Polynomials small at N equidistant points in (-1,1) are small on subinterval

You need to know: Polynomial, degree of a polynomial, inverse tangent function $\arctan(x)$ .

Background: Let $P_{n,N}$ be the set of polynomials P of degree at most n with $|P(x_k)|\leq 1, k=1,2,\dots,N$ , where $x_k = -1 + (2k-1)/N, \, k=1,2,\dots,N$ is the sequence of N equidistant points on $(-1,1)$ . Let $K_{n,N}(x) = \max\limits_{P\in P_{n,N}}|P(x)|$ .

The Theorem: On 22nd November 2002, Evguenii Rakhmanov submitted to the Annals of Mathematics a paper in which he proved the existence of constant C such that inequality $K_{N,n}(x) \leq C \log \frac{\pi}{\arctan \left(\frac{N}{n}\sqrt{r^2-x^2}\right)}$ holds for all $n<N$ and all $x\in(-r,r)$ , where $r=\sqrt{1-\frac{n^2}{N^2}}$ .

Short context: The Theorem implies that if a degree n polynomial is bounded at N pre-specified discrete points then its maximal possible absolute value $K_{N,n}(x)$ is bounded on any compact subinterval of $(-r,r)$ . The bound is essentially sharp. It is useful in approximation theory, where we approximate a function by a polynomial at some discrete points and need to prove that this approximation works well on some interval.

Links: The original paper is available here. See also Section 7.1 of this book for an accessible description of the Theorem.

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A version of Banach’s fixed point theorem holds for partially ordered sets

You need to know: Complete metric space X, continuous map $f:X \to X$ , notation $f^n(x)$ for $f(f(\dots f(x)\dots))$ with f repeated n times, partially ordered set.

Background: Let $(T, \leq)$ be a partially ordered set. A map $f:T \to T$ is called monotone if it is either order-preserving ( $x \leq y$ implies $f(x) \leq f(y)$ ) or order-reversing ( $x \leq y$ implies $f(y) \leq f(x)$ ). A point $x\in T$ is called a fixed point of f if $f(x)=x$ .

The Theorem: On 19th June 2002, André Ran and Martine Reurings submitted to the Proceedings of the AMS a paper in which they proved the following result. Let T be a partially ordered set such that for every pair $x,y \in T$ there exist $u,v \in T$ such that $u\leq x \leq v$ and $u\leq y \leq v$ . Furthermore, let d be a metric on T such that $(T, d)$ is a complete metric space. If $f:T \to T$ is a continuous, monotone map such that (i) $\exists c\in(0,1): d(f(x),f(y))\leq c\cdot d(x,y), \, \forall x \geq y$ , and (ii) $\exists x_0\in T:\, x_0 \leq f(x_0) \,\text{or}\,x_0 \geq f(x_0)$ , then f has a unique fixed point $x^*$ . Moreover, $\lim\limits_{n\to\infty} f^n(x) = x^*$ for every $x\in T$ .

Short context: The 1922 Banach’s Fixed Point Theorem states that, if $(X,d)$ is a non-empty complete metric space and $f:X\to X$ is such that (*) $d(f(x),f(y))\leq c\cdot d(x,y), \, \forall x,y$ for some $c\in(0,1)$ , then f has a unique fixed point $x^*$ , and $\lim\limits_{n\to\infty} f^n(x) = x^*$ for every $x\in X$ . It is a fundamental result in mathematics with countless applications. The Theorem proves a version of it for partially ordered sets, in which condition (*) is required to hold only for ordered pairs $x,y$ . It has applications to, for example, matrix equations.

Links: The original paper is available here.

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Ergodic averages, taken along cubes whose sizes tend to infinity, converge in L^2

You need to know: Set ${\mathbb Z}^k$ of vectors $x=(x_1,\dots,x_k)$ with integer coordinates, addition in ${\mathbb Z}^k$ , set ${\mathbb N}$ of natural numbers, $\limsup$ notation.

Background: The upper density $d(A)$ of set $A \subset {\mathbb N}$ is $d(A)=\limsup\limits_{N\to \infty} \frac{1}{N}|A \cap \{1,2,\dots,N\}|$ . For any $A \subset {\mathbb Z}^k$ and $x\in {\mathbb Z}^k$ the translate $A+x$ is $\{y: \, y=a+x, \, a\in A \}$ . Let $t(A)$ be the minimal number of translates of A needed to fully cover ${\mathbb Z}^k$ . Set A is called syndetic if $t(A)<\infty$ . Also, denote $V_k \subset {\mathbb Z}^k$ the set of vectors $x=(x_1,\dots,x_k)$ with all coordinates $0$ or $1$ .

The Theorem: On 16th June 2002, Bernard Host and Bryna Kra submitted to the Annals of Mathematics a paper in which they proved that for any $A \subset {\mathbb N}$ with $d(A) > \delta > 0$ and integer $k \geq 1$ , the set of $n=(n_1, n_2, \dots, n_k) \in {\mathbb Z}^k$ such that $d\left(\bigcap\limits_{\epsilon\in V_k}\left(A + \sum\limits_{i=1}^k\epsilon_i n_i\right)\right) \geq \delta^{2^k}$ is syndetic.

Short context: The Theorem, as stated above, is a combinatorial reformulation of a deep theorem is the field of ergodic theory, which establishes $L^2$ -convergence of so-called “ergodic averages taken along cubes whose sizes tend to infinity”. The details of this original formulation are too difficult to be presented here.

Links: The original paper is available here. See also Section 5.1 of this book for an accessible description of the Theorem.

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The covering number of a uniformly bounded set of functions is exponential in its shattering dimension

You need to know: Probabilty measure $\mu$ on set $\Omega$ , norm $L_2(\mu)$ for functions on $\Omega$ .

Background: Let $\Omega$ be a set, and let $\mu$ be a probability measure on $\Omega$ . Let $B_1$ be the set of all functions $f:\Omega \to [-1,1]$ , and let A be any subset of $B_1$ . Denote $N(A,t,\mu)$ the covering number of A, that is, the minimal number of functions whose linear combination can approximate any function in A within an error t in the $L_2(\mu)$ norm.

We say that a subset $\sigma$ of $\Omega$ is t-shattered by A if there exists a function h on $\sigma$ such that, given any decomposition $\sigma=\sigma_1 \cup \sigma_2$ with $\sigma_1 \cap \sigma_2 = \emptyset$ , one can find a function $f \in A$ with $f(x) \leq h(x)$ if $x \in \sigma_1$ but $f(x) \geq h(x) + t$ if $x \in \sigma_2$ . The shattering (or combinatorial) dimension $vc(A,t)$ of A is the maximal cardinality of a set t-shattered by A.

The Theorem: On 10th December 2001, Shahar Mendelson and Roman Vershynin submitted to Inventiones Mathematicae a paper in which they proved that $N(A, t,\mu) \leq \left(\frac{2}{t}\right)^{K \cdot vc(A,ct)}, \, 0<t<1,$ where K and c are positive absolute constants.

Short context: It is well-known that the covering numbers of a set are exponential in its linear algebraic dimension. The Theorem extends this result to shattering dimension, solving so-called Talagrand’s entropy problem. This result is especially useful because shattering dimension never exceeds linear algebraic dimension, and is often much smaller than it.

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

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The analytic capacity is semiadditive

You need to know: Compact set, notation $A\setminus B = \{x: x \in A, \, x\not\in B\}$ for set difference, $\sup$ notation for supremum, complex plane ${\mathbb C}$ , absolute value $|.|$ of a complex number, analytic function, notation $f(\infty)$ for $\lim\limits_{z\to\infty} f(z)$ .

Background: The analytic capacity of a compact subset E of complex plane ${\mathbb C}$ is defined as $\gamma(E) = \sup|f'(\infty)|$ where the supremum is taken over all analytic functions $f:C\setminus E \to C$ such that $|f|\leq 1$ on $C\setminus E$ , and $f'(\infty)=\lim\limits_{z\to\infty} z(f(z)-f(\infty))$ .

The Theorem: On 13th September 2001, Xavier Tolsa submitted to Acta Mathematica a paper in which he proved the existence of an absolute constant C such that $\gamma(E \cup F) \leq C(\gamma(E) + \gamma(F))$ for all compact sets E and F. In other words, the analytic capacity is semiadditive.

Short context: The question whether analytic capacity is semiadditive was raised by Vitushkin in 1960’s, who showed that a positive answer to this question would have important applications to the theory of rational approximation. The Theorem answers this question affirmatively. More generally, Tolsa proved that semiadditivity holds for any countable collection of sets $E_i$ with compact union.

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

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The best constant in the centered Hardy–Littlewood maximal inequality

You need to know: Integration, $\sup$ notation, Lebesgue measure.

Background: Let ${\cal L}^1({\mathbb R})$ be the set of all functions $f:{\mathbb R}\to {\mathbb R}$ such that the integral $\int_{-\infty}^{+\infty} |f(x)| dx$ exists and finite. We will denote $\|f\|_1$ the value of this integral. The centered Hardy-Littlewood maximal operator is the map which assigns to any function $f \in {\cal L}^1({\mathbb R})$ a function $M_f (t) = \sup\limits_{\epsilon>0}\frac{1}{2\epsilon}\int_{t-\epsilon}^{t+\epsilon} |f(x)| dx$ . For any $\delta>0$ , denote $|\{M_f > \delta\}|$ the Lebesgue measure of set of real numbers t such that $M_f(t) > \delta$ .

The Theorem: On 16th August 2001, Antonios Melas submitted to the Annals of Mathematics a paper in which he proved that for every $f \in {\cal L}^1({\mathbb R})$ and for every $\delta>0$ we have $|\{M_f > \delta\}| \leq \frac{11+\sqrt{61}}{12}\frac{\|f\|_1}{\delta}$ , and the constant $\frac{11+\sqrt{61}}{12} \approx 1.5675208$ in this inequality is the best possible.

Short context: The statement that inequality $|\{M_f > \delta\}| \leq C\frac{\|f\|_1}{\delta}$ holds with some constant C is known as the centered Hardy–Littlewood maximal inequality, and has various applications in the theories of differentiation and integration (in particular, it is related to Lebesgue differentiation theorem, fractional integration theorem, and more). Let us denote $C^*$ the best possible (smallest) constant C such that the inequality holds. Carbery and Soria suggested that $C^*=1.5$ , but this was disproved by Aldaz in 1998. Few years later, Melas proved that $\frac{11+\sqrt{61}}{12} \leq C^* \leq \frac{5}{3}$ and conjectured that the lower bound is the actual value of $C^*$ . The Theorem confirms this conjecture.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 3.3 of this book for an accessible description of the Theorem.

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Rational polygon whose set of non-ergodic directions has dimension 1/2

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

Background: Let Q be a rational polygon, that is, polygon whose angles measured in radians are rational multiples of $\pi$ . A “billiard” in Q is a point moving inside Q with constant speed, with the usual rule that the angle of incidence equals the angle of reflection. Its trajectory is called equidistributed if, for every subset $M \subset Q$ of area $S(M)$ , $\lim\limits_{T\to\infty}\frac{f_M(T)}{T} = \frac{S(M)}{S(Q)}$ where $S(Q)$ is the area of Q, and $f_M(T)$ is the time the point spent inside $M$ during the time interval $[0,T]$ . A direction, parametrized by angle $\alpha\in[0,2\pi)$ , leading to equidistributed trajectory is called ergodic, and all other directions are called non-egrodic. Let $\text{NE}(Q)\subset [0,2\pi)$ denotes the set of non-egrodic directions in Q, and let $\text{dim}(\text{NE}(Q))$ be its Hausdorff dimension.

The Theorem: On 23rd July 2001, Yitwah Cheung and Michael Boshernitzan submitted to the Annals of Mathematics a paper in which they proved the existence of a rational polygon Q with $\text{dim}(\text{NE}(Q))=1/2$ .

Short context: A fundamental theorem proved by Kerckhoff, Masur, and Smillie in 1986 states that, in any rational polygon Q, set $\text{NE}(Q)$ has Lebesgue measure $0$ . In 1992, Masur proved a much stronger result that in fact $\text{dim}(\text{NE}(Q)) \leq 1/2$ . The Theorem shows that constant $1/2$ in this inequality is the best possible.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 3.7 of this book for an accessible description of the Theorem.

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Regular n-gon minimizes the logarithmic capacity among all n-gons with a fixed area

You need to know: Basic geometry, polygon, regular polygon, notation $|AB|$ for the length of line interval with endpoints A and B. n-gon is a polygon with n vertices. We assume that polygon contains its boundary (vertices and sides).

Background: Let E be a polygon, or, more generally, a closed bounded subset of the plane. For any k points $A_1, \dots A_k \in E$ , denote $p(E, A_1, A_2, \dots, A_k) = \left(\prod\limits_{i=1}^{k-1}\prod\limits_{j=i+1}^k |A_iA_j|\right)^{\frac{2}{k(k-1)}}$ . Then let $p_k(E) := \max\limits_{A_1, A_2, \dots, A_k \in E} p(E, A_1, A_2, \dots, A_k)$ , and $p(E) = \lim\limits_{k\to\infty} p_k(E).$ Number $p(E)$ is called transfinite diameter, or logarithmic capacity of set E.

The Theorem: On 25th May 2001, Alexander Solynin and Victor Zalgaller submitted to the Annals of Mathematics a paper in which they proved that, for any n-gon $E_n$ , $p(E_n) \geq p(E^*_n)$ , where $E^*_n$ is the regular n-gon with the same area as $E_n$ .

Short context: Number $p(E)$ has many names (logarithmic capacity, transfinite diameter, conformal radius, Chebyshev constant), indicative of its numerous applications. In 1951, Pólya and Szegö conjectured that regular n-gon minimizes the logarithmic capacity among all n-gons with a fixed area and proved this conjecture for $n=3$ and $n=4$ . The Theorem confirms this conjecture for all n.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 4.2 of this book for an accessible description of the Theorem.

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Any sequence of Lipschitz maps has a common point of differentiability

You need to know: Banach space, norm $\|.\|_X$ in Banach space X, dimension of a Banach space, bounded linear functions between Banach spaces, small $o(.)$ notation.

Background: Let $X,Y$ be Banach spaces. A function $f:X \to Y$ is called Lipschitz continuous if there is a constant $K \geq 0$ such that $\|f(x)-f(y)\|_Y \leq K \|x-y\|_X$ for all $x,y \in X$ . A function $f:X\to Y$ is called Fréchet differentiable at $x_0 \in X$ if there is a bounded linear function $T:X \to Y$ such that $f(x_0 + u) = f(x_0) + T(u) + o(\|u\|_X) \, \text{as} \, u \to 0.$

The Theorem: On 22nd May 2001, Joram Lindenstrauss and David Preiss submitted to the Annals of Mathematics a paper in which they proved the existence of infinite dimensional Banach space X such that every sequence $f_1, f_2, \dots,$ of Lipschitz continuous functions on X has a point $x_0 \in X$ such that all $f_i$ are Fréchet differentiable at $x_0$ .

Short context: A well-known open question is to investigate which Banach spaces X have the property that every countable collection of Lipschitz functions on X has a common point of Fréchet differentiability. Before 2001, this property was known to only hold for finite dimensional spaces. The Theorem proves that it also holds for some Banach spaces of infinite dimension.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 3.1 of this book for an accessible description of the Theorem.

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Sinai conjecture on orbit recurrence of nonregular quadratic maps is true

You need to know: Limits, the concept of (Lebesgue) almost every.

Background: For a quadratic polynomial $f:{\mathbb R}\to {\mathbb R}$ and initial point $x_0 \in {\mathbb R}$ , consider infinite sequence defined by induction: $x_n=f(x_{n-1})$ , $n=1,2,\dots$ . Let I be the longest interval which f maps into itself. If, for almost every $x_0 \in I$ , the sequence above converges to a cycle, f is called regular.

The Theorem: On 6th October 2000, Artur Ávila and Carlos Moreira submitted to arxiv a paper in which they proved that, for almost every $a\in[-1/4, 2]$ such that quadratic polynomial $f_a(x)=a-x^2$ is not regular, and $x_0=0$ , the set of n such that $|x_n| < 1/n^\gamma$ is finite if $\gamma > 1$ and infinite if $\gamma < 1$ .

Short context: The sequence $x_n$ defined above is an example of (an orbit of) a dynamical system. If $x_n \approx 0 = x_0$ , then $x_{n+1} \approx x_1$ , $x_{n+2} \approx x_2$ , $\dots$ and the sequence is close to being periodic for some time. The Theorem provides an exact rate how close $x_n$ can be to $0$ , answering a long-standing conjecture of Sinai. This result helps to understand to what extend this simple dynamical system exhibits a periodic behaviour, and is the step towards understanding approximate periodicity in more complex dynamical systems.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 5.4 of this book for an accessible description of the Theorem.

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