The Furstenberg’s conjecture on dimension of sum of invariant sets in true

You need to know: Closed subset of $[0,1]$ , multiplication of a set by a constant $s\cdot A = \{s\cdot a: a \in A\}$ , sum of two sets $A + B = \{a + b : a \in A; b \in B\}$ , Hausdorff dimension $\text{dim}(A)$ of set $A \subset [0,1]$ .

Background: For real number $x$ , let $\left \lfloor{x}\right \rfloor$ be the largest integer not exceeding x, and let $\text{frac}(x)=x-\left \lfloor{x}\right \rfloor$ . For integer m, let $T_m:[0,1)\to[0,1)$ be a function given by $T_m(x)=\text{frac}(mx)$ . We say that set $S \subset [0,1]$ is invariant under $T_m$ if $T_m(x) \in S$ for every $x \in S$ .

The Theorem: On 11th October 2009, Michael Hochman and Pablo Shmerkin submitted to arxiv a paper in which they proved the following result. Let $X; Y \subseteq [0,1]$ be closed sets which are invariant under $T_2$ and $T_3$ , respectively. Then, for any $s \neq 0$ , $\text{dim}(X + s\cdot Y ) = \min\{1, \text{dim} X + \text{dim} Y\}$ .

Short context: The Theorem confirms a long-standing conjecture of Furstenberg made in late 1960th. See here for a theorem resolving another related conjecture of Furstenberg in a later work.

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

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Real polynomial interval maps with only real critical points have connected isentropes

You need to know: Eucledean space ${\mathbb R}^n$ , connected subset of ${\mathbb R}^n$ , polynomial, critical point of a polynomial, non-degenerate critical point, notation $f^n$ for the n-th iterate of f.

Background: Let $I=[a,b]$ be an interval in ${\mathbb R}$ , and let $\partial I = \{a,b\}$ be the set of its endpoints. Let $P^d$ be the set of real polynomials of degree d+1 mapping I (and $\partial I$ ) into itself, with precisely d non-degenerate critical points, each of which is contained in the interior of I. For $\epsilon=\pm 1$ , let $P^d_\epsilon \subset P^d$ be the set of polynomials which are increasing (respectively decreasing) at the left endpoint of I when $\epsilon=1$ (respectively $\epsilon=-1$ ). For each polynomial $P(x)=\sum\limits_{i=0}^{d+1}a_ix^i$ one can associate a point $a(P) = (a_0, \dots, a_{d+1})\in {\mathbb R}^{d+2}$ . A subset $S \subseteq P^d_\epsilon$ is called connected if $\{a(P), P \in S\}$ is a connected subset of ${\mathbb R}^{d+2}$ .

Given $f:I \to I$ , assume that I can be decomposed into finitely many subintervals $I_0,\dots, I_m$ on which f is monotone. The smallest number $m+1$ of such intervals is called the lap number $l(f)$ of f. The topological entropy $h_{top}(f)$ of f is $h_{top}(f)=\lim\limits_{n\to\infty}\frac{1}{n}\log l(f^n)$ .

The Theorem: On 20th May 2005, Henk Bruin and Sebastian van Strien submitted to arxiv and the Journal of the AMS a paper in which they proved that, for $\epsilon \in \{-1,+1\}$ , each positive integer d, and each $h_0 \geq 0$ , the set $\{f \in P^d_\epsilon: h_{top}(f) = h_0\}$ (called the isentrope) is connected.

Short context: The lap number $l(f)$ is one of the ways to measure how “complex” is f, and the topological entropy $h_{top}(f)$ measures the “dynamic complexity” of iterates of f. The isentrope $\{f \in P^d_\epsilon: h_{top}(f) = h_0\}$ is the set of polynomials with fixed dynamic complexity. In 1992, Milnor conjectured that this set is connected. Since then, the conjecture was extensively studied and proved for quadratic and cubic polynomials. The Theorem proves the conjecture in full.

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

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The product of typical interval exchange transformations is uniquely ergodic

You need to know: Direct product $\times$ of sets, Borel probability measure on interval $I=[0,1)$ or unit square $I \times I$ . Also, see this previous theorem description for the notion of interval exchange transformation (IET) on I, and explanation what is meant by “almost every” IET.

Background: The product of maps $f:I \to I$ and $g:I \to I$ is map $f \times g: I \times I \to I \times I$ given by $(f \times g)(x,y)=(f(x), g(y))$ . Let J be either interval $I=[0,1)$ or unit square $I \times I$ . A map $f:J \to J$ is called uniquely ergodic if there exists a unique Borel probability measure $\mu$ on J such that $\mu(f^{-1}(A))=\mu(A)$ for all measurable $A \subset J$ .

The Theorem: On 14th May 2005, Jon Chaika submitted to arxiv a paper in which he proved that, for almost every pair $f,g$ of IETs their product $f \times g$ is uniquely ergodic.

Short context: Interval exchange transformations (IETs in short) are basic examples of measure-preserving transformations $f:I \to I$ (that is, such that $\mu(f^{-1}(A))=\mu(A)$ for all measurable $A \subset I$ ), which are central objects of study in dynamical systems. The fundamental work of Masur and Veech established that almost every IET is uniquely ergodic. The Theorem extends this result to the product of almost every IETs.

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

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The Bennett–Carbery–Tao multilinear Kakeya conjecture is true

You need to know: Euclidean space ${\mathbb R}^n$ , lines in ${\mathbb R}^n$ , volume $\text{Vol}(S)$ of $S\subset {\mathbb R}^n$ . In addition, you need the concept of Hausdorff dimension of subsets of ${\mathbb R}^n$ for the context section.

Background: A cylinder of radius R around a line $L \subset {\mathbb R}^n$ is the set of all points $x \in {\mathbb R}^n$ within a distance R of the line L. The line L is called the core of the cylinder. Suppose that we have a finite collection of cylinders $T_{j,a}\subset{\mathbb R}^n$ , where $1 \leq j \leq n$ , and $1\leq a \leq A$ for some integer A. Each cylinder $T_{j,a}$ has radius 1 and the angle between the core of $T_{j,a}$ and the $x_j$ -axis is at most $(100n)^{-1}$ . Let $I=\bigcap\limits_{j=1}^{n}\bigcup\limits_{a=1}^{A}T_{j,a}$ be the set of points that belong to at least one cylinder in each direction.

The Theorem: On 14th November 2008, Larry Guth submitted to arxiv a paper in which he proved that for each dimension n there is a constant $C=C(n)$ such that $\text{Vol}(I) \leq CA^{n/(n-1)}$ for any collection of cylinders as above.

Short context: Famous Kakeya conjecture states that every compact set $K \subset {\mathbb R}^n$ , which contains a line segment of unit length in every direction, must have Hausdorff dimension equal to n. This conjecture is important but difficult, and researchers study various “easier” versions first. In 1999, Wolff suggested to study its finite field analogue, which is then proved by Dvir, see here. In 2006, Bennett, Carbery, and Tao formulated “mutlilinear” analogue of the Kakeya conjecture, which says that cylinders pointing in different directions cannot overlap too much. The Theorem (more exactly, its generalisation proved by Guth in the same paper) finishes the proof of this conjecture.

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

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Centered Hardy–Littlewood maximal inequality in R^d has no uniform bound

You need to know: Euclidean space ${\mathbb R}^d$ , integration in ${\mathbb R}^d$ , supremum $\sup$ , notation $|V|$ for the volume (Lebesgue measure) of set $V \subset {\mathbb R}^d$ .

Background: For $x=(x_1,\dots,x_d) \in {\mathbb R}^d$ , denote $||x||_\infty=\max\limits_{1\leq i \leq d}|x_i|$ . For $x\in {\mathbb R}^d$ and $r>0$ , let $Q(x,r)=\{y\in {\mathbb R}^d : ||y-x||_\infty \leq r\}$ . Let ${\cal L}^1({\mathbb R}^d)$ be the set of all functions $f:{\mathbb R}^d\to {\mathbb R}$ such that $\|f\|_1=\int_{{\mathbb R}^d} |f(x)| dx <\infty$ . For any $f \in {\cal L}^1({\mathbb R}^d)$ , let $M_f (x) = \sup\limits_{r>0}\frac{1}{|Q(x,r)|}\int_{Q(x,r)} |f(y)| dy$ . Let $c_d$ be the lowest constant $c$ such that inequality $\alpha|\{x\in {\mathbb R}^d : M_f(x) > \alpha\}| \leq c \|f\|_1$ holds for all ${\cal L}^1({\mathbb R}^d)$ and all $\alpha>0$ .

The Theorem: On 11th May 2008, Jesus Aldaz submitted to arxiv a paper in which he proved that $\lim\limits_{d\to\infty}c_d = \infty$ .

Short context: Inequality $\alpha|\{x\in {\mathbb R}^d : M_f(x) > \alpha\}| \leq c \|f\|_1$ is known as the centered Hardy–Littlewood maximal inequality for cubes in ${\mathbb R}^d$ , and has various applications in the theories of differentiation and integration. In 1983, Stein and Strömberg asked if it holds with the uniform bound, that is, with some constant c independent of the dimension. The Theorem gives negative answer to this question. Before 2008, it was only known that $c_1=\frac{11+\sqrt{61}}{12}$ and that $c_d \geq \left(\frac{1+2^{1/d}}{2}\right)^d$ for all d (note that $\left(\frac{1+2^{1/d}}{2}\right)^d<2$ for all d).

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

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De Giorgi’s conjecture fails in dimensions at least 9

You need to know: Euclidean space ${\mathbb R}^n$ , hyperplane in ${\mathbb R}^n$ , function $u:{\mathbb R}^n \to {\mathbb R}$ (function in n variables), partial derivatives of such function, differential equation.

Background: A level set of a function $u:{\mathbb R}^n \to {\mathbb R}$ is a set of the form $L_c(f)=\{(x_1, \dots, x_n)\,|\,u(x_1, \dots, x_n)=c\}$ for some constant $c\in{\mathbb R}$ . For a twice-differentiable function $u:{\mathbb R}^n \to {\mathbb R}$ , its Laplacian $\Delta u$ is $\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \dots + \frac{\partial^2 u}{\partial x_n^2}$ .

The Theorem: On 23rd January 2008, Manuel del Pino, Michał Kowalczyk, and Juncheng Wei submitted to the Annals of Mathematics a paper in which they proved that, for any $n\geq 9$ , there exists a solution $u:{\mathbb R}^n \to {\mathbb R}$ to differential equation $\Delta u=u^3-u$ such that (i) $|u|<1$ ; (ii) $\frac{\partial u}{\partial x_n}>0$ for every $x=(x_1, \dots, x_n)\in{\mathbb R}^n$ ; but (iii) the level sets of u are not hyperplanes.

Short context: Differential equation $\Delta u=u^3-u$ originates in the theory of phase transition, is important and well-studied. In 1978, De Giorgi conjectured that if $u:{\mathbb R}^n \to {\mathbb R}$ is a solution to this equation satisfying (i) and (ii), then all level sets of u are hyperplanes, at least if $n \leq 8$ . This conjecture, when true, allows to write down a formula for u easily. It is known to hold in dimensions $n\leq 3$ , and, under some additional conditions, in all dimensions $n \leq 8$ , see here. The Theorem proves that it fails in all dimensions $n \geq 9$ .

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

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First moment concentration of log-concave measures implies exponential concentration

You need to know: Euclidean space ${\mathbb R}^n$ , norm $\|x\|$ of $x\in{\mathbb R}^n$ , convex set $D\subset {\mathbb R}^n$ , convex function on D, probability measure $\mu$ on ${\mathbb R}^n$ , integration with respect to $\mu$ .

Background: A probability measure $\mu$ on ${\mathbb R}^n$ is called absolutely continuous if there exists function $g:{\mathbb R}^n \to {\mathbb R}$ such that $\mu(A)=\int_A g(x)dx$ for every measurable $A\subset {\mathbb R}^n$ . If set $D=\{x\in{\mathbb R}^n\,|\,g(x)>0\}$ is convex, and $-\log(g(x))$ is a convex function on D, then $\mu$ is called log-concave. Given a function $f:{\mathbb R}^n \to {\mathbb R}$ , we write $E_\mu(f)=\int_{{\mathbb R}^n}f d\mu$ , and $||f||_{L^1(\mu)} = E_\mu(|f|)$ . Function f is called 1-Lipschitz if $|f(x)-f(y)| \leq \|x-y\|$ for all $x,y \in {\mathbb R}^n$ . We say that $\mu$ satisfies First-Moment concentration if there exists $D>0$ such that for every 1-Lipschitz f we have $||f-E_\mu(f)||_{L^1(\mu)} \leq \frac{1}{D}$ .

The Theorem: On 26th December 2007, Emanuel Milman submitted to arxiv a paper in which he proved that for every absolutely continuous log-concave probability measure $\mu$ on ${\mathbb R}^n$ satisfying First-Moment concentration, and every 1-Lipschitz $f:{\mathbb R}^n \to {\mathbb R}$ , inequality $\mu(|f-E_\mu(f)| \geq t) \leq c \exp(-CDt)$ holds for all $t>0$ , where c and C are universal constants.

Short context: Let $x\in{\mathbb R}^n$ be selected at random with respect to measure $\mu$ , and f be some useful function we want to compute. In general, because x is random, $f(x)$ is also random and unpredictable. However, if the conclusion of the theorem (known as exponential concentration) holds, then $f(x) \approx E_\mu(f)$ with very high probability. This property looks much stronger than “just” first moment concentration. The Theorem states, however, that these concentrations are in fact equivalent! In fact, in the same paper authors studied several other notions of concentrations, some looks very week and some very strong, and proved that they are all equivalent for the log-concave $\mu$ .

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

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The Hausdorff dimension of the set of singular pairs is 4/3

You need to know: Euclidean space ${\mathbb R}^2$ , norm $||{\bf x}||=\sqrt{x_1^2+x_2^2}$ of vector ${\bf x}=(x_1, x_2) \in {\mathbb R}^2$ , Hausdorff dimension of a subset of ${\mathbb R}^2$ .

Background: Vector ${\bf x}=(x_1, x_2) \in {\mathbb R}^2$ is called singular pair if for every $\delta > 0$ there exists $T_0$ such that for all $T>T_0$ there exist vector ${\bf p}=(p_1, p_2)$ with integer coordinates and integer $q$ , $0<q<T$ , such that $||q{\bf x}-{\bf p}||<\frac{\delta}{\sqrt{T}}$ .

The Theorem: On 27th September 2007, Yitwah Cheung submitted to the Annals of Mathematics a paper in which they proved that the set of all singular pairs in ${\mathbb R}^2$ has Hausdorff dimension $\frac{4}{3}$ .

Short context: Singular pair is a pair of real numbers $(x_1,x_2)$ that can be simultaneously approximated by rational numbers $\frac{p_1}{q}, \frac{p_2}{q}$ with the same not too large denominator, see here for a progress in a related Littlewood conjecture. All points on any line $a_1x_1+a_2x_2+b=0$ with rational coefficients $a_1, a_2, b$ are known to be singular pairs. The set of points on such lines has Hausdorff dimension $1$ . The Theorem shows that the set of all singular pairs is much large.

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

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Every triangle with all angles at most 100 degrees has a periodic billiard path

You need to know: Polygon, angles measured in radians, angle of incidence, angle of reflection, limits, Hausdorff dimension of a set.

Background: A billiard path in a triangle T is a point moving inside T with constant speed, with the usual rule that the angle of incidence equals the angle of reflection. Such a path is fully described by point’s initial position and initial direction to move. A billiard path is called periodic if, at some time moment, the point trajectory starts to repeat itself.

The Theorem: On 8th September 2007, Richard Evan Schwartz submitted to the Experimental mathematics a paper in which he proved that every triangle with all angles at most one hundred degrees has a periodic billiard path.

Short context: Billiard path is a simple but fundamental example of a dynamical system. More that two-hundred-year-old triangular billiards conjecture predicts that every triangle has a periodic billiard path. The conjecture is relatively easy for rational triangles (those whose angles are all rational multiples of $\pi$ ), and also for right triangles. In 1775, Fagnano proved it for acute triangles. However, there was essentially no progress for general (possibly not rational) obtuse triangles. The Theorem proves the conjecture for all triangles with largest angle at most 100 degrees.

Links: The original paper is available here.

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Nonlinear complex same-degree polynomials with infinite orbit intersection must have a common iterate

You need to know: Complex numbers, set ${\mathbb C}[X]$ of polynomials $f(x)$ in complex variable $x$ with complex coefficients, degree $\text{deg}(f)$ of a polynomial f, notation $f^n(z)$ for $f(f(\dots f(z)\dots))$ , where f is repeated n times.

Background: For $f \in {\mathbb C}[X]$ , and initial point $x_0 \in {\mathbb C}$ , the orbit $O_f(x_0)$ is the set $\{x_0, f(x_0), f(f(x_0)), \dots, f^n(x_0), \dots\}$ . We say that same degree polynomials $f, g \in {\mathbb C}[X]$ have a common iterate if latex $f^n = g^n$ for some n.

The Theorem: On 14th May 2007, Dragos Ghioca, Thomas Tucker, and Michael Zieve submitted to arxiv a paper in which they proved the following result. Let $x_0, y_0 \in {\mathbb C}$ and $f, g \in {\mathbb C}[X]$ with $\text{deg}(f) = \text{deg}(g) > 1$ . If $O_f(x_0) \cap O_g(y_0)$ is infinite, then f and g have a common iterate.

Short context: The study of orbits of polynomial maps is one of the main topics in complex dynamics. One natural question to ask is under what conditions two orbits may have infinite intersections. This may obviously be the case if the polynomials have the common iterate. The Theorem states that, for same-degree non-linear polynomials, this obvious sufficient condition is in fact necessary. The Theorem has applications in arithmetic geometry.

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

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