There are (C_k+o(1))N^2/log(N)^k k-term arithmetic progressions of primes at most N

You need to know: Prime numbers, arithmetic progression, logarithm, limit, infinite product.

Background: We denote by $o(1)$ any function $f(N)$ such that $\lim\limits_{N\to\infty} f(N)=0$ . Let ${\cal P}$ denote the set of all prime numbers.

The Theorem: On 21st September 2010, Ben Green, Terence Tao, and Tamar Ziegler submitted to arxiv a paper which, among other results, finishes the proof of the following theorem. For any fixed integer $k\geq 2$ , the number of k-tuples of primes $p_1<p_2<\dots<p_k \leq N$ which lie in arithmetic progression is $(C_k+o(1))\frac{N^2}{\log^k N}$ , with $C_k=\frac{1}{2(k-1)}\prod\limits_{p\in{\cal P}} \beta_p$ , where $\beta_p=\frac{1}{p}\left(\frac{p}{p-1}\right)^{k-1}$ if $p\leq k$ and $\beta_p=\left(1-\frac{k-1}{p}\right)\left(\frac{p}{p-1}\right)^{k-1}$ if $p\geq k$ .

Short context: In a paper submitted in 2004, Green an Tao proved that, for any k, the set of primes contains infinitely many arithmetic progressions of length k. The Theorem counts how many such progressions exist. In a paper submitted in 2006, Green and Tao proved it for $k=4$ and also proved that the general case (and in fact much more general theorem called “The generalised Hardy-Littlewood conjecture in finite complexity case”) would follow from two new conjectures they formulated. They then, together with Ziegler, proved these conjectures, which finishes the proof of the Theorem.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 10.8 of this book for an accessible description of the Theorem.

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For integers a>b>0, the the greatest prime factor of a^n-b^n grows superlinearly in n

You need to know: Prime numbers, logarithms, limits. In addition, you need the definition of Lehmer sequence (see here) to understand the last sentences in the context.

Background: For any integer $m$ let $P(m)$ denote the greatest prime factor of $m$ , with the convention that $P(m)=1$ when $m$ is $1$ , $0$ , or $-1$ .

The Theorem: On 6th August 2010, Cameron Stewart submitted to arxiv a paper in which he proved that for any integers $a>b>0$ there is a constant $N=N(a,b)$ , such that $P(a^n-b^n) > n\exp\left(\frac{\log n}{104 \log\log n}\right)$ for all $n\geq N$ .

Short context: In 1886, Bang proved that $P(a^n-1)\geq n+1$ for any integers $a>1$ and $n>2$ . Can this lower bound be improved? In 1965, Erdős conjectured that $\frac{P(2^n-1)}{n}\to\infty$ as $n\to\infty$ . The Theorem (with $b=1$ ) proves Erdős conjecture (for any $a>1$ in place of $2$ ) and gives the first improvement on Bang’s bound for 124 years. For general $a>b>0$ , Zsigmondy proved in 1892 that $P(a^n-b^n)\geq n+1$ for $n>2$ , and Schinzel asked in 1962 if $P(a^n-b^n)\geq 2n$ for large n. The Theorem provides a much better lower bound than $2n$ . In fact, Stewart proved, much more generally, that the same lower bound holds for $P(u_n)$ , where $u_n$ is the n-th term of Lehmer sequence. Previously, it was only known (as a corollary from this theorem) that $P(u_n) \geq n-1$ for $n>30$ .

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

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A positive proportion of elliptic curves over Q ordered by height have rank 0

You need to know: For integers a,b, notation $a|b$ if b is divisible by a, and $a\nmid b$ if not, notation ${\mathbb Q}$ for the set of rational numbers, elliptic curve over ${\mathbb Q}$ , rank $\text{rank}(E)$ of elliptic curve E, limit superior $\limsup$ , limit inferior $\liminf$ , notation $|S|$ for the number of elements in any finite set S.

Background: Any elliptic curve E over ${\mathbb Q}$ can be written in the form $y^2 = x^3+ax+b$ , where $a,b$ are integers such that if $p^4|a$ for some prime p, then $p^6\nmid b$ . The height of E is $\max\{4|a^3|, 27b^2\}$ . For any h>0, let $S(h)$ be the set of elliptic curves of height at most h. Also, let $R_0$ be the set of all elliptic curves having rank $0$ .

The Theorem: On 1st July 2010 Manjul Bhargava and Arul Shankar submitted to arxiv a paper in which they proved that (i) $\limsup\limits_{h\to\infty}\left(\frac{1}{|S(h)|}\sum\limits_{E\in S(h)}\text{rank}(E)\right) \leq \frac{7}{6}$ , and (ii) $\liminf\limits_{h\to\infty}\frac{|S(h)\cap R_0|}{|S(h)|} >0$ .

Short context: Quantity $\frac{1}{|S(h)|}\sum\limits_{E\in S(h)}\text{rank}(E)$ is the average rank of elliptic curves of height at most h. In an earlier work, Bhargava and Shankar proved for the first time that the average rank is bounded from above by some constant C. Moreover, they show that one can take $C=1.5$ . Part (i) of the Theorem improves this to $C=\frac{7}{6}<1.17$ . Part (ii) of the Theorem states that a positive proportion of elliptic curves have rank $0$ .

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

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For most cubic forms F, the solution set of F(x,y)=z^l, gcd(x,y)=1, l>=4 is finite

You need to know: Notation $\text{gcd}(x,y)$ for the greatest common divisor of integers x and y.

Background: Let $F(x,y)=ax^3+bx^2y+cxy^2+dy^3$ be a binary cubic form with integer coefficients. F is called irreducible if it cannot be written as $(a'x+b'y)(c'x^2+d'xy+e'y^2)$ for some integers $a',b',c',d',e'$ . Number $\Delta_F=18abcd-4b^3d+b^2c^2-4ac^3-27a^2d^2$ is called the discriminant of F. Let $S_F$ be the set of primes dividing $2\Delta_F$ , and let $U_F$ be the set of nonzero integers u with the property that if u is divisible by some prime p then $p\in S_F$ . Let ${\cal F}$ be the set of binary cubic forms F such that equation $F(x,y) \in U_F$ has no solutions in integers x and y.

The Theorem: On 16th June 2010, Michael Bennett and Sander Dahmen submitted to the Annals of Mathematics a paper in which they proved that, for any $F \in {\cal F}$ , equation $F(x,y) = z^l$ has at most finitely many solutions in integers $(x,y,z,l)$ , such that $\text{gcd}(x,y)=1$ and $l\geq 4$ .

Short context: A special case of the 1995 theorem of Darmon and Granville states that, for any irreducible binary cubic form F and for any fixed $l\geq 4$ , equation $F(x,y) = z^l$ has at most finitely many solutions in co-prime integers $(x,y,z)$ . The Theorem states that, for forms $F\in{\cal F}$ , this equation has at most finitely many solutions even if $l$ is also a variable. For example, the authors show that $F(x,y)=3x^3-ax^2y-(a+9)xy^2-3y^3$ belong to ${\cal F}$ for infinitely many a, which provides an infinite family of forms for which the Theorem is applicable. Moreover, the authors provides a heuristic argument which suggests that “almost all” binary cubic forms belong to ${\cal F}$ .

Links: The original paper is available here.

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The average rank of elliptic curves over Q ordered by height is at most 1.5

You need to know: For integers a,b, notation $a|b$ if b is divisible by a, and $a\nmid b$ if not, notation ${\mathbb Q}$ for the set of rational numbers, elliptic curve over ${\mathbb Q}$ , rank $\text{rank}(E)$ of elliptic curve E, limit superior $\limsup$ , notation $|S|$ for the number of elements in any finite set S.

The Theorem: On 4th June 2010 Manjul Bhargava and Arul Shankar submitted to arxiv a paper in which they proved that $\limsup\limits_{h\to\infty}\left(\frac{1}{|S(h)|}\sum\limits_{E\in S(h)}\text{rank}(E)\right) \leq 1.5$ .

Short context: Quantity $\frac{1}{|S(h)|}\sum\limits_{E\in S(h)}\text{rank}(E)$ is the average rank of elliptic curves of height at most h, and the Theorem states that average rank of all elliptic curves over Q ordered by height is at most 1.5. It is an old conjecture that 50% of all elliptic curves over Q have rank $0$ and 50% have rank 1, which would imply that the average rank is 0.5. However, before 2010, no-one could prove that the average rank is bounded from above by any finite constant whatsoever.

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

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The positive density conjecture for integer Apollonian circle packings is true

You need to know: Circles in the plane, radius of a circle, tangent circles, point of tangency, curvature of a circle ( $1/r$ where r is the radius).

Background: A set of four mutually tangent circles in the plane with distinct points of tangency is called a Descartes configuration. Given a Descartes configuration, one
can construct four new circles, each of which is tangent to three of the given ones.
Continuing to repeatedly fill the interstices between mutually tangent circles with
further tangent circles, we arrive at an infinite circle packing. It is called a bounded Apollonian circle packing (ACP). It is known that if the original four circles have integer curvatures, all of the circles in the packing will have integer curvatures as well. In this case, the ACP is called integer. For an integer ACP ${\cal P}$ , let $\kappa({\cal P},X)$ denote the number of distinct integers up to X occurring as curvatures in the packing. Let $X_{\cal P}$ be the curvature of the largest circle in ${\cal P}$ .

The Theorem: On 21st January 2010, Jean Bourgain and Elena Fuchs submitted to the Journal of the AMS a paper in which they proved that for any integer ACP ${\cal P}$
there exist a constant $c>0$ depending on ${\cal P}$ such that $\kappa({\cal P},X) \geq c X$ for all $X \geq X_{\cal P}$ .

Short context: Apollonian circle packing is named after Apollonius of Perga, who lived more than 2000 years ago, and is studied by many researchers since that. Important research directions are counting circles with the curvature at most X (see here), as well as the number $\kappa({\cal P},X)$ of distinct integers up to X occurring as curvatures. Graham et.al. proved in 2003 that $\kappa({\cal P},X) \geq c'\sqrt{X}$ for some constant $c'$ and conjectured that this can be improved to $\kappa({\cal P},X) \geq c X$ . The Theorem proves this conjecture. It also makes an important step towards even more general local-global conjecture, see here.

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

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The Schmidt conjecture in simultaneous Diophantine approximation is true

You need to know: Notations: ${\mathbb N}$ for the set of positive integers, $||x||$ for the distance from real number x to the nearest integer. In addition, you need to know the concept of Hausdorff dimension of set $A\subset {\mathbb R}^2$ to understand the context.

Background: Let S be the set of pairs of real numbers $(i,j)$ such that $0\leq i,j \leq 1$ and $i+j=1$ . For $(i; j)\in S$ , let $\text{Bad}(i; j)$ denote the set of points $(x; y)\in {\mathbb R}^2$ for which there exists a positive constant $c=c(x,y)$ such that $\max\{||qx||^{1/i}, ||qy||^{1/j}\} > c/q$ for all $q \in {\mathbb N}$ .

The Theorem: On 15th January 2010, Dzmitry Badziahin, Andrew Pollington, and Sanju Velani submitted to arxiv and the Annals of Mathematics a paper in which they proved that for any finite number $(i_1; j_1); \dots ; (i_d; j_d)$ of pairs from S, the set $\bigcap\limits_{t=1}^d \text{Bad}(i_t; j_t)$ is non-empty.

Short context: A real number x is said to be badly approximable if there exists a constant $c(x)>0$ such that $||qx|| > c(x)/q$ for all $q \in {\mathbb N}$ . Sets $\text{Bad}(i; j)$ are natural generalisations containing pairs $(x; y)$ of simultaneously badly approximable numbers. The Theorem confirms a 1983 conjecture of Schmidt. In fact, Schmidt conjectured it for specific values $d=2$ , $i_1=1/3$ , and $j_1=2/3$ , and even this remained open. Any counterexample to this conjecture would imply the famous Littlewood conjecture (see here), but the Theorem states that there is no counterexample. In fact, Badziahin, Pollington, and Velani proved a much stronger result that the set $\bigcap\limits_{t=1}^d \text{Bad}(i_t; j_t)$ is not only non-empty, but has Hausdorff dimension 2 – the same as the whole plane. In a later work, this result has been extended to higher dimensions.

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

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The density Hales-Jewett theorem holds with an explicit bound

You need to know: Let ${\mathbb N}$ be the set of positive integers $\{1,2,3,\dots\}$ , big O notation.

Background: For positive integers k,n, let $[k]=\{1,2,\dots,k\}$ and let $[k]^n$ be the set of vectors $x=(x_1, \dots, x_n)$ such that each $x_i\in[k]$ . Let $y=(y_1, \dots, y_n)$ be a vector such that each $y_i$ either belongs to $[k]$ or equal to the wildcard value *. Assume that at least one coordinate of y takes the wildcard value *. For $j=1,2,\dots, k$ , let $y^j \in [k]^n$ be a vector obtained from y by setting all the wildcards * equal to j. The set $y^1, y^2, \dots, y^n$ is called a combinatorial line. Let $A:{\mathbb N} \times {\mathbb N} \to {\mathbb N}$ be a function such that $A(k,1)=2$ for all k, $A(1,n)=2n$ for all n, and $A(k,n) = A(k-1, A(k,n-1))$ for all $k, n>1$ .

The Theorem: On 20th October 2009, a large group of mathematicians submitted to arxiv a paper in which they proved the following result. For every positive integer k and every real number $\delta>0$ , there exists a positive integer $N=N(k,\delta)$ such that if $n\geq N$ and A is any subset of $[k]^n$ with at least $\delta k^n$ elements, then A contains a combinatorial line. Moreover, one may take $N(3,\delta) = A(3,O(1/\delta^2))$ and $N(k, \delta) = A(k+1,O(1/\delta))$ for $k \geq 4$ .

Short context: In 1975, Szemerédi proved that for any $k\in {\mathbb N}$ and any $\delta > 0$ , there exists a positive integer $M = M(k, \delta)$ such that every subset of the set $\{1, 2, \dots ,M\}$ of size at least $\delta M$ contains an arithmetic progression of length k. In 1991, Furstenberg and Katznelson proved the statement of the Theorem without “moreover” part, and called it “the density Hales-Jewett theorem”. This theorem greatly generalises (and easily implies) the Szemerédi theorem. However, the proof of Furstenberg and Katznelson does not imply any explicit algorithm how to actually compute $N=N(k,\delta)$ given $k$ and $\delta$ . The Theorem provides such an algorithm. It also has much simpler proof. It is proved by a large group of mathematicians in an online discussion, which is called a Polymath project.

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

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Any nondegenerate analytic submanifold of R^n is of Khintchine type for divergence

You need to know: Notation ${\mathbb R}^+$ for positive real numbers, Euclidean space ${\mathbb R}^n$ of vectors $x=(x_1,\dots,x_n)$ , norm $||x||_\infty = \max_i |x_i|$ , set ${\mathbb Z}^n$ of vectors with integer coordinates, affine subspace of ${\mathbb R}^n$ , proper subspace of ${\mathbb R}^n$ , open subset U of ${\mathbb R}^n$ , analytic map $f: U \to {\mathbb R}^m$ , the notion of (Lebesgue) almost all.

Background: For $y\in {\mathbb R}^n$ , let $||y||=\min\limits_{x \in {\mathbb Z}^n}||y-x||_\infty$ . For a decreasing function $\psi: {\mathbb R}^+ \to {\mathbb R}^+$ , let $S_n(\psi)$ be the set of points $y\in {\mathbb R}^n$ such that inequality $||m\cdot y||<\psi(m)$ holds for infinitely many positive integers m. Let $M=\{(x_1, \dots, x_d, f_1(x), \dots, f_m(x)) \in {\mathbb R}^n: x=(x_1, \dots, x_d) \in U)\}$ , where U is an open subset of ${\mathbb R}^d$ and $f=(f_1, \dots, f_m): U \to {\mathbb R}^m$ is an analytic map. We will call M analytic submanifold of ${\mathbb R}^n$ . M is called nondegenerate if it is not contained in a proper affine subspace of ${\mathbb R}^n$ . M is called of Khintchine type for divergence if for any $\psi$ such that $\sum\limits_{k=1}^\infty \psi(k)^n = \infty$ , almost all points on M belong to $S_n(\psi)$ .

The Theorem: On 2nd April 2009, Victor Beresnevich submitted to arxiv a paper in which he proved that any nondegenerate analytic submanifold of ${\mathbb R}^n$ is of Khintchine type for divergence.

Short context: If $x\in S_n(\psi)$ , then all coordinates of x can be simultanuously approximated by rational numbers with the same denominator, where function $\psi$ controls the error of approximation. In 1924, Khintchin proved that almost all points in ${\mathbb R}^n$ belong to $S_n(\psi)$ . However, submanifolds of ${\mathbb R}^n$ has measure $0$ , so Khintchin’s theorem says nothing about points on them. The Theorem states that coordinates of almost all points of any nondegenerate analytic submanifold of ${\mathbb R}^n$ can be simultanuously approximated. This result was known earlier only for $n=2$ .

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

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Any C^(2) nondegenerate planar curve is of Khintchine type for convergence

You need to know: Differentiable function, (k times) continuously differentiable function, sets of measure $0$ in ${\mathbb R}$ and ${\mathbb R}^2$ , notion of “almost all”, sum of infinite series, standard notations ${\mathbb R}^+$ for positive real numbers, ${\mathbb Z}$ for integers, ${\mathbb N}$ for natural numbers, $f'(x)$ for derivative, $f''(x)$ for the second derivative.

Background: Let $\psi: {\mathbb R}^+ \to {\mathbb R}^+$ be a decreasing function. Pair $(\alpha,\beta) \in {\mathbb R}^2$ is called simultaneously $\psi$ -approximable if there are infinitely many $n \in {\mathbb N}$ such that $\max\{||n\alpha||,||n\beta||\} < \psi(n)$ , where $||x||=\min\{|x-m|: m \in {\mathbb Z}\}$ denotes the distance from x to the nearest integer.

The Theorem: On 13th December 2005, Robert Vaughan and Sanju Velani submitted to the Inventiones mathematicae a paper in which they proved the following result. Let $\psi: {\mathbb R}^+ \to {\mathbb R}^+$ be a decreasing function with $\sum\limits_{n=1}^\infty \psi(n)^2 < \infty$ . Let f be a twice continuously differentiable function on some interval $(a, b)$ , such that $f''(x) \neq 0$ for almost all $x\in(a,b)$ . Then for almost all $x\in(a,b)$ the pair $(x, f(x))$ is not simultaneously $\psi$ -approximable.

Short context: Let $S(\psi)$ be the set of all pairs $(\alpha,\beta) \in {\mathbb R}^2$ that are simultaneously $\psi$ -approximable. A planar curve is called of Khintchine type for divergence if almost all points on it belong to $S(\psi)$ whenever $\sum\limits_{n=1}^\infty \psi(n)^2 = \infty$ . It is called of Khintchine type for convergence if almost all points on it does not belong to $S(\psi)$ whenever $\sum\limits_{n=1}^\infty \psi(n)^2 < \infty$ . The Theorem proves that all smooth nondegenerate planar curves are of Khintchine type for convergence, complementing an earlier result that they are of Khintchine type for divergence.

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

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