There exist lattices with exponentially large kissing numbers

You need to know: Set ${\mathbb Z}$ of integers, Euclidean space ${\mathbb R}^n$ , basis for ${\mathbb R}^n$ , norm $||x||=\sqrt{\sum_{i=1}^n x_i^2}$ of $x=(x_1, \dots, x_n) \in {\mathbb R}^n$ .

Background: A lattice $L$ in ${\mathbb R}^n$ is a set of the form $L =\left\{\left.\sum\limits_{i=1}^n a_i v_i\,\right\vert\, a_i \in {\mathbb Z}\right\}$ , where $v_1, \dots, v_n$ is a basis for ${\mathbb R}^n$ . Let $\lambda_1(L)$ be the length of the shortest non-zero vector in L. The kissing number $\tau(L)$ of L is the number of vectors of length $\lambda_1(L)$ in L. The lattice kissing number $\tau_n^l$ in dimension n is the maximum value of $\tau(L)$ over all lattices $L$ in ${\mathbb R}^n$ .

The Theorem: On 3rd February 2018, Serge Vlăduţ submitted to arxiv a paper in which he proved the existence of constant $c>0$ such that $\tau_n^l \geq e^{cn}$ for any $n\geq 1$ .

Short context: The kissing number in ${\mathbb R}^n$ is the highest number of equal nonoverlapping spheres in ${\mathbb R}^n$ that can touch another sphere of the same size. Determining the kissing number in various dimensions is an active area of research, see here and here. The lattice kissing number corresponds to the case when spheres form a “regular pattern”. It was a long-standing open problem whether the lattice kissing number grows exponentially with dimension. The Theorem resolves this problem affirmatively.

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

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The De Bruijn-Newman constant is non-negative

You need to know: Sum of infinite series, integration. Notations: ${\mathbb R}$ is the set of real numbers, ${\mathbb C}$ is the set of complex numbers.

Background: Let $\Phi: {\mathbb R} \to {\mathbb R}$ be a function defined as a sum of infinite series: $\Phi(u) = \sum\limits_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3\pi n^2 e^{5u})\exp(-\pi n^2 e^{4u})$ . For each $t\in{\mathbb R}$ , let $H_t: {\mathbb C} \to {\mathbb C}$ be a function given by $H_t(z) = \int\limits_0^\infty e^{tu^2}\Phi(u)\cos(zu)du$ . Newman showed that there exists a finite constant $\Lambda$ (known as the de Bruijn–Newman constant) such that the zeros of $H_t$ are all real if and only if $t\geq \Lambda$ .

The Theorem: On 18th January 2018, Brad Rodgers and Terence Tao submitted to arxiv a paper in which they proved that $\Lambda \geq 0$ .

Short context: The importance of the de Bruijn–Newman constant is that it is connected to the famous Riemann hypothesis, see here. Specifically, the Riemann hypothesis is equivalent to the inequality $\Lambda \leq 0$ . In 1976, Newman conjectured the complementary bound $\Lambda \geq 0$ , and noted that this conjecture states that if the Riemann hypothesis is true, it is only “barely so”. The Theorem proves the Newman’s conjecture.

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

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If k large, p prime, gcd(n,d)=1, and n(n+d)…(n+(k-1)d)=y^p, then yd=0 or p<=exp(10^k)

You need to know: Prime numbers, greatest common divisor $\text{gcd}(n,d)$ of positive integers $n,d$ , perfect power.

Background: Arithmetic progression of length k is the sequence $n, n+d, \dots n+(k-1)d$ .

The Theorem: On 4th September 2017, Michael Bennett and Samir Siksek submitted to arxiv and the Annals of Mathematics a paper in which they proved the existence of constant $k_0$ such that if $k\geq k_0$ is a positive integer, then any solution in integers to equation $n(n+d)(n+2d)\dots(n + (k-1)d) = y^p$ with $\text{gcd}(n,d)=1$ and prime exponent $p$ satisfies either $y=0$ or $d=0$ , or $p\leq \exp(10^k)$ .

Short context: In 1975, Erdős and Selfridge, answering a question posed by Liouville in 1857, proved that the product of two or more consecutive nonzero integers can never be a perfect power. Erdős further conjectured that the same is true for the products of consecutive terms in arithmetic progression $n, n+d, \dots n+(k-1)d$ , as soon as k sufficiently large and $\text{gcd}(n,d)=1$ . The Theorem, together with Faltings’ theorem, implies that, for every fixed $k\geq k_0$ , there may be at most finitely many arithmetic progressions of length k that violates the Erdős’ prediction.

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

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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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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 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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Elliptic curve with discriminant D has at most O(|D|^(0.2007…+e)) integer points

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

The Theorem: On 11th May 2004, Harald Helfgott and Akshay Venkatesh 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 $\beta=\frac{4\sqrt{3}\log(2+\sqrt{3})-6\log 2 -3\log 3}{12\log 2}=0.2007...$ .

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 1992, Schmidt proved that $N_{a,b} \leq C_\epsilon |\Delta_{a,b}|^{1/2+\epsilon}$ . The Theorem improves this bound significantly. The bound in the Theorem remained unimproved for over decade, until Bhargava et.al. proved that $N_{a,b} \leq C_\epsilon |\Delta_{a,b}|^{0.1117...+\epsilon}$ .

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

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Schwartz functions on the real line have explicit Fourier interpolation

You need to know: Set ${\mathbb R}$ of real numbers, even function $f:{\mathbb R} \to {\mathbb R}$ (such that $f(x)=f(-x)$ for all $x\in{\mathbb R}$ ), derivative, k-th derivative $f^{(k)}(x)$ , integration, set ${\mathbb C}$ of complex numbers, Fourier transform $\hat{f}(t)=\int_{-\infty}^{\infty} f(x) e^{-2\pi i t x} dx$ of an integrable function $f:{\mathbb R} \to {\mathbb R}$ , absolute convergence of infinite series.

Background: Function $f:{\mathbb R}\to{\mathbb R}$ is called a Schwartz function if there exist all derivatives $f^{(k)}(x)$ for all $k=1,2,3,\dots$ and for all $x\in{\mathbb R}$ , and, for every k and $\gamma\in{\mathbb R}$ , there is a constant $C(k,\gamma)$ such that $|x^\gamma f^{(k)}(x)| \leq C(k,\gamma), \, \forall x\in {\mathbb R}$ .

The Theorem: On 1st January 2017, Danylo Radchenko and Maryna Viazovska submitted to arxiv a paper in which they proved the existence of a collection of even Schwartz functions $a_n:{\mathbb R}\to{\mathbb R}$ with the property that for any even Schwartz function $f:{\mathbb R} \to {\mathbb R}$ and any $x\in{\mathbb R}$ we have $f(x)=\sum\limits_{n=0}^\infty a_n(x)f(\sqrt{n})+\sum\limits_{n=0}^\infty \hat{a_n}(x)\hat{f}(\sqrt{n})$ , where the right-hand side converges absolutely.

Short context: The classical Whittaker-Shannon interpolation formula states that if the Fourier transform $\hat{f}$ of function $f:{\mathbb R} \to {\mathbb R}$ is supported in $[-w/2,w/2]$ , then $f(x)=\sum\limits_{n=-\infty}^\infty f(n/w)\text{sinc}(wx-n)$ , where $\text{sinc}(x) = \sin(\pi x)/(\pi x)$ . The formula has numerous applications, in particular it allows to construct a “nice” continuous function which approximates a given sequence of real numbers. However, it does not work for functions whose Fourier transform has unbounded support. The Theorem provides a similar formula which works for arbitrary Schwartz functions. In particular, it implies that if $f:{\mathbb R} \to {\mathbb R}$ is an even Schwartz function such that $f(\sqrt{n})=\hat{f}(\sqrt{n})=0$ for $n=0,1,2,\dots$ , then $f(x)=0$ for all $x\in{\mathbb R}$ .

Links: The original paper is available here, journal version is here.

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The dimension of intersection of Lagrange spectrum with a half-line may assume any value in [0,1]

You need to know: Notations: ${\mathbb R}$ for the set of real numbers, ${\mathbb Q}$ for the set of rational numbers, ${\mathbb R}\setminus {\mathbb Q}$ for the set of irrational numbers, $\text{dim}(A)$ for the Hausdorff dimension of set $A \subseteq {\mathbb R}$ .

Background: For $\alpha \in {\mathbb R}\setminus {\mathbb Q}$ , let $k(\alpha)$ be the supremum of all $k>0$ , for which the inequality $\left|\alpha-\frac{p}{q}\right|<\frac{1}{kq^2}$ holds for infinitely many rational numbers $\frac{p}{q}$ . The Lagrange spectrum is the set $L= \{k(\alpha)\,|\,\alpha\in {\mathbb R}\setminus {\mathbb Q}, k(\alpha) <+\infty\}$ of all possible finite values of $k(\alpha)$ . For any $t \in {\mathbb R}$ , let $d(t) = \text{dim}(L \cap (-\infty, t))$ . A function $f:{\mathbb R}\to[0,1]$ is called surjective if for every $x\in[0,1]$ there exists $t\in {\mathbb R}$ such that $f(t)=x$ .

The Theorem: On 17th December 2016, Carlos Moreira submitted to arxiv a paper in which he proved that $d(t)$ is a continuous non-decreasing surjective function from ${\mathbb R}$ to $[0,1]$ , such that $\max\{t\in {\mathbb R}\,|\,d(t)=0\}=3$ and $d(\sqrt{12}-\delta)=1$ for some $\delta>0$ .

Short context: Approximating irrational numbers by rationals is an old topic in number theory. In 1891, Hurwitz proved that every irrational number $\alpha$ can be approximated by infinitely many rational numbers $\frac{p}{q}$ with accuracy $\left|\alpha-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}$ . The constant $\sqrt{5}$ in this Theorem is the best possible which works for all $\alpha$ . Function $k(\alpha)$ defined above is the best constant which works for any specific $\alpha$ , and it measures “how well” $\alpha$ can be approximated by rationals. Hurwitz Theorem states that $k(\alpha)\geq \sqrt{5}$ for all $\alpha \in {\mathbb R}\setminus {\mathbb Q}$ . This is the best possible because $k\left(\frac{1+\sqrt{5}}{2}\right)=\sqrt{5}$ . In terms of Lagrange spectrum L, this means that $\sqrt{5}$ is the smallest element of L. Properties of L are critical to understand rational approximation, and the Theorem (which answers a question asked by Bugeaud in 2008) deeply enrich our understanding of L.

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

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