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.

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 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.

Go to the list of all theorems

Vertical versus horizontal isoperimetric inequality holds in Heisenberg groups

You need to know: Set ${\mathbb Z}$ of integers, set ${\mathbb N}$ of positive integers, notation $A\times B$ for the set of pairs $(x,y)$ with $x\in A$ and $y\in B$ , group G, notation $g^{-1}$ for an inverse element of $g \in G$ , notation $1$ for the identity element, notation $g^n$ for $g\cdot g \cdot \dots \cdot g$ ( $n$ times) and $g^{-n}$ for $(g^{-1})^n$ , commutator $[g,h]=ghg^{-1}h^{-1}$ for $g,h \in G$ , set of generators of a group, presentation of a group in the form $<S|R>$ , where S is a set of generators and R is a set of relations, notation $|A|$ for the cardinality (the number of elements) of finite set A.

Background: For every $k \in {\mathbb N}$ , the $k$ -th discrete Heisenberg group, denoted $H^{2k+1}_{\mathbb Z}$ , is the group with $2k+1$ generators $a_1, b_1, \dots, a_k, b_k, c$ and relations $[a_1,b_1]=\dots=[a_k,b_k]=c$ and $[a_i, a_j]=[b_i, b_j]=[a_i, b_j ]=[a_i, c]=[b_i, c] =1$ for every distinct $i,j \in \{1,\dots,k\}$ . The horizontal perimeter of a finite $\Omega \subseteq H^{2k+1}_{\mathbb Z}$ is $|\partial_h \Omega|$ , where $\partial_h \Omega = \{(x,y)\in \Omega \times (H^{2k+1}_{\mathbb Z} \setminus \Omega)\,:\,x^{-1}y \in S_k\}$ , where $S_k=\{a_1, b_1, a_1^{-1}, b_1^{-1}, \dots, a_k, b_k, a_k^{-1}, b_k^{-1}\}$ . The vertical perimeter of $\Omega$ is $|\partial_v \Omega| = \left(\sum\limits_{t=1}^\infty \frac{|\partial_v^t \Omega|^2}{t^2}\right)^{\frac{1}{2}}$ , where $\partial_v^t \Omega = \{(x,y)\in \Omega \times (H^{2k+1}_{\mathbb Z} \setminus \Omega)\,:\,x^{-1}y \in \{c^t, c^{-t}\}\}$ .

The Theorem: On 3rd January 2017, Assaf Naor and Robert Young submitted to arxiv a paper in which they proved the existence of a universal constant $C<\infty$ such that inequality $|\partial_v \Omega| \leq \frac{C}{k}|\partial_h \Omega|$ holds for every integer $k\geq 2$ and every finite subset $\Omega \subseteq H^{2k+1}_{\mathbb Z}$ .

Short context: The inequality proved in the Theorem is called vertical versus horizontal isoperimetric inequality and has applications in the embedding theory for metric spaces and in the analysis of complexity of algorithms for combinatorial problems, such as the Sparsest Cut Problem (see here).

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

Go to the list of all theorems

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.

Go to the list of all theorems

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.

Go to the list of all theorems

Tarski’s circle squaring problem has a constructive solution

You need to know: Euclidean plane ${\mathbb R}^2$ , notation $A+v$ for the translation of set $A\subset {\mathbb R}^2$ by a vector $v\in {\mathbb R}^2$ , open subset of ${\mathbb R}^2$ , countable union, countable intersection, and set difference ( $B \setminus A = \{x\in B, \, x\not\in A\}$ ) of sets. You also need to know what is the Axiom of choice to fully understand the context.

Background: A subset $A\subset {\mathbb R}^2$ is a Borel set if it can be formed from open sets through the operations of countable union, countable intersection, and set difference. We call two sets $A,B \subset {\mathbb R}^2$ equidecomposable by translations if there are partitions $A = A_1 \cup \dots \cup A_m$ and $B = B_1 \cup \dots \cup B_m$ , such that $B_i = A_i + v_i$ , $i=1,\dots,m$ , for some vectors $v_1, \dots, v_m \in {\mathbb R}^2$ . If, moreover, all $A_i$ (and thus $B_i$ ) are Borel sets, we say that A and B are equidecomposable by translations with Borel parts.

The Theorem: On 17th December 2016, Andrew Marks and Spencer Unger submitted to arxiv a paper in which they proved that a circle and a square of the same area on the plane are equidecomposable by translations with Borel parts.

Short context: In 1990, Laczkovich, answering a 1925 question of Tarski, proved that circle and a square of the same area are equidecomposable by translations. In a paper submitted in 2015, Grabowski, Máthé, and Pikhurko proved that this is possible even using only Lebesgue measurable pieces (that is, those having a well-define area). However, both results use axiom of choice and the resulting pieces $A_i$ are impossible to construct explicitly. The Theorem states that the circle can be squared with only Borel pieces. The proof does not use the axiom of choice. If such a proof is possible, we say that a problem has a constructive solution.

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

Go to the list of all theorems

There is no Diophantine quintuple

You need to know: Perfect square

Background: A set of $m$ distinct positive integers $\{a_1, a_2, \dots, a_m\}$ is called a Diophantine $m$ -tuple if $a_ia_j+1$ is a perfect square for all $1\leq i<j\leq m$ . In particular, it is called Diophantine quadruple, quintuple, and sextuple for $m=4$ , $m=5$ , and $m=6$ , respectively.

The Theorem: On 13th October 2016, Bo He, Alain Togbè, and Volker Ziegler submitted to arxiv a paper in which they proved that there is no Diophantine quintuple.

Short context: More than two thousands years ago, Diophantus noticed that the set of rational numbers $\{\frac{1}{16}, \frac{33}{16}, \frac{17}{4}, \frac{105}{16}\}$ has the property that the product of any two of them plus one is a square of a rational number. Later, Fermat found positive integers $\{1,3,8,120\}$ with this property, and Euler proved that there are infinitely many such quadruples. A long-standing folklore conjecture predicts that no five positive integers with this property exist. As a partial progress, Dujella proved in 2004 that there is no Diophantine sextuple and that there can be at most finitely many Diophantine quintuples. The Theorem confirms the conjecture in full.

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

Go to the list of all theorems

The elliptic Harnack inequality on a graph is stable under rough isometries

You need to know: Graph, connected graph, infinite graph, degree of a vertex of a graph, bounded degree graph (for which there is a constant $C<\infty$ such that every vertex has degree at most C), path connecting 2 vertices, length of a path, notation $d_G(x,y)$ for the length of the shortest path connecting vertices $x$ and $y$ in graph G.

Background: Let G be a connected graph with infinite vertex set V. For each pair $x, y \in V$ we define a conductance $C_{xy}\geq 0$ such that $C_{xy} = C_{yx}$ and also $C_{xy} = 0$ unless x and y are connected by edge. The graph G together with the conductances $C_{xy}$ is denoted $(G,C)$ and called a weighted graph. For $x\in V$ , let $\mu_G(x)=\sum\limits_{y\in V}C_{xy}$ . For $A\subset V$ , let $\mu_G(A)=\sum\limits_{x \in A}\mu_G(x)$ . A function $h:A\to {\mathbb R}$ is called harmonic on A if $h(x)=\sum\limits_{y\in V}h(y)C_{xy}$ for all $x\in A$ . For $x\in V$ and $r>0$ , let $B_G(x,r)$ be the set of $y\in V$ with $d_G(x,y)<r$ . We say that the elliptic Harnack inequality holds for $(G,C)$ if there exists $c_1$ such that whenever $x_0\in V$ , $r \geq 1$ , and h is non-negative and harmonic in $B_G(x_0, 2r)$ , then $h(x) \leq c_1 h(y)$ for all $x,y \in B_G(x_0,r)$ . We say that weighed graphs $(G,C)$ and $(H,C')$ with vertex sets $V_G$ and $V_H$ are roughly isometric if there is a function $\phi:V_G\to V_H$ and constants $C_1>0$ and $C_2,C_3>1$ such that (i) for every $y\in V_H$ there exists $x\in V_G$ , such that $d_H(y,\phi(x))\leq C_1$ , (ii) $C_2^{-1}(d_G(x,y)-C_1) \leq d_H(\phi(x), \phi(y)) \leq C_2(d_G(x,y)+C_1)$ for all $x,y \in V_G$ , and (iii) $C_3^{-1} \mu_G(B_G(x,r)) \leq \mu_H(B_H(\phi(x),r)) \leq C_3 \mu_G(B_G(x,r))$ for all $x\in V_G$ and $r>0$ .

The Theorem: On 5th October 2016, Martin Barlow and Mathav Murugan submitted to arxiv a paper in which they proved the following result. Let $(G,C)$ and $(H,C')$ be two connected bounded degree weighed graphs that are roughly isometric. Then the elliptic Harnack inequality holds for $(G,C)$ if and only if it holds for $(H,C')$ .

Short context: The elliptic Harnack inequality was proved by Moser in 1961 for some partial differential equations, but since that turned out to be useful in many other applications, for example for weighted graphs. The Theorem proves that this inequality is stable under rough isometries, resolving a long-standing open question. While we state the Theorem only for weighed graphs here, Barlow and Murugan actually proved it in much more general setting.

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

Go to the list of all theorems

The Furstenberg’s conjecture on the intersections of invariant sets is true

You need to know: Set ${\mathbb Q}$ of rational numbers, 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 25th September 2016, Pablo Shmerkin submitted to arxiv a paper in which he proved the following result. Let $p,q \geq 2$ be positive integers such that $\frac{\log p}{\log q}\not\in{\mathbb Q}$ . Let $A; B \subseteq [0,1]$ be closed sets which are invariant under $T_p$ and $T_q$ , respectively. Then for all real numbers $u$ and $v$ , $\text{dim}((u\cdot A + v) \cap B) \leq \max\{0, \text{dim}(A) + \text{dim}(B) - 1\}$ .

Short context: The Theorem confirms a long-standing conjecture of Furstenberg made in 1969. In fact, the Furstenberg’s conjecture corresponds to the case $u=1$ and $v=0$ of the Theorem. On 26th September 2016, Meng Wu submitted to arxiv a paper with independent and different proof of the same result. Also, see here for an earlier theorem resolving another related conjecture of Furstenberg.

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

Go to the list of all theorems

The Pomerance’s conjecture is true: the threshold for making squares is sharp

You need to know: Perfect square, limits, notation P for the probability, independent events, selection uniformly at random.

Background: We say that a finite sequence S of positive integers has a square dependence if it has a subset $A\subset S$ such that the product $\prod\limits_{n\in A}n$ of all integers in A is a perfect square. For $x>1$ , let us select integers $a_1,a_2,\dots$ independently uniformly at random from $[1,x]$ , and let T be the smallest integer such that the sequence $a_1,a_2,\dots, a_T$ has a square dependence.

The Euler-Mascheroni constant is the limit $\gamma=\lim\limits_{n\to\infty}\left(-\log n+\sum\limits_{k=1}^n\frac{1}{k}\right) = 0.577...$ . Integer n is called y-smooth if all of its prime factors are at most y. Let $\Psi(x,y)$ be the number of y-smooth integers up to x, $\pi(y)$ be the number of primes up to y, and let $J(x) = \min\limits_{2\leq y \leq x} \frac{\pi(y)x}{\Psi(x, y)}$ .

The Theorem: On 12th August 2016, Paul Balister, Béla Bollobás, and Robert Morris submitted to arxiv a paper in which they proved that for any $\epsilon>0$ , we have $\lim\limits_{x\to\infty} P\left((e^{-\gamma}-\epsilon)J(x) \leq T \leq (e^{-\gamma}+\epsilon)J(x)\right) = 1$ , where $\gamma$ is the Euler-Mascheroni constant.

Short context: How many random integers between 1 and x we should select such that the product of some selected integers is a perfect square? This question is important for understanding the performance of fastest known factorisation algorithms. In 1994, Pomerance conjectured that the number T of integers needed for this exhibits a sharp threshold, that is, $\lim\limits_{x\to\infty} P\left((1-\epsilon)f(x) \leq T \leq (1+\epsilon)f(x)\right) = 1$ for some function $f(x)$ and any $\epsilon>0$ . See here for the best partial result towards this conjecture before 2016. The Theorem confirms the conjecture in full.

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

Go to the list of all theorems

For all n>n_a, there are at most 2n-2 lines in R^n with common angle a

You need to know: Euclidean space ${\mathbb R}^n$ , origin in ${\mathbb R}^n$ , lines in ${\mathbb R}^n$ , angle between lines, the inverse trigonometric function of cosine $\arccos$ .

Background: A set of lines through the origin in ${\mathbb R}^n$ is called equiangular if any pair of lines defines the same angle. Let $N(n)$ denotes the maximum cardinality of an equiangular set of lines in ${\mathbb R}^n$ . Let $N_\theta(n)$ denotes the maximum number of equiangular lines in ${\mathbb R}^n$ with common angle $\theta$ , where $\theta$ does not depend on dimension.

The Theorem: On 21st June 2016, Igor Balla, Felix Dräxler, Peter Keevash, and Benny Sudakov submitted to arxiv a paper in which they proved that for any angle $\theta \in (0,\pi/2)$ , $\theta \neq \arccos \frac{1}{3}$ , there is a constant $n_\theta$ , such that $N_\theta(n)\leq 1.93n$ for all $n \geq n_\theta$ .

Short context: Equiangular sets of lines appear naturally in many areas of mathematics, and the problem of estimating the maximum size of such sets has been studied starting from at least the work of Haantjes in 1948, who proved that $N(3)=N(4)=6$ . In 1973, Lemmens and Seidel formulated a problem of estimating $N_\theta(n)$ for fixed $\theta$ , and proved that $N_\theta(n) = 2n-2$ for $\theta=\arccos \frac{1}{3}$ and sufficiently large n. The Theorem proves a stronger upper bound for all $\theta \neq \arccos \frac{1}{3}$ . This implies that, for all large n, $N_\theta(n)$ is maximised at $\theta=\arccos \frac{1}{3}$ .

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

Go to the list of all theorems