A new proof from the Australian science fiction writer Greg Egan and a 2011 proof anonymously posted online are now being hailed as significant advances on a puzzle mathematicians have been studying for at least 25 years.
Tag: math.CO
🔖 The Erdős discrepancy problem [2017] | Terence Tao | YouTube
🔖 The Erdős Discrepancy Problem (6.09.2017) | Terence Tao | YouTube
The discrepancy of a sequence f(1), f(2), ... of numbers is defined to be the largest value of |f(d) + f(2d) + ... + f(nd)| as n and d range over the natural numbers. In the 1930s, Erdős posed the question of whether any sequence consisting only of +1 and -1 could have bounded discrepancy. In 2010, the collaborative Polymath5 project showed (among other things) that the problem could be effectively reduced to a problem involving completely multiplicative sequences. Finally, using recent breakthroughs in the asymptotics of completely multiplicative sequences by Matomaki and Radziwiłł, as well as a surprising application of the Shannon entropy inequalities, the Erdős discrepancy problem was solved in 2015. In his talk TT will discuss this solution and its connection to the Chowla and Elliott conjectures in number theory.
🔖 The Erdős Discrepancy Problem | Terence Tao | YouTube
🔖 Tao’s resolution of the Erdős discrepancy problem | AMS | K. Soundararajan
This article gives a simplified account of some of the ideas behind Tao’s resolution of the Erdős discrepancy problem.
http://dx.doi.org/10.1090/bull/1598 | PDF
🔖 The Erdős discrepancy problem | Polymath1Wiki
🔖 The Entropy Decrement Method and the Erdos Discrepancy Problem | Simons Institute for the Theory of Computing
Tuesday, April 11th, 2017 9:30 am – 10:30 am
Structure vs. Randomness
Speaker: Terry Tao, UCLAWe discuss a variant of the density and energy increment arguments that we call an "entropy decrement method", which can be used to locate a scale in which two relevant random variables share very little mutual information, and thus behave somewhat like independent random variables. We were able to use this method to obtain a new correlation estimate for multiplicative functions, which in turn was used to establish the Erdos discrepancy conjecture that any sequence taking values in {-1,+1} had unbounded sums on homogeneous arithmetic progressions.
🔖 [1509.05363] The Erdos discrepancy problem by Terence Tao | arXiv
We show that for any sequence f:N→{−1,+1} taking values in {−1,+1}, the discrepancy
supn,d∈N∣∣∣∣∑j=1nf(jd)∣∣∣∣
of f is infinite. This answers a question of Erdős. In fact the argument also applies to sequences f taking values in the unit sphere of a real or complex Hilbert space. The argument uses three ingredients. The first is a Fourier-analytic reduction, obtained as part of the Polymath5 project on this problem, which reduces the problem to the case when f is replaced by a (stochastic) completely multiplicative function g. The second is a logarithmically averaged version of the Elliott conjecture, established recently by the author, which effectively reduces to the case when g usually pretends to be a modulated Dirichlet character. The final ingredient is (an extension of) a further argument obtained by the Polymath5 project which shows unbounded discrepancy in this case.
👓 Terence Tao’s Answer to the Erdős Discrepancy Problem | Quanta Magazine
Using crowd-sourced and traditional mathematics research, Terence Tao has devised a solution to a long-standing problem posed by the legendary Paul Erdős.
The article does a reasonable job of laying out some of the problem and Tao’s solution to it. I was a bit bothered by the idea of “magical” in the title, but it turns out it’s a different reference than the one I was expecting.
Gems And Astonishments of Mathematics: Past and Present—Lecture One
Lecture notes
For those who may have missed last night’s first lecture, I’m linking to a Livescribe PDF document which includes the written notes as well as the accompanying audio from the lecture. If you view it in Acrobat Reader version X (or higher), you should be able to access the audio portion of the lecture and experience it in real time almost as if you had been present in person. (Instructions for using Livescribe PDF documents.)
We’ve covered the following topics:
- Class Introduction
- Erdős Discrepancy Problem
- n-cubes
- Hilbert’s Cube Lemma (1892)
- Schur (1916)
- Van der Waerden (1927)
- Sylvester’s Line Problem (partial coverage to be finished in the next lecture)
- Ramsey Theory
- Erdős (1943)
- Gallai (1944)
- Steinberg’s alternate (1944)
- DeBruijn and Erdős (1948)
- Motzkin (1951)
- Dirac (1951)
- Kelly & Moser (1958)
- Tao-Green Proof
- Homework 1 (homeworks are generally not graded)
Over the coming days and months, I’ll likely bookmark some related papers and research on these and other topics in the class using the class identifier MATHX451.44 as a tag in addition to topic specific tags.
Course Description
Mathematics has evolved over the centuries not only by building on the work of past generations, but also through unforeseen discoveries or conjectures that continue to tantalize, bewilder, and engage academics and the public alike. This course, the first in a two-quarter sequence, is a survey of about two dozen problems—some dating back 400 years, but all readily stated and understood—that either remain unsolved or have been settled in fairly recent times. Each of them, aside from presenting its own intrigue, has led to the development of novel mathematical approaches to problem solving. Topics to be discussed include (Google away!): Conway’s Look and Say Sequences, Kepler’s Conjecture, Szilassi’s Polyhedron, the ABC Conjecture, Benford’s Law, Hadamard’s Conjecture, Parrondo’s Paradox, and the Collatz Conjecture. The course should appeal to devotees of mathematical reasoning and those wishing to keep abreast of recent and continuing mathematical developments.
Suggested Prerequisites
Some exposure to advanced mathematical methods, particularly those pertaining to number theory and matrix theory. Most in the class are taking the course for “fun” and the enjoyment of learning, so there is a huge breadth of mathematical abilities represented–don’t not take the course because you feel you’ll get lost.
I’ve written some general thoughts, hints, and tips on these courses in the past.
Renovated Classrooms
I’d complained to the UCLA administration before about how dirty the windows were in the Math Sciences Building, but they went even further than I expected in fixing the problem. Not only did they clean the windows they put in new flooring, brand new modern chairs, wood paneling on the walls, new projection, and new white boards! I particularly love the new swivel chairs, and it’s nice to have such a lovely new environment in which to study math.
Category Theory for Winter 2019
As I mentioned the other day, Dr. Miller has also announced (and reiterated last night) that he’ll be teaching a course on the topic of Category Theory for the Winter quarter coming up. Thus if you’re interested in abstract mathematics or areas of computer programming that use it, start getting ready!