## Most popular topics on Stack Exchange sites

This is an attempt to create a bird’s eye-view picture of Stack Exchange network by picking the most-used tag on every site. The sites are ordered by their size, measured by the number of questions they have so far — so, newer sites are toward the bottom of the list. For each site, I included the description of its audience, as seen on the site list, and the description of the top tag. The tag name is linked to the page with more information, such as tag wiki, and the revision history with the list of contributors.

#### Stack Overflow

For professional and enthusiast programmers.

Top tag javascript: JavaScript (not to be confused with Java) is a dynamic, weakly-typed language used for both client-side and server-side scripting. Use this tag for questions regarding ECMAScript and its various dialects/implementations (excluding ActionScript and Google-Apps-Script). Unless another tag for a framework/library is also included, a pure JavaScript answer is expected.

#### Mathematics

For people studying math at any level and professionals in related fields.

Top tag calculus: For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

#### Super User

For computer enthusiasts and power users.

Top tag windows-7: For questions specific to Windows 7. Use [windows] instead for questions involving Windows in general.

For Ubuntu users and developers.

Top tag 14.04: Fifth LTS (Long Term Support) release of Ubuntu, code-named “Trusty Tahr”. Released on 17th April, 2014. Will go End Of Life (EOL) April 2019. Only use this tag if your question is version-specific.

#### Server Fault

Top tag linux: Linux is the generic term for a UNIX-like open source operating system based on the Linux kernel.

#### Stack Overflow на русском

For программистов.

Top tag php: PHP — скриптовый язык программирования общего назначения, активно применяемый для разработки веб-приложений. Используйте эту метку, если у Вас возникли вопросы по применению данного языка или о самом языке.

#### TeX – LaTeX

For users of TeX, LaTeX, ConTeXt, and related typesetting systems.

Top tag tikz-pgf: TikZ is a higher-level drawing language built on top of the PGF graphics framework. For questions specifically about the PGF layer use {pgf-core} instead. Both tags are possible on the same question. The tag {diagrams} is also compatible with this tag.

#### Unix & Linux

For users of Linux, FreeBSD and other Un*x-like operating systems.

Top tag linux: These questions are about Linux in general — NOT specific to a particular distribution. If the question just happens to be in a Linux environment, please specify your Linux distribution in the body of your question, but do NOT use the /linux tag.

#### Cross Validated

For people interested in statistics, machine learning, data analysis, data mining, and data visualization.

Top tag r: Use this tag for any *on-topic* question that (a) involves R either as a critical part of the question or expected answer, & (b) is not *just* about how to use R.

#### Physics

For active researchers, academics and students of physics.

Top tag quantum-mechanics: Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy and the uncertainty principle and is generally used in single body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.

#### English Language & Usage

For linguists, etymologists, and serious English language enthusiasts.

Top tag single-word-requests: This tag is for questions seeking a single word that fits a meaning. To ensure your question is not closed as off-topic, please be specific about the intended use of the word. YOU MUST INCLUDE A SAMPLE SENTENCE demonstrating how the word would be used. Please use the “phrase-requests” tag instead if you seek more than just a single word.

#### Geographic Information Systems

For cartographers, geographers and GIS professionals.

Top tag qgis: QGIS is a cross-platform GIS application licensed under the GNU General Public License.

For power users of Apple hardware and software.

Top tag macos: macOS is the current marketing name for Apple’s Macintosh Operating System. This OS was previously known as OS X, and Mac OS X before that (which itself succeeded the ‘classic’ Mac OS).

#### Electrical Engineering

For electronics and electrical engineering professionals, students, and enthusiasts.

Top tag arduino: Be sure to use the Arduino Stack Exchange for questions that are more Arduino and less electronics.

#### MathOverflow

For professional mathematicians.

Top tag ag.algebraic-geometry: for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

#### WordPress Development

Top tag custom-post-types: Custom Post Types extend the WordPress back-end to support additional, non-Post content.

For passionate videogamers on all platforms.

Top tag minecraft: Mojang’s exploration and survival based sandbox game in almost endless, procedurally generated worlds. This tag is for (vanilla) Minecraft on the PC. See the tag wiki for related tags. Please note that questions requesting technical support for modded Minecraft crash issues are off-topic.

#### SharePoint

For SharePoint enthusiasts.

Top tag 2013: For questions completely specific to all editions of SharePoint 2013 and **not past or future versions**

#### Stack Overflow em Português

Top tag php: PHP é uma linguagem de script interpretada do lado do servidor de código aberto, amplamente utilizada no desenvolvimento web. Use esta tag para perguntas sobre classes, métodos, funções, sintaxe e uso geral desta linguagem. Não use esta tag se o PHP é usado circunstancialmente mas não tem relação com o problema da pergunta.

Top tag 7: Version tags should be used only when strictly necessary, for questions that apply to a version only, and not to merely say “I am using Drupal 7 in my site.”

#### Salesforce

For Salesforce administrators, implementation experts, developers and anybody in-between.

Top tag apex: Questions relating to Apex, the native programming language for the Force.com platform. Use it for general questions on syntax, errors, constructs, and rules of use. Most questions should include a code *excerpt* to help answerers understand specifically what has gone wrong or why you need help.

#### Magento

For users of the Magento e-Commerce platform.

Top tag magento-1.9: Magento Community version 1.9

For database professionals who wish to improve their database skills and learn from others in the community.

Top tag sql-server: All versions of Microsoft SQL Server (not MySQL). Please also add a version-specific tag like sql-server-2016 if that is relevant to the question.

#### Software Engineering

For professionals, academics, and students working within the systems development life cycle.

Top tag java: Java is a high-level, platform-independent, object-oriented programming language originally developed by Sun Microsystems. Java is currently owned by Oracle, which purchased Sun in 2010.

#### Code Review

For peer programmer code reviews.

Top tag java: Java (not to be confused with JavaScript) is a class-based, object-oriented, strongly typed, reflective language and run-time environment (JRE). Java programs are compiled to bytecode and run in a virtual machine (JVM) enabling a “write once, run anywhere” (WORA) methodology.

#### Android Enthusiasts

For enthusiasts and power users of the Android operating system.

Top tag applications: For questions about Android apps in general. (If possible, use a specific application’s tag instead.) Note that Development is off-topic here.)

#### Mathematica

For users of Wolfram Mathematica.

Top tag plotting: Questions on creating visualizations from functions or data using high-level constructors such as Plot, ListPlot, Histogram, etc.

#### Science Fiction & Fantasy

For science fiction and fantasy enthusiasts.

Top tag story-identification: Use for help identifying a story and/or its creator(s), including novels, movies, comic books, entire TV series, etc. Use with another tag to specify which type of media, eg. [short-stories]. For identifying a single episode of a known series, whether TV, film, book, or comic, use [episode-identification].

#### Information Security

For information security professionals.

Top tag encryption: Encryption is the process of transforming plaintext using a cipher to make it unreadable to anyone except those possessing the key.

#### English Language Learners

For speakers of other languages learning English.

Top tag meaning: This tag is for questions which a dictionary cannot answer about what a word means. If the question is about the meaning of a word that can’t be understood outside its phrase or sentence, the “meaning-in-context” tag should be also used; for the meaning of a phrase, use the “phrase-meaning” tag instead.

#### Game Development

For professional and independent game developers.

Top tag unity: Unity is a cross-platform game creation system that focuses on easy art pipeline process. It consists of a game engine and an integrated development environment. The game engine’s scripting is built on Mono.

#### Home Improvement

For contractors and serious DIYers.

Top tag electrical: Distribution and use of electricity throughout the home. Electrical standards vary greatly worldwide, so providing your location in your question or your profile helps ensure you get answers that are relevant to you. Posting photographs of wiring junctions or drawing a wiring diagram and posting it is also highly recommended.

#### Blender

For people who use Blender to create 3D graphics, animations, or games.

Top tag python: Python is an object-oriented programming language. In Blender, it is used as a general purpose scripting language and to create add-ons to extend Blender’s functionality.

#### Webmasters

For pro webmasters.

Top tag seo: Search Engine Optimization (SEO) is the process of improving the visibility of web content in search engines using an understanding of search engines’ processes and algorithms. Also known as “natural” or “organic” search, it is distinct from paid web advertising.

#### Stack Overflow en español

For programadores y profesionales de la informática.

Top tag java: Java (no se debe confundir con JavaScript) es un lenguaje de programación de propósito general que soporta programación orientada a objetos. Para correr depende de una Máquina Virtual de Java (JVM). La Plataforma de Java es el nombre de un conjunto de herramientas para desarrollar y ejecutar programas de Java. Use esta etiqueta para preguntas referentes al lenguaje de programación Java o las herramientas de la plataforma de Java.

#### Travel

For road warriors and seasoned travelers.

Top tag visas: Token showing authorization to apply to enter the territory for which it was issued. Don’t forget to include your citizenship when asking!

#### User Experience

For user experience researchers and experts.

Top tag usability: The extent to which a product can be used by specified users to achieve specified goals with effectiveness, efficiency, and satisfaction in a specified context of use.

#### Mi Yodeya

For those who base their lives on Jewish law and tradition and anyone interested in learning more.

Top tag halacha: NOTE: Like Wikipedia, this site makes no guarantee of validity, and does not offer professional (particularly rabbinic) advice. Treat information from this site like it came from a crowd of your friends. // Jewish law. Specifically, the legal process beginning with the written Torah and continuing through the Mishna, Talmud, and the legal codes (e.g., Rambam, Tur, Shulchan Aruch).

#### Web Applications

For power users of web applications.

Top tag gmail: For questions about the Gmail service as accessed by a desktop or mobile browser. Questions about native smartphone apps are off-topic on Web Apps.

#### Chemistry

For scientists, academics, teachers and students.

Top tag organic-chemistry: Organic chemistry is the study of the structure and reactivity of organic molecules, typically containing carbon and hydrogen in addition to other elements. This tag should be applied to all questions relating to organic molecules and their properties (structure of organic molecules, spectroscopic properties, reaction mechanisms, stereochemistry etc), along with other suitable tags to help narrow down the scope of the question.

#### Role-playing Games

For gamemasters and players of tabletop, paper-and-pencil role-playing games.

Top tag dnd-5e: Dungeons & Dragons 5th edition, by Wizards of the Coast. D&D 5e is a heroic fantasy RPG inspired by the mechanics and settings of all previous D&D editions. Previously code-named D&D Next.

#### Graphic Design

For Graphic Design professionals, students, and enthusiasts.

#### Computer Science

For students, researchers and practitioners of computer science.

Top tag algorithms: Questions related to design and analysis of algorithms

For academics and those enrolled in higher education.

Top tag publications: Questions related to academic publications including online and traditional journals, books, and conference proceedings.

#### Photography

For professional, enthusiast and amateur photographers.

Top tag lens: A photographic lens is used to focus the light onto film or a digital sensor. Many cameras have interchangeable lens systems, allowing the photographer to choose the type of lens to use.

#### Personal Finance & Money

For people who want to be financially literate.

Top tag united-states: For questions that relate to the laws, practices, and products of the United States.

#### Raspberry Pi

For users and developers of hardware and software for Raspberry Pi.

Top tag raspbian: Raspbian is a GNU/Linux operating system derived from Debian. Version numbering follows that of Debian and the current stable is 8 (Jessie). Raspbian is the most widely used Pi based distro and the one recommended by the Foundation, who distribute their own images of it.

For professional and amateur chefs.

Top tag baking: Questions about cooking by dry heat without direct exposure to a flame, typically in an oven.

#### Movies & TV

For movie and tv enthusiasts.

Top tag plot-explanation: Seeking to understand a story plot better, or to clear up confusion about certain aspects or plot points.

#### Biology

For biology researchers, academics, and students.

Top tag human-biology: This tag is for questions about the general biological features of human beings (as opposed to the biology of non-humans).

#### The Workplace

For members of the workforce navigating the professional setting.

Top tag professionalism: Questions referring to how one presents themselves and interacts with others in a professional environment

#### Bitcoin

For Bitcoin crypto-currency enthusiasts.

Top tag transactions: Transactions are signed messages regarding the transfer or generation of bitcoins. They are broadcasted through the network and, if accepted, integrated in the blockchain.

#### Cryptography

For software developers, mathematicians and others interested in cryptography.

Top tag encryption: Encryption is the process of transforming plaintext using a cipher into ciphertext to make it unreadable to anyone except those possessing the key. Decryption is the process of transforming that ciphertext back into plaintext, using the key.

#### Motor Vehicle Maintenance & Repair

For mechanics and DIY enthusiast owners of cars, trucks, and motorcycles.

Top tag engine: Internal Combustion Engine. Most engines in road going vehicles are 4-cycle internal combustion gasoline engines.

#### Japanese Language

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

Top tag grammar: 文法. A collective term for syntax (the way sentences are put together) and morphology (forms of words, including the way new words are put together). Often used to describe function words such as particles, to describe word endings, and to talk about general sentence structure.

#### Software Recommendations

For people seeking specific software recommendations.

Top tag windows: Software to run on the Microsoft Windows family of operating systems.

#### Signal Processing

For practitioners of the art and science of signal, image and video processing.

Top tag image-processing: In general, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame.

#### Worldbuilding

For writers/artists using science, geography and culture to construct imaginary worlds and settings.

Top tag science-based: For questions that require answers based on hard science, not magic or pseudo-science, but do not require scientific citations. Consider, alternatively, the hard-science and reality-check tags. Avoid using this tag as the only tag on a question.

For administrators, end users, developers and designers for ExpressionEngine® CMS.

Top tag expresso-store: Exp:resso Store is an e-commerce module. You can find more details about Store at https://exp-resso.com/store/

#### スタック・オーバーフロー

For プログラマーとプログラミングに熱心の人.

Top tag javascript: JavaScriptとは、プログラミング言語のひとつである。Javaと名前が似ているが、異なるプログラミング言語である。
オブジェクト指向のスクリプト言語であることを特徴とする。

#### Русский язык

For лингвистов, этимологов, и энтузиастов русского языка.

Top tag пунктуация: Для вопросов, связанных со знаками препинания, а также пунктуационными правилами.

#### Arduino

For developers of open-source hardware and software that is compatible with Arduino.

Top tag arduino-uno: The Arduino Uno is the most common of the Arduino boards. It is based on the ATmega328P microcontroller.

#### Emacs

For those using, extending or developing Emacs.

Top tag org-mode: is an Emacs major mode for keeping notes, maintaining TODO lists, planning projects, and authoring documents with a fast and effective plain-text system. Because Org mode is a vast subject area, other tags must accompany the org-mode tag.

#### Music: Practice & Theory

For musicians, students, and enthusiasts.

Top tag guitar: For questions about guitar playing in general. For questions specifically about electric guitars, see “electric-guitar”. For questions specifically about acoustic guitars with or without pickup systems, see “acoustic-guitar”.

#### Bicycles

For people who build and repair bicycles, people who train cycling, or commute on bicycles.

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

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

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

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

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## 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 ${\displaystyle L_n(x) = \left(\frac{d}{dx} - 1\right)^n \frac{x^n}{n!}}$ because it’s so easy to remember:

1. Begin with ${x^n}$
2. Repeat “subtract the derivative” ${n}$ times
3. 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.

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.

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

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  ${|b| \ge \sqrt{1/2}}$:

(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?

## Iterating the logistic map: limsup of nonperiodic orbits

Last time we found that when a sequence with ${x_1\in (0,1)}$ and ${x_{n+1} = 4x_n(1-x_n)}$ does not become periodic, its upper limit ${\limsup x_n}$ must be at least ${\approx 0.925}$. This time we’ll see that ${\limsup x_n}$ can be as low as ${(2+\sqrt{3})/4\approx 0.933}$ and determine for which ${x_1}$ it is equal to 1.

The quadratic polynomial ${f(x)=4x(1-x)}$ maps the interval ${[0,1]}$ onto itself. Since the linear function ${g(x) = 1-2x}$ maps ${[0,1]}$ onto ${[-1,1]}$, it follows that the composition ${h=g\circ f\circ g^{-1}}$ maps ${[-1,1]}$ onto ${[-1,1]}$. This composition is easy to compute: ${h(x) = 2x^2-1 }$.

We want to know whether the iteration of ${f}$, starting from ${x_1}$, produces numbers arbitrarily close to ${1}$. Since ${f\circ f \circ \cdots \circ f = g^{-1}\circ h \circ h \circ \cdots \circ h\circ g}$ the goal is equivalent to finding whether the iteration of ${h}$, starting from ${g(x_1)}$, produces numbers arbitrarily close to ${g(1) = -1}$. To shorten formulas, let’s write ${h_n}$ for the ${n}$th iterate of ${h}$, for example, ${h_3 = h\circ h\circ h}$.

So far we traded one quadratic polynomial ${f}$ for another, ${h}$. But ${h}$ satisfies a nice identity: ${h(\cos t)=2\cos^2 t-1 = \cos(2t)}$, hence ${h_n(\cos t) = \cos (2^n t)}$ for all ${n\in\mathbb N}$. It’s convenient to introduce ${\alpha = \frac{1}{\pi}\cos^{-1}(1-2x_1)}$, so that ${ h_n(g(x_1)) = h_n(\cos 2\pi \alpha ) = \cos(2^n\cdot 2\pi \alpha) }$.

The problem becomes to determine whether the numbers ${2^n\cdot 2\pi \alpha}$ come arbitrarily close to ${\pi}$, modulo an integer multiple of ${2\pi}$. Dividing by ${2\pi}$ rephrases this as: does the fractional part of ${2^n \alpha}$ come arbitrarily close to ${1/2}$?

A number that is close to ${1/2}$ has the binary expansion beginning either with ${0.01111111\dots}$ or with ${0.10000000\dots}$. Since the binary expansion of ${2^n\alpha}$ is just the binary expansion of ${\alpha}$ shifted ${n}$ digits to the left, the property ${\limsup x_n=1}$ is equivalent to the following: for every ${k\in\mathbb N}$ the binary expansion of ${\alpha}$ has infinitely many groups of the form “1 followed by k zeros” or “0 followed by k ones”.

A periodic expansion cannot have the above property; this, ${\alpha}$ must be irrational. The property described above can then be simplified to “irrational and has arbitrarily long runs of the same digit”, since a long run of ${0}$s will be preceded by a ${1}$, and vice versa.

For example, combining the pairs 01 and 10 in some non-periodic way, we get an irrational number ${\alpha}$ such that the fractional part of ${2^n\alpha}$ does not get any closer to 1/2 than ${0.01\overline{10}_2 = 5/12}$ or ${0.10\overline{01}_2 = 7/12}$. Hence, ${\cos 2^n 2\pi \alpha \ge -\sqrt{3}/2}$, which leads to the upper bound ${x_n\le (2+\sqrt{3})/4\approx 0.933}$ for the sequence with the starting value ${x_1=(1-\cos\pi\alpha)/2}$.

Let us summarize the above observations about ${\limsup x_n}$.

Theorem: ${\limsup x_n=1}$ if and only if (A) the number ${\alpha = \frac{1}{\pi}\cos^{-1}(1-2x_1)}$ is irrational, and (B) the binary expansion of ${\alpha}$ has arbitrarily long runs of the same digit.

Intuitively, one expects that a number that satisfies (A) will also satisfy (B) unless it was constructed specifically to fail (B). But to verify that (B) holds for a given number is not an easy task.

As a bonus, let’s prove that for every rational number ${y\in (-1,1)}$, except 0, 1/2 and -1/2, the number ${\alpha = \frac{1}{\pi}\cos^{-1}y}$ is irrational. This will imply, in particular, that ${x_1=1/3}$ yields a non-periodic sequence. The proof follows a post by Robert Israel and requires a lemma (which could be replaced with an appeal to Chebyshev polynomials, but the lemma keeps things self-contained).

Lemma. For every ${n\in \mathbb N}$ there exists a monic polynomial ${P_n}$ with integer coefficients such that ${P_n(2 \cos t) = 2\cos nt }$ for all ${t}$.

Proof. Induction, the base case ${n=1}$ being ${P_1(x)=x}$. Assuming the result for integers ${\le n}$, we have ${2 \cos (n+1)t = e^{i(n+1)t} + e^{-i(n+1)t} }$ ${ = (e^{int} + e^{-int})(e^{it} + e^{-it}) - (e^{i(n-1)t} + e^{-i(n-1)t}) }$ ${ = P_n(2 \cos t) (2\cos t) - P_{n-1}(2\cos t) }$
which is a monic polynomial of ${2\cos t}$. ${\Box}$

Suppose that there exists ${n}$ such that ${n\alpha \in\mathbb Z}$. Then ${2\cos(\pi n\alpha)=\pm 2}$. By the lemma, this implies ${P_n(2\cos(\pi \alpha)) =\pm 2}$, that is ${P_n(2y)=\pm 2}$. Since ${2y}$ is a rational root of a monic polynomial with integer coefficients, the Rational Root Theorem implies that it is an integer. ${\Box}$

## A limsup exercise: iterating the logistic map

Define the sequence ${\{x_n\}}$ as follows: ${x_1=1/3}$ and ${x_{n+1} = 4x_n(1-x_n)}$ for ${n=1,2,\dots}$. What can we say about its behavior as ${n\rightarrow\infty}$?

The logistic map ${f(x)=4x(1-x)}$ leaves the interval [0,1] invariant (as a set), so ${0\le x_n\le 1}$ for all ${n}$. There are two fixed points: 0 and 3/4.

Can ${x_n}$ ever be 0? If ${n}$ is the first index this happens, then ${x_{n-1}}$ must be ${1}$. Working backwards, we find ${x_{n-2}=1/2}$, and ${x_{n-3}\in \{1/2 \pm \sqrt{2}/4\}}$. But this is impossible since all elements of the sequence are rational. Similarly, if ${n}$ is the first index when ${x_n = 3/4}$, then ${x_{n-1}=1/4}$ and ${x_{n-2}\in \{1/2\pm \sqrt{3}/4\}}$, a contradiction again. Thus, the sequence never stabilizes.

If ${x_n}$ had a limit, it would have to be one of the two fixed points. But both are repelling: ${f'(x) = 4 - 8x}$, so ${|f'(0)|=4>1 }$ and ${|f'(3/4)| = 2 > 1}$. This means that a small nonzero distance to a fixed point will increase under iteration. The only way to converge to a repelling fixed point is to hit it directly, but we already know this does not happen. So the sequence ${\{x_n\}}$ does not converge.

But we can still consider its upper and lower limits. Let’s try to estimate ${S = \limsup x_n}$ from below. Since ${f(x)\ge x}$ for ${x\in [0,3/4]}$, the sequence ${\{x_n\}}$ increases as long as ${x_n\le 3/4}$. Since we know it doesn’t have a limit, it must eventually break this pattern, and therefore exceed 3/4. Thus, ${S\ge 3/4}$.

This can be improved. The second iterate ${f_2(x)=f(f(x))}$ satisfies ${f_2(x)\ge x}$ for ${x}$ between ${3/4}$ and ${a = (5+\sqrt{5})/8 \approx 0.9}$. So, once ${x_n>3/4}$ (which, by above, happens infinitely often), the subsequence ${x_n, x_{n+2}, x_{n+4},\dots}$ increases until it reaches ${a}$. Hence ${S\ge a}$.

The bound ${\limsup x_n\ge a}$ is best possible if the only information about ${x_1}$ is that the sequence ${x_n}$ does not converge. Indeed, ${a}$ is a periodic point of ${f}$, with the corresponding iteration sequence ${\{(5+ (-1)^n\sqrt{5})/8\}}$.

Further improvement is possible if we recall that our sequence is rational and hence cannot hit ${a}$ exactly. By doubling the number of iterations (so that the iterate also fixes ${a}$ but also has positive derivative there) we arrive at the fourth iterate ${f_4}$. Then ${f_4(x)\ge x}$ for ${a\le x\le b}$, where ${b }$ is the next root of ${f_4(x)-x}$ after ${a}$, approximately ${0.925}$. Hence ${S\ge b}$.

This is a colorful illustration of the estimation process (made with Sage): we are covering the line ${y=x}$ with the iterates of ${f}$, so that each subsequent one rises above the line the moment the previous one falls. This improves the lower bound on ${S}$ from 0.75 to 0.9 to 0.92.

Although this process can be continued, the gains diminish so rapidly that it seems unlikely one can get to 1 in this way. In fact, one cannot because we are not using any properties of ${x_1}$ other than “the sequence ${x_n}$ is not periodic.” And it’s not true that ${\limsup x_n = 1}$ for every non-periodic orbit of ${f}$. Let’s return to this later.