T. Gallai's proof has been outlined by P. Erdös in his submission of the problem to The American Mathematical Monthly in 1943. Solution Given the set Π of noncollinear points, consider the set of lines Σ that pass through at least two points of Π. Such lines are said to be connecting. Among the connecting lines, those that pass through exactly two points of Π are called ordinary.
Links
🔖 Sylvester’s Problem, Steinberg’s Solution | Cut the Knot
R. Steinberg's was actually the first published solution to Syvester's problem, Solution Given the set Π of noncollinear points, consider the set of lines Σ that pass through at least two points of Π. Such lines are said to be connecting. Among the connecting lines, those that pass through exactly two points of Π are called ordinary. We consider the configuration in the projective plane.
🔖 Sylvester–Gallai theorem | Wikipedia
The Sylvester–Gallai theorem in geometry states that, given a finite number of points in the Euclidean plane, either
* all the points lie on a single line; or
* there is a line which contains exactly two of the points.
It is named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944.A line that contains exactly two of a set of points is known as an ordinary line. According to a strengthening of the theorem, every finite point set (not all on a line) has at least a linear number of ordinary lines. There is an algorithm that finds an ordinary line in a set of n points in time proportional to n log n in the worst case.
🔖 The Erdős discrepancy problem [2017] | Terence Tao | YouTube
Terence Tao (YouTube)🔖 The Erdős Discrepancy Problem (6.09.2017) | Terence Tao | YouTube
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
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
Terence Tao (arxiv.org)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.
🔖 Sign patterns of the Mobius and Liouville functions | Terence Tao
Kaisa Matomäki, Maksym Radziwiłł, and I have just uploaded to the arXiv our paper “Sign patterns of the Liouville and Möbius functions”. This paper is somewhat similar to our previous p…
🔖 [1501.04585] Multiplicative functions in short intervals | arXiv
We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the M\"obius function we show that there are cancellations in the sum of μ(n) in almost all intervals of the form [x,x+ψ(x)] with ψ(x)→∞ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of xϵ-smooth numbers in intervals of the form [x,x+c(ε)x−−√], recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of λ(n)λ(n+1), with λ(n) Liouville's function, is non-trivially bounded in absolute value by 1−δ for some δ>0. This settles an old folklore conjecture and constitutes progress towards Chowla's conjecture. Fourth, we show that a (general) real-valued multiplicative function f has a positive proportion of sign changes if and only if f is negative on at least one integer and non-zero on a positive proportion of the integers. This improves on many previous works, and is new already in the case of the M\"obius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.
👓 Terence Tao’s Answer to the Erdős Discrepancy Problem | Quanta Magazine
Erica Klarreich (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.
In the middle of the lecture last night, I was thinking to myself that this problem seems like a mixture of combinatorics, integer partitions and coding theory. Something about this article reminds me of that fact again. Most of the references I’m seeing however are directly to number theory and don’t relate to the integer partition piece–perhaps worth delving into to see what shakes out.
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.
👓 What is Applied Category Theory? | Azimuth
Tai-Danae Bradley has a new free “booklet” on applied category theory. It was inspired by the workshop Applied Category Theory 2018, which she attended, and I think it makes a great com…
👓 On the recently removed paper from the New York Journal of Mathematics | Terence Tao
In the last week or so there has been some discussion on the internet about a paper (initially authored by Hill and Tabachnikov) that was initially accepted for publication in the Mathematical Inte…
I wish there were more on the math here or at least some solid discussion of the actual science. The huge number of comments make me just think that this is gasoline, however well intentioned it may be.