c_n|b_n-b_{n+1}|}$ ; that is, ${d_n}$ differs from ${c_n}$ only by sign. In particular, ${d_n\rightarrow 0}$ . Summation by parts yields

$\displaystyle { \sum_{n=1}^N d_n(b_n-b_{n+1}) = \sum_{n=2}^N (d_{n}-d_{n-1})b_n + d_1b_1-d_Nb_{N+1} }$

As ${N\rightarrow\infty}$ , the left hand side of does not have a limit since ${\sum d_n(b_n-b_{n+1})}$ diverges. On the other hand, ${d_1b_1-d_Nb_{N+1}\rightarrow d_1b_1}$ since ${d_N\rightarrow 0}$ while ${b_{N+1}}$ stays bounded. Therefore, ${\lim_{N\rightarrow\infty} \sum_{n=2}^N (d_{n}-d_{n-1})b_n}$ does not exist.

Let ${a_n= d_n-d_{n-1}}$ . The series ${\sum a_n}$ converges (by telescoping, since ${\lim_{n\rightarrow\infty} d_n}$ exists) but ${\sum a_nb_n}$ diverges, as shown above.

In terms of functional analysis the preservation of absolute convergence is essentially the statement that ${(\ell_1)^* = \ell_\infty}$ . Notably, the ${\ell_\infty}$ norm of ${\{b_n\}}$ , i.e., ${\sup |b_n|}$ , is the number that controls by how much ${\sum |a_nb_n|}$ can exceed ${\sum |a_n|}$ .

I don’t have a similar quantitative statement for the case of convergence. The BV space has a natural norm too, ${\sum |b_n-b_{n-1}|}$ (interpreting ${b_0}$ as ${0}$ ), but it’s not obvious how to relate this norm to the values of the sums ${\sum a_n}$ and ${\sum a_nb_n}$ .

Compact sets in Banach spaces

In a Euclidean space, a set is compact if and only if it is closed and bounded. This fails in all infinite-dimensional Banach spaces (and in particular in Hilbert spaces) where the closed unit ball is not compact. However, one still has a simple description of compact sets:

A subset of a Banach space is compact if and only if it is closed, bounded, and flat.

By definition, a set is flat if for every positive number r it is contained in the r-neighborhood of some finite-dimensional linear subspace.

Notes:

The r-neighborhood of a set consists of all points whose distance to the set is less than r.
In a finite-dimensional subspace every subset is vacuously flat.

Necessity: Suppose K is a compact set. Every compact set is closed and bounded, this is true in all metric spaces. Given a positive number r, let F be a finite set such that K is contained in the r-neighborhood of F; the existence of such F follows by covering K with r-neighborhoods of points and choosing a finite subcover. Then the linear subspace spanned by F is finite-dimensional and demonstrates that K is flat.

Sufficiency: to prove K is compact, we must show it’s complete and totally bounded. Completeness follows from being a closed subset of a complete space, so the issue is total boundedness. Given r > 0, let M be a finite-dimensional subspace such that K is contained in the (r/2)-neighborhood of M. For each point of K, pick a point of M at distance less than r/2 from it. Let E be the set of all such points in M. Since K is bounded, so it E. Being a bounded subset of a finite-dimensional linear space, E is totally bounded. Thus, there exists a finite set F such that E is contained in the (r/2)-neighborhood of F. Consequently, K is contained in the r-neighborhood of F, which shows its total boundedness.

It’s worth noting that the equivalence of compactness with “flatness” (existence of finite-dimensional approximations) breaks down for linear operators in Banach spaces. While in a Hilbert space an operator is compact if and only if it is the norm-limit of finite-rank operators, some Banach spaces admit compact operators without a finite-rank approximation; that is, they lack the Approximation Property.

Laguerre polynomials under 1

Laguerre polynomials have many neat definitions; I am partial to $L_n(x) = \left(\frac{d}{dx} - 1\right)^n \frac{x^n}{n!}$ because it’s so easy to remember:

Begin with ${x^n}$
Repeat “subtract the derivative” ${n}$ times
Normalize so the constant term is 1.

For example, for n=3 this process goes as ${x^3}$ to ${x^3-3x^2}$ to ${x^3 -6x^2 + 6x}$ to ${x^3-9x^2+18x -6}$ , which normalizes to ${-\frac16x^3+\frac{3}{2}x^2 -3x +1}$ . This would make a mean exercise on differentiating polynomials: every mistake snowballs throughout the computation.

What would happen if we added the derivative instead? Nothing really new: this change is equivalent to reversing the direction of the x-axis, so we’d end up with ${L_n(-x)}$ . Incidentally, this shows that the polynomial ${L_n(-x)}$ has positive coefficients, which means the behavior of ${L_n}$ for negative ${x}$ is boring: the values go up as ${x}$ becomes more negative. Laguerre polynomials are all about the interval ${[0,\infty)}$ on which they are orthogonal with respect to the weight ${\exp(-x)}$ and therefore change sign often.

But when I look at the plot shown above, it’s not the zeros that draw my attention (perhaps because the x-axis is not shown) but the horizontal line ${y=1}$ , the zero-degree polynomial. The coefficients of ${L_n}$ have alternating signs; in particular, ${L_n(0)=1}$ and ${L_n'(0)=-n}$ . So, nonconstant Laguerre polynomials start off with the value of 1 and immediately dive below it. All except the linear one, ${L_1(x)=1-x}$ , eventually recover and reach 1 again (or so it seems; I don’t have a proof).

The yellow curve that is the first to cross the blue line is the 5th degree Laguerre polynomial. Let’s see if any of the other polynomials rises about 1 sooner…

Still, nobody beats ${L_5}$ (and the second place is held by ${L_4}$ ). By the way, the visible expansion of oscillations is approximately exponential; multiplying the polynomials by ${\exp(-x/2)}$ turns the envelopes into horizontal lines:

Back to the crossing of y=1 line. The quantity to study is the smallest positive root of ${L_n - 1}$ , denoted ${r(n)}$ from now on. (It is the second smallest root overall; as discussed above, this polynomial has a root at x=0 and no negative roots.) For n=2,3,4,5,6, the value of ${r(n)}$ is ${4, 3, 6-2\sqrt{3}, (15-\sqrt{105})/2, 6}$ which evaluates to 4, 3, 2.536…, 2.377…, and 6 respectively. I got these with Python / SymPy:

from sympy import *
x = Symbol('x')
[Poly(laguerre(n, x) - 1).all_roots()[1] for n in range(2, 7)]

For higher degrees we need numerics. SymPy can still help (applying .evalf() to the roots), but the process gets slow. Switching to NumPy’s roots method speeds things up, but when it indicated than ${r(88)}$ and a few others are in double digits, I became suspicious… a closer check showed this was a numerical artifact.

Conjecture: ${r(5) \le r(n) \le r(6)}$ for all ${n}$ . Moreover, ${3 < r(n) < 6}$ when ${n \ge 7}$ .

Here is a closer look at the exceptional polynomials of degrees 3, 4, 5 and 6, with 1 subtracted from each:

The first local maximum of ${L_n}$ shifts down and to the left as the degree n increases. The degree n=5 is the last for which ${L_n}$ exceeds 1 on the first attempt, so it becomes the quickest to do so. On the other hand, n=6 fails on its first attempt to clear the bar, and its second attempt is later than for any subsequent Laguerre polynomial; so it sets the record for maximal ${r(n)}$ .

Evaluating high-degree Laguerre polynomials is a numerical challenge: adding large terms of alternating signs can reduce accuracy dramatically. Here is a plot of the degree 98 polynomial (minus 1): where is its first positive root?

Fortunately, SymPy can evaluate Laguerre polynomials at rational points using exact arithmetics since the coefficients are rational. For example, when it evaluates the expression laguerre(98, 5) > 1 to True, that’s a (computer-assisted) proof that ${r(98) < 5}$ , which one could in principle "confirm" by computing the same rational value of ${L_{98}(5) }$ by hand (of course, in this situation a human is far less trustworthy than a computer) . Evaluation at the 13 rational points 3, 3.25, 3.5, … , 5.75, 6 is enough to certify that ${r(n) < 6}$ for ${n}$ up to 200 (with the aforementioned exception of ${r(6) = 6}$ ).

The lower bounds call for Sturm’s theorem which is more computationally expensive than sampling at a few rational points. SymPy offers a root-counting routine based on this theorem (it counts the roots within the given closed interval):

for n in range(2, 101):
    num_roots = count_roots((laguerre(n,x)-1)/x, 0, 3)
    print('{} roots for n = {}'.format(num_roots, n))

Division by x eliminates the root at 0, so we are left with the root count on (0,3] — which is 1 for n=3,4 and 2 for n=5. The count is zero for all other degrees up to 100, confirming that ${r(n) > 3}$ for ${n \ge 6}$ .

So, the conjecture looks solid. I don’t have a clue to its proof (nor do I know if it’s something known). The only upper bound on ${L_n}$ that I know is Szegő’s ${|L_n(x)|\le \exp(x/2)}$ for ${x\ge 0}$ , which is not helping here.

Complex Cantor sets

Every real number in the interval [0,1] can be written in binary as ${\sum_{k=1}^\infty c_k(1/2)^k}$ where each coefficient ${c_k}$ is either 0 or 1. Another way to put this: the set of all possible sums ${\sum_{k=1}^\infty c_kb^k}$ for b = 1/2 is a line segment.

line12

What is this set for other values of “base” b, then? Let’s stick to |b| < 1 for now, so that the series converges. Nothing interesting happens for real b between 1/2 and 1; the segment grows longer, to length b/(1-b). When b is between 0 and 1, we get Cantor sets, with the classical middle-third set being the case b = 1/3.

cantor13

There is no need to consider negative b, because of a symmetry between b and -b. Indeed, up to scaling and translation, the coefficients can be taken from {-1, 1} instead of {0, 1}. Then it’s obvious that changing the sign of b is the same as flipping half of coefficients the other way — does not change the set of possible sums.

Let’s look at purely imaginary b, then. Here is b = 0.6i

j06

Why so rectangular? The real part is the sum of ${c_kb^k}$ over even k, and the imaginary part is the sum over odd k. Each of these yields a Cantor type set as long as ${|b|^2 < 1/2}$ . Since the odd- and even-numbered coefficients are independent of each other, we get the product of two Cantor sets. Which changes into a rectangle when ${|b| \ge \sqrt{1/2}}$ :

jsqrt12

(I didn’t think a full-size picture of a solid rectangle was necessary here.)

This is already interesting: the phase transition from dust to solid (connected, and even with interior) happens at different values in the real and imaginary directions: 1/2 versus ${\sqrt{1/2}}$ . What will happen for other complex values? Using complex conjugation and the symmetry between b and -b, we reduce the problem to the quarter-disk in the first quadrant. Which still leaves a room for a lot of things to happen…

It’s clear that for |b| < 1/2 we get a totally disconnected set — it is covered by 2 copies of itself scaled by the factor of |b|, so its Hausdorff dimension is less than 1 when |b| is less than 1/2. Also, the argument of b is responsible for rotation of the scaled copies, and it looks like rotation favors disconnectivity… but then again, the pieces may link up again after being scaled-rotated a few times, so the story is not a simple one.

The set of bases b for which the complex Cantor set is connected is a Manderbrot-like set introduced by Barnsley and Harrington in 1985. It has the symmetries of a rectangle, and features a prominent hole centered at 0 (discussed above). But it actually has infinitely many holes, with “exotic” holes being tiny islands of disconnectedness, surrounded by connected sets. This was proved in 2014 by Calegari, Koch, Walker, so I refer to Danny Calegari’s post for an explanation and more pictures (much better looking than mine).

Besides “disconnected to connected”, there is another phase transition: empty interior to nonempty interior. Hare and Sidorov proved that the complex Cantor set has nonempty interior when ${|b| > 2^{-1/4}}$ ; their path to the proof involved a MathOverflow question The Minkowski sum of two curves which is of its own interest.

The pictures were made with a straightforward Python script, using expansions of length 20:

import matplotlib.pyplot as plt
import numpy as np
import itertools
n = 20
b = 0.6 + 0.3j
c = np.array(list(itertools.product([0, 1], repeat=n)))
w = np.array([b**k for k in range(n)]).reshape(1, -1)
z = np.sum(c*w, axis=1)
plt.plot(np.real(z), np.imag(z), '.', ms=4)
plt.axis('equal')
plt.show()

Since we are looking at partial sums anyway, it’s not necessary to limit ourselves to |b| being less than 1. Replacing b by 1/b only scales the picture, so the place to look for new kinds of pictures is the unit circle. Let’s try a 7th root of unity:

The set above looks sparse because many points overlap. Let’s change b to something non-algebraic:

What’s with the cusps along the perimeter?