The primes contain arbitrarily long multivariate polynomial progressions

You need to know: Prime number, polynomial in r variables, degree of a polynomial, notation ${\mathbb Z}^r$ for the set of vectors $\textbf{m}=(m_1,\dots,m_r)$ with r integer components $m_i$ .

Background: Let ${\mathbb Z}[m_1, \dots, m_r]$ denote the set of polynomials in r variables with integer coefficients.

The Theorem: On 25th March 2016, Terence Tao and Tamar Ziegler submitted to arxiv a paper in which they proved the following result. Let $d, r$ be natural numbers,
$P_1,\dots ,P_k \in {\mathbb Z}[m_1, \dots, m_r]$ be polynomials of degree at most d, such that $P_i-P_j$ has degree exactly d for all $1\leq i<j\leq k$ . Suppose that for each prime p there exist $n\in{\mathbb Z}$ and $\textbf{m}\in {\mathbb Z}^r$ such that $n+P_1(\textbf{m}),\dots, n + P_1(\textbf{m})$ are all not divisible by p. Then there exist infinitely many natural numbers $n, m_1,\dots, m_r$ such that $n+P_1(m_1,\dots, m_r),\dots, n+P_k(m_1,\dots, m_r)$ are simultaneously prime.

Short context: For $n,m \in {\mathbb Z}$ , the sequence $n, n+m,\dots , n+(k-1)m$ is called an arithmetic progression. By analogy, we can call sequence $n+P_1(m),\dots , n+P_k(m)$ for polynomials $P_1,\dots ,P_k$ in one variable polynomial progression, and the sequence $n+P_1(\textbf{m}),\dots, n+P_k(\textbf{m})$ as in the Theorem – multivariate polynomial progression. In an earlier work, Green and Tao proved that primes contains arbitrary long arithmetic progressions (with $m\neq 0$ ). Later, Tao and Ziegler generalised this to polynomial progressions, under the condition that $P_1(0)=\dots=P_k(0)=0$ . The Theorem removes this condition, and also generalises the result to the multivariate case, but introduces a new condition that “ $P_i-P_j$ has degree exactly d”. Proving any result of this kind without any additional condition is difficult, because the case $k=2$ with $P_1-P_2=2$ would imply famous twin prime conjecture.

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

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The Leech lattice gives the densest packing of congruent balls in 24-dimensional space

You need to know: Euclidean space ${\mathbb R}^n$ , norm $\|x\|=\sqrt{\sum_{i=1}^nx_i^2}$ of $x=(x_1,\dots,x_n)\in {\mathbb R}^n$ , open ball ball $B(x,R)=\{y\in {\mathbb R}^n: ||y-x||<R\}$ with centre $x\in {\mathbb R}^n$ and radius $R>0$ , origin $O=(0,0,\dots,0)\in{\mathbb R}^n$ , notation $|V|$ for the volume of set $V\subset {\mathbb R}^n$ , limit superior $\limsup$ , notation $n!=1\cdot 2 \cdot \dots \cdot n$ .

Background: Packing of congruent balls (also called sphere packing) in ${\mathbb R}^n$ is an (infinite) set S of non-overlapping open balls with the same radii. The upper density of a sphere packing S is $\Delta(S) := \limsup\limits_{R\to\infty} \frac{|S\cap B(O,R)|}{|B(O,R)|}$ . If the limit (as opposite to limsup) exists, $\Delta(S)$ is called the density of S.

The Theorem: On 21st March 2016, Henry Cohn, Abhinav Kumar, Stephen Miller, Danylo Radchenko, and Maryna Viazovska submitted to arxiv a paper in which they proved that no packing of congruent balls in ${\mathbb R}^{24}$ can have upper density greater than $\frac{\pi^{12}}{12!}$ .

Short context: Finding the densest possible packing of congruent balls in ${\mathbb R}^n$ is a fundamental problem in geometry. Because there is a packing in ${\mathbb R}^{24}$ (called the Leech lattice packing) with density $\frac{\pi^{12}}{12!}$ , the Theorem completely solves this problem in dimension $n=24$ . Earlier, it was only known that the Leech lattice is optimal among lattice packings. For general packings, the densest one was known, before 2016, only in dimensions $n\leq 3$ . Earlier in 2016, Viazovska solved this problem in dimension $n=8$ . The proof of the Theorem extends Viazovska’s arguments to the case $n=24$ .

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

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The Markoff-Hurwitz equation has (c+o(1))(log R)^b integer solutions up to R

You need to know: Set ${\mathbb Z}$ of integers, perfect square, o(1) notation.

Background: For integer parameters $n\geq 3$ , $a\geq 1$ , and $k\in {\mathbb Z}$ consider the equation $x_1^2+x_2^2+\dots+x_n^2=ax_1x_2\dots x_n+k$ . This is called generalized Markoff-Hurwitz equation (the Markoff-Hurwitz equation corresponds to the case $k=0$ ). For $R>0$ , let $M_{n,a,k}(R)$ be the number of integer solutions $(x_1, \dots, x_n)$ to this equation with $|x_i|\leq R$ for $1\leq i \leq n$ .

The Theorem: On 20th March 2016, Alex Gamburd, Michael Magee, and Ryan Ronan submitted to arxiv a paper in which they proved that if $n\geq 3$ , $a\geq 1$ , and $k\in {\mathbb Z}$ are such that $k-n+2$ and $k-n-1$ are not perfect squares, then either $\lim\limits_{R\to\infty}M_{n,a,k}(R)<\infty$ or $M_{n,a,k}(R)=(c+o(1))(\log R)^\beta$ , where $c=c(n,a,k)>0$ and $\beta=\beta(n)>0$ are positive constants.

Short context: The generalized Markoff-Hurwitz equation contains many important and well-studied equations as special cases. For example, with $n=a=3$ and $k=0$ , it reduces to the equation $x_1^2+x_2^2+x_3^2=3x_1x_2x_3$ , whose solutions in positive integers are known as Markoff triples, and have applications in number theory, group theory, and geometry. The Theorem establishes an asymptotic formula for the number of integer solutions of this equation.

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

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The E8 lattice gives the densest packing of congruent balls in 8-dimensional space

The Theorem: On 14th March 2016, Maryna Viazovska submitted to arxiv a paper in which she proved that no packing of congruent balls in ${\mathbb R}^8$ can have upper density greater than $\frac{\pi^4}{384}$ .

Short context: Finding the densest possible packing of congruent balls in ${\mathbb R}^n$ is a fundamental problem in geometry, which, before 2016, was solved only in dimensions $n\leq 3$ . Because there is a packing in ${\mathbb R}^8$ (called the $E_8$ -lattice packing) with density $\frac{\pi^4}{384}$ , the Theorem completely solves this problem in dimension $n=8$ . The proof is based on this theorem of Cohn and Elkies. In a later work, a similar method was used to solve this problem in dimension $n=24$ as well.

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

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The restriction conjecture for paraboloids is true for p>2(3n+1)/(3n-3), n>=2

You need to know: Notations: ${\mathbb C}$ for the set of complex numbers, $|z|$ for the absolute value of complex number z, $i = \sqrt{-1}$ , $e^{ix} = \cos(x)+i\sin(x)$ , ${\mathbb R}^n$ for n-dimensional Euclidean space, $B^{n-1}$ for the unit ball in ${\mathbb R}^{n-1}$ , $|w|^2=\sum\limits_{i=1}^{n-1}w_i^2$ for $w=(w_1, \dots, w_{n-1}) \in {\mathbb R}^{n-1}$ , $||f||_{L^p(\Omega)} = \left(\int_\Omega |f(\omega)|^p d\omega\right)^{1/p}$ , where $p\geq 1$ , $\Omega\subseteq {\mathbb R}^n$ , and $f:\Omega \to {\mathbb C}$ .

Background: The extension operator for the paraboloid is the operator E which puts to every function $f:B^{n-1} \to {\mathbb C}$ with $||f||_{L^p(B^{n-1})}<\infty$ into correspondence a function $E_f: {\mathbb R}^{n} \to {\mathbb C}$ given for every $x=(x_1, \dots, x_n)$ by $E_f(x) = \int_{B^{n-1}}e^{i(x_1w_1+\dots+x_{n-1}w_{n-1}+x_n|w|^2)}f(w)dw$ .

The Theorem: On 14th March 2016, Larry Guth submitted to arxiv a paper in which he proved the following result. Let $n\geq 2$ and let $p>2\frac{3n-1}{3n-3}$ if n odd and $p>2\frac{3n+2}{3n-2}$ if n is even. Then there exists a constant $C=C(p)$ such that $||E_f||_{L^p({\mathbb R}^n)} \leq C ||f||_{L^p(B^{n-1})}$ for every f.

Short context: In 1979, Stein conjectured that the conclusion of the Theorem holds for all $p>\frac{2n}{n-1}$ and $n\geq 2$ . This is known as the restriction conjecture for paraboloids, and is an important conjecture in harmonic analysis. The Theorem proves this conjecture for all $n\geq 2$ under assumption $p>2\frac{3n-1}{3n-3}$ (with slightly less restrictive assumption $p>2\frac{3n+2}{3n-2}$ if n is even).

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

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A problem posed by Erdős and Szekeres in 1950 has a positive answer

You need to know: Rational and irrational numbers, complex numbers, notation $i$ for $\sqrt{-1}$ , absolute value $|z|$ of complex number $z$ , limit inferior $\liminf$ .

Background: For any real $x$ , by $e^{ix}$ we mean $\cos(x)+i\sin(x)$ .

The Theorem: On 16th February 2016, Artur Avila, Svetlana Jitomirskaya, and Christoph Marx submitted to arxiv a paper in which they proved, among other results, that for all irrational $\alpha$ , one has $\liminf\limits_{n\to\infty}\max\limits_{|z|=1}\prod\limits_{k=1}^n\left|z-e^{2\pi i k \alpha}\right|<\infty$ .

Short context: The Theorem gives a positive answer to a long-standing open problem posed by Erdős and Szekeres in 1950. Interestingly, the authors needed to solve this elementary-looking number theoretic problem on a way to deriving deep results related to spectral theory of so-called Harper’s model used in statistical mechanics.

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

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Bernoulli convolution is absolutely continuous for l=1-10^(-50) and 3/4>=p>=1/4

You need to know: Basic probability theory, notation $\text{Pr}$ for probability, random variable, distribution of a random variable, independent identically distributed (i.i.d.) random variables, polynomial, degree of a polynomial, monic polynomial. You also need Lebesgue measure and Hausdorff dimension to understand the context.

Background: Let $\lambda,p\in(0,1)$ be real numbers, and let $\xi_0, \xi_1, \xi_2, \dots$ be a sequence of i.i.d. random variables with $\text{Pr}(\xi_n=1)=p$ and $\text{Pr}(\xi_n=-1)=1-p$ for all n. By Bernoulli convolution we mean the random variable $X_{\lambda,p}=\sum\limits_{n=0}^\infty \xi_n \lambda^n$ . $X_{\lambda,p}$ is called absolutely continuous if there exists function $f:{\mathbb R}\to{\mathbb R}$ such that $\text{Pr}(a\leq X_{\lambda,p}\leq b)=\int_a^b f(x)dx$ whenever $a\leq b$ . The Mahler measure of any polynomial $P(x)=a\prod\limits_{i=1}^d(x-z_i)$ is $M(P)=a\prod\limits_{j:|z_j|>1}|z_j|$ . A real number $r$ is called algebraic if $P(r)=0$ for some polynomial P with rational coefficients. The (unique) monic polynomial $P_r$ of smallest degree with this property is called the minimal polynomial of r. The Mahler measure of r is $M_r=M(P_r)$ .

The Theorem: On 31st January 2016, Péter Varjú submitted to arxiv a paper in which he proved that for every $\epsilon>0$ and $p\in(0,1)$ , there is a constant $c=c(\epsilon,p)>0$ such that for any algebraic number $\lambda$ satisfying $1>\lambda>1 - c\min(\log M_\lambda, (\log M_\lambda)^{-1-\epsilon})$ , the Bernoulli convolution $X_{\lambda,p}$ is absolutely continuous.

Short context: Bernoulli convolutions has been introduced by Jessen and Wintner in 1935, and studied by Erdős and many others since that. The main research question is for which values of $\lambda$ and $p$ the Bernoulli convolution is absolutely continuous. In 1939, Erdős noticed that $X_{\lambda,1/2}$ is not absolutely continuous for $\lambda<1/2$ , and also for $\lambda \in E$ for some non-empty set $E \subset [1/2,1)$ . However, Solomyak proved in 1995 that set E has Lebesgue measure $0$ . Moreover, Shmerkin, building on this theorem, proved in 2014 that E has Hausdorff dimension $0$ . Despite on this, it was known very few explicit examples of $\lambda$ -s for which $X_{\lambda,p}$ is absolutely continuous, and none such examples was known for $p\neq 1/2$ . The Theorem (with explicit $c=c(\epsilon,p)$ ) provides a lot of such examples. For example, it implies that $X_{\lambda,p}$ is absolutely continuous for $\lambda=1-10^{-50}$ and $\frac{1}{4}\leq p \leq \frac{3}{4}$ .

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

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Bernoulli convolution v_l has dimension 1 outside a set of l of dimension 0

You need to know: Basic probability theory, random variable, distribution of a random variable, independent identically distributed (i.i.d.) random variables, infimum $\inf$ , supremum $\sup$ , limit superior $\limsup$ .

Background: For $\frac{1}{2}<\lambda<1$ , Bernoulli convolution $\nu_\lambda$ is the distribution of the real random variable $\sum\limits_{n=0}^\infty \pm \lambda^n$ , where the signs are chosen i.i.d. with equal probabilities. It is known that the limit $\lim\limits_{r\to 0+}\frac{\log \nu_\lambda([x-r,x+r])}{\log r}$ exists and is constant for $\nu_\lambda$ -almost every x. This constant is called dimension of $\nu_\lambda$ and in denoted $\text{dim}(\nu_\lambda)$ . The box dimension of set $S$ of real numbers is $\text{bdim}(S)=\limsup\limits_{r\to 0+}\frac{\log N_S(r)}{\log(1/r)}=0$ , where $N_S(r)$ is the minimal number of intervals of length $r$ needed to cover S. The packing dimension of S is $\text{pdim}(S)=\inf\left\{\sup\limits_n \text{bdim}(S_n) : S \subseteq \bigcup\limits_{n=1}^\infty S_n\right\}$ .

The Theorem: On 9th December 2012, Michael Hochman submitted to arxiv a paper in which he proved, among other results, that the set $\{\lambda \in (1/2,1) : \text{dim}(\nu_\lambda) <1\}$ has packing dimension $0$ . In other words, $\text{dim}(\nu_\lambda)=1$ outside a set of $\lambda$ of packing dimension $0$ .

Short context: Bernoulli convolutions has been introduced by Jessen and Wintner in 1935, and studied by Erdős and many others since that. The main question is for which values of $\lambda \in (1/2,1)$ the resulting probability measure $\nu_\lambda$ is “nice”, and having dimension 1 is one of the criteria for “niceness”. The Theorem states that the set of exceptional $\lambda$ -s for which $\nu_\lambda$ may be “not nice” is very small in a strong sense. For readers familiar with Hausdorff dimension, we note that having packing dimension $0$ implies Hausdorff dimension $0$ but not vice versa.

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

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If p>q>2, the maximal t for which the t-snowflake of L_q admits a bi-Lipschitz embedding into L_p is t=q/p

You need to know: Metric space.

Background: We say that a metric space $(X,\rho_X)$ admit a bi-Lipschitz embedding (or just “embeds” for short) into metric space $(Y,\rho_Y)$ if there exist constants $m,M>0$ and a function $f:X\to Y$ such that $m \rho_X(x,y) \leq \rho_Y(f(x),f(y)) \leq M \rho_X(x,y), \, \forall x,y \in X$ . For $\theta\in(0,1]$ , a $\theta$ -snowflake of metric space $(X,\rho_X)$ is the metric space on the same set X with distance $\rho(x,y)=(\rho_X(x,y))^\theta, \, \forall x,y \in X$ . By $L^p$ space we mean, for concreteness, space of functions $f:[0,1]\to{\mathbb R}$ with norm $||f||_p=\left(\int_0^1|f(x)|^pdx\right)^{1/p}<\infty$ .

The Theorem: On 13th January 2016, Assaf Naor submitted to arxiv a paper in which he proved that, for every $2<q<p$ , the maximal $\theta\in(0,1]$ for which the $\theta$ -snowflake of $L_q$ admits a bi-Lipschitz embedding into $L_p$ is equal to $\frac{q}{p}$ .

Short context: It is known that, if $2<q<p$ , then $L_q$ does not embed into $L_p$ . Hence, if $\theta^*=\theta^*(p,q)$ denotes the maximal $\theta\in(0,1]$ for which the $\theta$ -snowflake of $L_q$ embeds into $L_p$ , then $\theta^*<1$ . Quantifying by “how much” $\theta^*$ is bounded away from 1 gives an important quantitative refinement of non-embeddability $L_q$ into $L_p$ . In 2004, Mendel and Naor proved that $\frac{q}{p}\leq \theta^*$ . In a paper submitted in 2014, Naor and Schechtman proved that $\theta^* \leq 1-\frac{(p-q)(q-2)}{2p^3}$ and conjectured that $\theta^*=\frac{q}{p}$ . The Theorem confirms this conjecture.

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

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If p>q>2, and the t-snowflake of L_q admits a bi-Lipschitz embedding into L_p, then t is bounded away from 1

You need to know: Metric space.

The Theorem: On 25th August 2014, Assaf Naor and Gideon Schechtman submitted to arxiv a paper in which they, among other results, proved that, for every $2<q<p$ , if $\theta\in(0,1]$ is such that the $\theta$ -snowflake of $L_q$ admits a bi-Lipschitz embedding into $L_p$ , then necessarily $\theta \leq 1-\frac{(p-q)(q-2)}{2p^3}$ .

Short context: It is known that, if $2<q<p$ , then $L_q$ does not embed into $L_p$ . Hence, if $\theta$ -snowflake of $L_q$ embeds into $L_p$ , then $\theta<1$ . Quantifying by “how much” $\theta$ is bounded away from 1 gives an important quantitative refinement of non-embeddability $L_q$ into $L_p$ . However, before 2014, no estimate in the form $\theta \leq 1-\delta(p,q)$ for any explicit function $\delta(p,q)$ has been known. The Theorem provides the first such estimate. In fact, the authors conjectured that the inequality in the Theorem can be improved to $\theta \leq \frac{q}{p}$ , which would be the best possible. In a later work, Naor proved this conjecture.

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

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