The Liouville function has super-linear block growth

You need to know: Prime factorisation of positive integers, limits.

Background: For a positive integer n, let $\Omega(n)$ be the number of prime factors of n, counted with multiplicity. Function $\lambda(n)=(-1)^{\Omega (n)}$ is known as the Liouville function. The block complexity $P_{\lambda}(n)$ of $\lambda(n)$ is the number of sign patterns of size n that are taken by consecutive values of $\lambda(n)$ .

The Theorem: On 2nd August 2017, Nikos Frantzikinakis and Bernard Host submitted to arxiv a paper in which they proved, among other results, that $\lim\limits_{n\to\infty}\frac{P_{\lambda}(n)}{n}=\infty$ .

Short context: The Chowla conjecture predicts, as one may naturally expect, that all possible sign patterns of size n are taken by the Liouville function. In other words, $P_{\lambda}(n)$ is conjectured to be $2^n$ . However, the best lower bound for $P_{\lambda}(n)$ before 2017 was only $P_{\lambda}(n)\geq n+5$ for $n\geq 3$ . The Theorem significantly improves this bound.

Links: Free arxiv version of the original paper is here, journal version is here.

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There are fifteen types of convex pentagons that can tile the plane

You need to know: Convex set on the plane, convex polygon.

Background: We say that convex polygon P can tile the plane if the plane can be covered by identical copies of P (possibly translated and rotated) with no overlaps and no gaps. For a convex pentagon, denote $\alpha_1, \dots, \alpha_5$ the sizes of its angles (in radians) and $l_1, l_2, \dots, l_5$ the lengths of its sides, enumerated counterclockwise starting from a fixed vertex.

The Theorem: On 1st August 2017, Michael Rao submitted to arxiv a paper in which he proved that a convex pentagon can tile the whole plane by identical copies if and only if it belongs to one of the following fifteen types: (1) $\alpha_1+\alpha_2+\alpha_3 = 2\pi$ ; (2) $\alpha_1+\alpha_2+\alpha_4 = 2\pi$ and $l_1=l_4$ ; (3) $\alpha_1 = \alpha_3 = \alpha_4 = \frac{2}{3}\pi$ , $l_1=l_2$ , and $l_4=l_3+l_5$ ; (4) $\alpha_1 = \alpha_3 = \frac{1}{2}\pi$ , $l_1=l_2$ , and $l_3=l_4$ ; (5) $\alpha_1=\frac{1}{3}\pi$ , $\alpha_3=\frac{2}{3}\pi$ , $l_1=l_2$ , and $l_3=l_4$ ; (6) $\alpha_1+\alpha_2+\alpha_4 = 2\pi$ , $\alpha_1= 2\alpha_3$ , $l_1=l_2=l_5$ , and $l_3 = l_4$ ; (7) $2\alpha_2+\alpha_3 = 2\alpha_4+\alpha_1 = 2\pi$ , and $l_1=l_2=l_3=l_4$ ; (8) $2\alpha_1+\alpha_2=2\alpha_4 + \alpha_3 = 2\pi$ , and $l_1 = l_2 = l_3 = l_4$ ; (9) $\alpha_5 = \frac{\pi}{2}$ , $\alpha_1+\alpha_4 = \pi$ , $2\alpha_2-\alpha_4 = 2\alpha_3+\alpha_4 = \pi$ , and $l_1 = l_2+l_4 = l_5$ ; (10) $\alpha_2 + 2\alpha_5 = 2\pi$ , $\alpha_3 + 2\alpha_4 = 2\pi$ , and $l_1=l_2=l_3=l_4$ ; (11) $\alpha_1 = \frac{\pi}{2}$ , $\alpha_3+\alpha_5 = \pi$ , $2\alpha_2+\alpha_3 = 2\pi$ , and $2l_1 + l_3 = l_4 = l_5$ ; (12) $\alpha_1 = \frac{\pi}{2}$ , $\alpha_3+\alpha_5 = \pi$ , $2\alpha_2+\alpha_3 = 2\pi$ , and $2l_1 = l_3+l_5 = l_4$ ; (13) $\alpha_1 = \alpha_3 = \frac{\pi}{2}$ , $2\alpha_2 + \alpha_4 = 2\alpha_5 + \alpha_4 = 2\pi$ , $l_3 = l_4$ , and $2l_3 = l_5$ ; (14) $\alpha_1 = \frac{\pi}{2}$ , $2\alpha_2 + \alpha_3 =2\pi$ , $\alpha_3 + \alpha_5 = \pi$ , and $2l_1 = 2l_3 = l_4 = l_5$ ; (15) $\alpha_1 = \frac{\pi}{3}$ , $\alpha_2 = \frac{3\pi}{4}$ , $\alpha_3 = \frac{7\pi}{12}$ , $\alpha_4 = \frac{\pi}{2}$ , $\alpha_5=\frac{5\pi}{6}$ , and $l_1 = 2l_2 = 2l_4 = 2l_5$ .

Short context: In the 1910s, Bieberbach suggested to determine all the convex domains which can tile the whole plane. It is easy to see that if convex set P tiles the plane that P must actually be a convex m-gon, $m\geq 3$ . It is also not difficult to verify that any triangle and any convex quadrilateral can tile the plane. In 1918, Reinhardt proved that no convex m-gon with $m>6$ can tile the plane, and a convex hexagon can tile the plane if and only if it belongs to one of the following three types: (i) $\alpha_1+\alpha_2+\alpha_3=2\pi$ , and $l_1 = l_4$ ; (ii) $\alpha_1 + \alpha_2 + \alpha_4 = 2\pi$ , $l_1 = l_4$ , and $l_3 = l_5$ ; (iii) $\alpha_1 = \alpha_3 = \alpha_5 = \frac{2}{3}\pi$ , $l_1 = l_2$ , $l_3 = l_4$ , and $l_5 = l_6.$ After this, only the case of pentagons remained open. From 1918 to 2015, various researchers discovered the 15 types of pentagons that can tile the plane. The Theorem proves that this list is complete, and thus finishes the solution to the Bieberbach’s problem.

Links: Free arxiv version of the original paper is here.

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There is no coarsely minimal infinite-dimensional Banach space

You need to know: Metric space, Banach space, infinite dimensional Banach space.

Background: We say that metric space X coarsely embeds into metric space Y if there is a function $f:X\to Y$ and non-decreasing functions $\rho_1, \rho_2 : [0,\infty)\to[0,\infty)$ such that (i) $\rho_1(d_X(x, y)) \leq d_Y (f(x), f(y)) \leq \rho_2(d_X(x, y))$ for all $x, y \in X$ , and $\lim\limits_{t\to\infty}\rho_1(t)=+\infty$ . A Banach space is called coarsely minimal if it coarsely embeds into every infinite-dimensional Banach space.

The Theorem: On 18th May 2017, Florent Baudier, Gilles Lancien, and Thomas Schlumprecht submitted to arxiv and the Journal of the AMS a paper in which they proved that there is no coarsely minimal infinite-dimensional Banach space

Short context: There was an important open problem asking weather the $l^2$ -space (the space of infinite sequences $x=(x_1, x_2, \dots, x_n, \dots)$ equipped with norm $|x| := \sqrt{\sum\limits_{i=1}^\infty x_i^2} < \infty$ ) is coarsely minimal. There were some evidence and partial results supporting that the answer may be positive. It is would be so, this would imply the solution of some other important problems in the field. The Theorem, however, states that the answer is negative in a strong sence: not only $l^2$ is not coarsely minimal, but, moreover, there are no coarsely minimal infinite-dimensional Banach spaces at all.

Links: Free arxiv version of the original paper is here, journal version is here.

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There are CN^(2p-4)(1+o(1)) meanders with at most 2N crossings and p minimal arcs

You need to know: Simple closed curve in the plane, transversal intersection of a curve and a line, topological configuration, binomial coefficient ${n\choose k}=\frac{n!}{k!(n-k)!}$ , small o notation.

Background: A meander is a topological configuration of a line and a simple closed curve in the plane intersecting transversally. We called crossings the points at which curve and lines intersects. There is always an even number of crossings which we denote $2N$ . An arc is the part AB of the curve between crossings A and B which do not contain any other crossings. If the interval AB on the line also does not contain any other crossings, the arc AB is called minimal. Let $M_p(N)$ be the number of meanders with exactly p minimal arcs and with at most $2N$ crossings.

The Theorem: On 15th May 2017, Vincent Delecroix, Elise Goujard, Peter Zograf, and Anton Zorich submitted to arxiv a paper in which they proved that $M_p(N)=C_pN^{2p-4}+o(N^{2p-4})$ as $N\to\infty$ (with $p\geq 3$ fixed), where $C_p=\frac{2^{p-3}}{\pi^{2p-4}p!(p-2)!}{{2p-2}\choose {p-1}}^2$ .

Short context: Meanders were studied already by Poincaré over 100 year ago, and appear in various areas of mathematics, and in applications, for example, in physics. A long-standing open problem is to derive the precise asymptotic how the number of meanders with $2N$ crossings grows with N. The Theorem solves a version of this problem when we also assume that the number p of minimal arcs is fixed.

Links: Free arxiv version of the original paper is here, journal version is here.

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A version of the OSSS inequality holds for monotonic probability measures

You need to know: Basic probability theory, the notion of algorithm.

Background: Let $[n]=\{1,2,\dots,n\}$ , and let $X=\{0,1\}^n$ be the set of vectors $w=(w_1,\dots,w_n)$ with each $w_i$ being either $0$ or $1$ . Assume that we select each $w\in X$ with some probability ${\mathbb P}(w)\geq 0$ , such that $\sum\limits_{w\in X}{\mathbb P}(w)=1$ . For $A \subseteq X$ , define ${\mathbb P}[A]=\sum\limits_{w\in A}{\mathbb P}(w)$ . For any function $f:X\to {\mathbb R}$ , define expectation $E_{\mathbb P}(f)=\sum\limits_{w\in X}{\mathbb P}(w)f(w)$ and variance $\text{Var}_{\mathbb P}(f)=E_{\mathbb P}(f^2)-E_{\mathbb P}(f)^2$ . For $f,g:X\to {\mathbb R}$ , let $\text{Cov}_{\mathbb P}(f,g)=E_{\mathbb P}(fg)-E_{\mathbb P}(f)E_{\mathbb P}(g)$ . A function $f:X\to{\mathbb R}$ is called increasing if $f(x)\geq f(y)$ whenever $x\geq y$ coordinate-wise. A probability ${\mathbb P}$ is monotonic if for any $i\in[n]$ , any $F\subset [n]$ , and any $x,y \in \{0,1\}^F$ satisfying $x\leq y$ , ${\mathbb P}[w_j=x_j, \, \forall j \in F]>0$ and ${\mathbb P}[w_j=y_j, \, \forall j \in F]>0$ , we have ${\mathbb P}[w_i=1 | w_j=x_j, \, \forall j \in F] \leq {\mathbb P}[w_i=1 | w_j=y_j, \, \forall j \in F]$ .

A decision tree T is an algorithm that takes $w\in X$ as an input, and then tries to determine $f(w)$ by querying the values of $w_i$ , $i\in[n]$ , one bit after the other. The bit it queries on step k may depends on the values of the bits obtained on steps $1,\dots,k-1$ . The algorithm stops when it can determine $f(\omega)$ from the bits it has. Let $\delta_i(f,T)$ be the probability that T, starting from random input $\omega$ , queries bit $w_i$ before it stops.

The Theorem: On 8th May 2017, Hugo Duminil-Copin, Aran Raoufi, and Vincent Tassion submitted to arxiv a paper in which they proved that for every increasing function $f:X\to[0,1]$ , any monotonic probability ${\mathbb P}$ , and any decision tree T, we have $\text{Var}_{\mathbb P}(f) \leq \sum\limits_{i=1}^n \delta_i(f,T)\text{Cov}_{\mathbb P}(f,w_i)$ .

Short context: The Theorem generalises the inequality proved in 2005 by O’Donnell, Saks, Schramm, and Servedio (known as the OSSS inequality) for the case when each bit $w_i$ of $w\in X$ is selected independently. The Theorem allows bit of $w$ to depend on each other, and instead imposes much weaker condition that the probability ${\mathbb P}$ is monotonic. The Theorem has applications in mathematical physics, specifically in the analysis of so-called lattice spin models and their random-cluster representations.

Links: Free arxiv version of the original paper is here, journal version is here.

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Every Besicovitch set in R^3 has dimension at least 5/2+e for some e>0

You need to know: Euclidean space ${\mathbb R}^3$ , unit line segment in ${\mathbb R}^3$ , direction of a line segment, compact sets in ${\mathbb R}^3$ , norm $||x||=\sqrt{x_1^2+x_2^2+x_3^2}$ of $x=(x_1,x_2,x_3)\in {\mathbb R}^3$ , diameter $\text{diam}\,U=\sup\limits_{x,y \in U}||x-y||$ of set $U\subset {\mathbb R}^3$ .

Background: A Besicovitch set in ${\mathbb R}^3$ is a compact set $X\subset {\mathbb R}^3$ that contains a unit line segment pointing in every direction.

For $\delta>0$ , a $\delta$ -cover of set $S \subset {\mathbb R}^3$ is a sequence of sets $U_1, U_2, \dots, U_i, \dots$ of diameters $\text{diam}\,U_i\leq \delta$ for all i such that $S \subset \bigcup\limits_{i=1}^\infty U_i$ . For $d\geq 0$ , let $H_\delta^d(S) = \inf \sum\limits_{i=1}^{\infty}(\text{diam}\,U_i)^d$ , where the infimum is taken over all $\delta$ -covers of S. Let $H^d(S)=\lim\limits_{\delta\to 0} H_\delta^d(S)$ . The Hausdorff dimension of S is $\text{dim}(S)=\inf\{d\geq 0\,:\,H^d(S)=0\}$ .

The Theorem: On 24th April 2017, Nets Katz and Joshua Zahl submitted to arxiv a paper in which they proved the existence of constant $\epsilon_0>0$ , such that every Besicovitch set in ${\mathbb R}^3$ has Hausdorff dimension at least $\frac{5}{2}+\epsilon _0$ .

Short context: The ( ${\mathbb R}^3$ case of) famous Kakeya conjecture states that every Besicovitch set in ${\mathbb R}^3$ must have Hausdorff dimension equal to 3. In 1995, Wolff proved that every such set must have Hausdorff dimension at least $\frac{5}{2}$ . For about 22 years, noone was able to improve the Wolff’s lower bound. The Theorem provides a small improvement.

Links: Free arxiv version of the original paper is here, journal version is here.

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Random multiplicative functions exhibit better than square root cancellation

You need to know: Basic probability theory, independent random variables, selection uniformly at random. Notations: ${\mathbb N}$ for the set of positive integers, ${\mathbb C}$ for the set of complex numbers, $|z|$ for the absolute value of complex number z, $\sum\limits_{n\leq x}$ for the sum over positive integers up to x, ${\mathbb E}$ for the expectation, small o notation.

Background: A function $f:{\mathbb N} \to {\mathbb C}$ is called completely multiplicative, if $f(1)=1$ and $f(x \cdot y) = f(x) \cdot f(y)$ for all positive integers x, y. To define such a function, it suffices to define $f(p)$ for primes p. We say that completely multiplicative $f$ is (Steinhaus) random is values $f(p)$ are selected independently, uniformly at random from the unit circle $\{z \in {\mathbb C}:\,|z|=1\}$ . For $0\leq q \leq 1$ , define $g_q(x)=\left(\frac{x}{1+(1-q)\sqrt{\log\log x}}\right)^q$ .

The Theorem: On 20th March 2017, Adam Harper submitted to arxiv a paper in which he proved the existence of positive constants $c,C$ , and $x_0$ , such that Steinhaus random multiplicative function f satisfies $c g_q(x) \leq {\mathbb E}\left|\sum\limits_{n\leq x}f(n)\right|^{2q} \leq Cg_q(x)$ for all $0\leq q \leq 1$ and all $x\geq x_0$ .

Short context: Many functions of central importance in number theory are multiplicative. In many applications, it is important to estimate moments of such functions, that is, expressions of the form $\left|\sum\limits_{n\leq x}f(n)\right|^{2q}$ . The Theorem estimates (up to a constant factor) the moments of a typical multiplicative function. With $q=1/2$ , it implies that $c\frac{\sqrt{x}}{(\log \log x)^{1/4}} \leq {\mathbb E}\left|\sum\limits_{n\leq x}f(n)\right|\leq C\frac{\sqrt{x}}{(\log \log x)^{1/4}}$ . This confirms a conjecture of Helson, who predicted that ${\mathbb E}\left|\sum\limits_{n\leq x}f(n)\right| = o(\sqrt{x})$ . In such cases we say that we have “better than square root cancellation”.

Links: Free arxiv version of the original paper is here, journal version is here.

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The Dichotomy conjecture is true: every CSP is either in P or is NP-complete

You need to know: Algorithm, running time of an algorithm, class P, class NP, NP-complete problems.

Background: Let $\Sigma$ be a finite set. Let ${\cal X}=\{x_1, \dots x_n\}$ be a set of n variables taking values in $\Sigma$ . An assignment $a=(a_1,\dots,a_n) \in \Sigma^n$ means that we assign to each variable $x_i$ the value $a_i$ . A relation R is a subset of $\Sigma^k$ , where $k$ is a positive integer. A constraint $C$ (corresponding to R) is a k-element subset $x_{i_1}, \dots x_{i_k}$ of ${\cal X}$ . We say that assignment $a\in \Sigma^n$ satisfies the constraint $C$ if $(a_{i_1}, \dots a_{i_k})\in R$ . Let $\Gamma$ be a finite set of relations. A constraint satisfaction problem $\text{CSP}(\Gamma)$ is: given $\Sigma$ , ${\cal X}$ , finite set $R_1, \dots, R_m$ of relations such that each $R_i \in \Gamma$ , and a finite set $C_1, C_2, \dots, C_t$ of constraints (each constraint $C_j$ corresponds to some relation $R_i$ ), is there an assignment $a\in \Sigma^n$ that satisfies all the constraints?

The Theorem: On 8th March 2017, Andrei Bulatov submitted to arxiv a paper in which he proved that, for every fixed $\Gamma$ , the problem $\text{CSP}(\Gamma)$ is either solvable in polynomial time or is NP-complete.

Short context: In 1975, Ladner proved that, if $P\neq NP$ , then there are problems in NP which are neither in P not NP-complete. However, the problems constructed in the proof of this theorem are rather artificial. In practise, most “natural” problems are either in P or NP-complete. Formalising this observation, Feder and Vardi in 1993 noticed that a large variety of “natural” problems are the special cases of $\text{CSP}(\Gamma)$ , and conjectured that in fact every $\text{CSP}(\Gamma)$ is either in P or NP-complete. This conjecture became known as the Dichotomy conjecture. The Theorem confirms this conjecture. In a well defined sense, the class of problems $\text{CSP}(\Gamma)$ is the largest subclass of NP in which such a dichotomy holds.

Links: Free arxiv version of the original paper is here.

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The set of congruent numbers equal to 1, 2, or 3 mod 8 has zero natural density in N

You need to know: Notations: ${\mathbb N}$ for the set of positive integers, $|S|$ for the number of elements in finite set S.

Background: We say that subset $S\subseteq{\mathbb N}$ has zero natural density in ${\mathbb N}$ if $\lim\limits_{n\to \infty}\frac{|S\cap\{1,2,\dots,n\}|}{n}=0$ . A positive integer $k\in {\mathbb N}$ is called a congruent number if it is the area of some right triangle with rational side lengths. In other words, $k=\frac{1}{2}ab$ for some rational numbers $a,b$ such that $\sqrt{a^2+b^2}$ is also a rational number. Let ${\cal C}_{123}$ be the set of all congruent numbers $k\in {\mathbb N}$ such that either $k-1$ , or $k-2$ , or $k-3$ is divisible by 8.

The Theorem: On 8th February 2017, Alexander Smith submitted to arxiv a paper in which he proved that the set ${\cal C}_{123}$ has zero natural dansity in ${\mathbb N}$ .

Short context: Determining whether or not a given number is congruent is called the congruent number problem and is one of the oldest unsolved problem of mathematics with over 1000 years history. We can also ask what proportion of positive integers are congruent numbers? In a paper submitted in 2016, Smith proved that the set of all congruent numbers have positive natural density. The Theorem states that the situation is completely different if we look only at numbers equal to 1, 2, or 3 mod 8.

Links: Free arxiv version of the original paper is here.

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Elliptic curve with discriminant D has at most O(|D|^(0.1117…+e)) integer points

You need to know: Notations: ${\mathbb Z}$ for the set of integers, ${\mathbb Q}$ for the set of rational numbers.

Background: Consider equation $y^2 = x^3+ax+b$ , where $a,b \in {\mathbb Z}$ are such that $\Delta_{a,b}=-16(4a^3+27b^2)\neq 0$ . A theorem of Siegel implies that this equation has only finitely many integer solutions $x,y \in {\mathbb Z}$ . Denote $N_{a,b}$ the number of such solutions.

The Theorem: On 10th January 2017, Manjul Bhargava, Arul Shankar, Takashi Taniguchi, Frank Thorne, Jacob Tsimerman, and Yongqiang Zhao submitted to arxiv a paper in which they proved that for every sufficiently small $\epsilon>0$ there is a constant $C_\epsilon<\infty$ , such that $N_{a,b} \leq C_\epsilon |\Delta_{a,b}|^{\beta+\epsilon}$ , where $\beta=0.1117...$ is an explicit constant.

Short context: The equation $y^2 = x^3+ax+b$ , with $a,b \in {\mathbb Z}$ and $\Delta_{a,b}=-16(4a^3+27b^2)\neq 0$ is called non-singular elliptic curve over ${\mathbb Q}$ in Weierstrass form with integer coefficients. Studying integer and rational points on such curve (that is, integer and rational solutions of the equation) is one of the important research directions in number theory. In 2006, Helfgott and Venkatesh proved that $N_{a,b} \leq C_\epsilon |\Delta_{a,b}|^{0.2007...+\epsilon}$ . The Theorem improves this bound significantly.

Links: Free arxiv version of the original paper is here, journal version is here.

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