Any C^(3) nondegenerate planar curve is of Khintchine type for divergence

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 24th September 2003, Victor Beresnevich, Detta Dickinson, and Sanju Velani submitted to the Annals of Mathematics 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 three times 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 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. The (two-dimensional version of) famous Dirichlet’s approximation theorem implies that $S(\psi)={\mathbb R}^2$ for $\phi(n)=\frac{1}{\sqrt{n}}$ (see here for the progress about related conjecture of Littlewood). In 1924, Khintchin proved that almost all pairs $(\alpha,\beta) \in {\mathbb R}^2$ belong to $S(\psi)$ if and only if $\sum\limits_{n=1}^\infty \psi(n)^2 = \infty$ . However, planar curves of the form $(x, f(x))$ has measure $0$ in ${\mathbb R}^2$ , so Khintchin’s theorem says nothing about simultaneous approximability on such curves. A curve is called of Khintchine type for divergence if almost all points on it belong to $S(\psi)$ whenever $\sum\limits_{n=1}^\infty \psi(n)^2 = \infty$ . The Theorem proves this property for a broad class of curves (which are called $C^{(3)}$ nondegenerate planar curves).

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.8 of this book for an accessible description of the Theorem.

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The Hopf condition for bilinear forms holds over arbitrary fields

You need to know: Factorial $n!$ of a non-negative integer n. Field.

Background: We say that field F with additive identity $0$ and multiplicative identity $1$ has characteristic not equal to 2 if $1+1 \neq 0$ . For a field F and positive integers r, s, and n, by sums-of-squares formula of type [r, s, n] over F we mean an identity of the form $\left(\sum\limits_{i=1}^r x_i^2\right)\cdot \left(\sum\limits_{i=1}^s y_i^2\right) = \sum\limits_{i=1}^n z_i^2$ , where each $z_i$ is a bilinear form, that is, expression of the form $z_i=\sum\limits_{j=1}^r\sum\limits_{k=1}^s c_{ijk}x_jy_k$ , with some coefficients $c_{ijk}\in F$ .

The Theorem: On 11th September 2003, Daniel Dugger and Daniel Isaksen submitted to arxiv a paper in which they proved that, if F is a field of characteristic not equal to 2, and a sums-of-squares formula of type [r, s, n] exists over F, then the numbers $\frac{n!}{i!(n-i )!}$ are even integers for all i such that $n-r < i < s$ .

Short context: For the special case when $F={\mathbb Q}$ is the field of rational numbers, the Theorem has been known since 1939, and the condition “the numbers $\frac{n!}{i!(n-i )!}$ are even integers for all i such that $n-r < i < s$ ” is known as the Hopf condition. This condition is one of the central tools in the problem of understanding for which r, s, and n, a sums-of-squares formula of type [r, s, n] exists. The Theorem establishes this condition over an arbitrary field (except of fields of characteristic 2 in which sums-of-squares formulas of type [r, s, n] trivially exist for all r, s, n).

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.5 of this book for an accessible description of the Theorem.

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Upper bound for the number of number fields with fixed degree and bounded discriminant

You need to know: Field, field ${\mathbb Q}$ of rational numbers, isomorphic fields, vector space, vector space over a field, dimension of a vector space, matrix, determinant of a matrix.

Background: Number field is a field F that contains ${\mathbb Q}$ and has finite dimension n when considered as a vector space over ${\mathbb Q}$ . Number n is called the degree of F. A set $e=\{e_1, e_2, \dots, e_n\}$ of n elements of F is called basis of F if every $x\in F$ can be written as $x=\sum_{i=1}^n c_i(x) e_i$ with coefficients $c_i(x) \in {\mathbb Q}$ . Sum $\text{Tr}(x)=\sum_{i=1}^nc_i(x\cdot e_i)$ does not depend on the choice of basis e and is called trace of x. If, for every $x\in F$ , all $c_i(x)$ are integers, e is called integral basis of F. The determinant of an $n\times n$ matrix with entries $\text{Tr}(e_i\cdot e_j)$ , $i=1,\dots,n$ , $j=1,\dots,n$ , does not depend on the choice of integral basis e and is called the discriminant of F. For $X>0$ , let $N_n(X)$ denotes the number of non-isomorphic number fields of degree n with absolute value of the discriminant at most X.

The Theorem: On 8th September 2003, Jordan Ellenberg and Akshay Venkatesh submitted to arxiv a paper in which they proved the existence of constant $B_n$ depending only on n and absolute constant C, such that inequality $N_n(X) \leq B_n X^{\exp(C\sqrt{\log n})}$ holds for all $n>2$ and all $X>0$ .

Short context: Counting number fields up to isomorphism is a basic and important open problem in the area. There is a conjecture that $N_n(X)$ grows as linear function of X for every fixed n, but, before 2003, this was known only for $n\leq 3$ (in later works – see here and here – Bhargava proved it for $n=4$ and $n=5$ ). For general n, the best upper bound was $N_n(X) \leq B_n X^{(n+2)/4}$ . The Theorem proves a bound which is significantly better for large n.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 6.2 of this book for an accessible description of the Theorem.

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Littlewood conjecture holds outside of a set of Hausdorff dimension 0

You need to know: For the Theorem: logarithm, limit, notations $|x|$ for the absolute value of $x$ , ${\mathbb R}^2$ for the Euclidean plane, and $\cup$ for the set union. In addition, you need to know $\liminf$ notation, Lebesgue measure, and Hausdorff dimension to understand the context.

Background: We say set set $S\subset {\mathbb R}^2$ has box dimension $0$ if $\lim\limits_{\epsilon\to 0+}\frac{\log N_S(\epsilon)}{\log(1/\epsilon)}=0$ , where $N_S(\epsilon)$ is the minimal number of squares of side length $\epsilon$ needed to cover S.

Let $T\subset {\mathbb R}^2$ be the set of all pairs $(\alpha, \beta)$ , for which there exists an $\epsilon>0$ such that inequality $\left|\alpha-\frac{a}{c}\right|\cdot \left|\beta-\frac{b}{c}\right| \geq \frac{\epsilon}{c^3}$ holds for all integers $a, b$ and $c\neq 0$ .

The Theorem: On 5th September 2003, Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss submitted to the Annals of Mathematics a paper in which they proved that $T$ can be written as a union of sets $S_1 \cup S_2 \cup S_3 \cup \dots$ , where each $S_i$ has box dimension $0$ .

Short context: In the 1930s, Littlewood conjectured that for any two real numbers $\alpha$ and $\beta$ , $\liminf\limits_{n\to\infty} n ||n\alpha|| ||n\beta|| = 0$ , where $||x||$ denotes the distance from real number x to the nearest integer. It is easy to see that set T is exactly the set of pairs $(\alpha, \beta)$ for which this conjecture does not hold. Hence, the conjecture predicts that T is an empty set. This remains open, but it follows from the 1909 Borel theorem that T has Lebesgue measure $0$ in ${\mathbb R}^2$ . The Theorem proves much stronger result on how “small” is T. In particular, it implies that T must have Hausdorff dimension $0$ .

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 6.5 of this book for an accessible description of the Theorem.

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For any b>0 exists k>0 such that for any finite set A of integers either|kA|>|A|^b or|A^k|>|A|^b

You need to know: Notation ${\mathbb Z}$ for the set of integers, notation $|A|$ for the size of set A.

Background: For $A,B\subset {\mathbb Z}$ , denote $A+B=\{a+b, \, a\in A, b\in B\}$ and $A\times B=\{a\cdot b, \, a\in A, b\in B\}$ . For $A\subset {\mathbb Z}$ and positive integer k, denote $kA= A + A + \dots + A$ and $A^k = A \times A \times \dots \times A$ , where A in the sum and in the product is repeated k times.

The Theorem: On 3rd September 2003, Jean Bourgain and Mei-Chu Chang submitted to arxiv a paper in which they proved that for any integer $b>0$ there is an integer $k = k(b) > 0$ such that $\max(|kA|,|A^k|)\geq|A|^b$ holds for any non-empty finite set $A \subset {\mathbb Z}$ .

Short context: In 1983, Erdős and Szemerédi proved the existence of positive constants c and $\epsilon$ such that inequality $\max(|A+A|, |A \cdot A|) \geq c|A|^{1+\epsilon}$ holds for any non-empty finite set $A \subset {\mathbb Z}$ (see here for a finite field analogue of this result). The Theorem states that if we consider k-fold sums and products for sufficiently large k, then exponent $1+\epsilon$ in the Erdős-Szemerédi theorem can be replaced by an arbitrary large constant.

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

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There exists c>0 such that every set of natural numbers of density cn^(1/2) is subcomplete

You need to know: Notation ${\mathbb N}$ for the set of natural numbers, notation $|B|$ for the number of elements in set B, arithmetic progression.

Background: Let ${\cal A}\subset {\mathbb N}$ be an infinite set of natural numbers (without repetitions). We say that ${\cal A}$ has density at least $f(n)$ if $|{\cal A}\cap\{1,2,\dots,n\}|\geq f(n)$ for all sufficiently large n. Let $S_A=\{\sum\limits_{x\in B} x\,|\,B\subset {\cal A}, |B|<\infty\}$ be the set of all possible sums of finite number of elements of ${\cal A}$ . We say that ${\cal A}$ is subcomplete if $S_A$ contains an infinite arithmetic progression.

The Theorem: On 19th March 2003, Endre Szemerédi and Van Vu submitted to the Annals of Mathematics a paper in which they proved the existence of constant $c>0$ such that every set ${\cal A}\subset {\mathbb N}$ with density at least $c\sqrt{n}$ is subcomplete.

Short context: We say that set ${\cal A}$ is complete if $S_A$ contains all sufficiently large integers. This notion was introduced by Erdős in the sixties. It is well studied, the central problem being to find sufficient conditions for completeness. In 1962, Erdős conjectured the existence of $c>0$ such that if (i) ${\cal A}$ has density at least $c\sqrt{n}$ and (ii) $S_A$ contains an element of every infinite arithmetic progression, then ${\cal A}$ is complete. In 1966, Folkman conjectured that every ${\cal A}$ satisfying (i) is subcomplete and deduced the Erdős conjecture from this. The Theorem confirms the Folkman conjecture and hence the Erdős conjecture.

Links: The original paper is available here. See also Section 6.1 of this book for an accessible description of the Theorem.

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Any positive proportion of primes contains a 3-term arithmetic progression

You need to know: Prime numbers, notation $|A|$ for size of set A, arithmetic progression, limit superior $\limsup$ .

Background: Let ${\mathbb N}$ be the set of positive integers, and let ${\cal P}$ be the set of primes. For $n\in {\mathbb N}$ , let $S_n=\{1,2,\dots,n\}$ , and let ${\cal P}_n = {\cal P}\cap S_n$ be the set of primes not exceeding n. We say that subset $A\subset N$ of integers has positive upper density if $\limsup\limits_{n\to\infty}\frac{|A \cap S_n|}{n} > 0$ . Similarly, we say that subset $A\subset {\cal P}$ of primes has positive upper density if $\limsup\limits_{n\to\infty}\frac{|A \cap {\cal P}_n|}{|{\cal P}_n|} > 0$ .

The Theorem: On 25th February 2003, Ben Green submitted to arxiv a paper in which he proved that every subset of ${\cal P}$ of positive upper density contains a 3-term arithmetic progression (3AP).

Short context: In 1939, Van der Corput proved that the set of primes contains infinitely many 3APs. In 1953, Roth proved that any subset of integers of positive upper density contains a 3AP. The Theorem provides a common generalisation of these two results. In a later work, Green and Tao proved that the same is true for arithmetic progressions of any length.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 5.5 of this book for an accessible description of the Theorem.

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The Erdős–Szemerédi sum-product theorem holds in finite fields

You need to know: Field, finite field $F_q$ with q elements, size $|A|$ of set A.

Background: For $A\subset F_q$ , denote $A+A=\{a+b, \, a\in A, b\in A\}$ and $A\cdot A=\{a\cdot b, \, a\in A, b\in A\}$ .

The Theorem: On 29th January 2003, Jean Bourgain, Nets Katz, and Terence Tao submitted to arxiv a paper in which they proved, among other results, that for any $\delta>0$ there exist positive constants c and $\epsilon$ such that for any prime q, and any $A\subset F_q$ such that $q^\delta<|A|<q^{1-\delta}$ , we have $\max(|A+A|, |A \cdot A|) \geq c|A|^{1+\epsilon}$ .

Short context: In 1983, Erdős and Szemerédi proved the existence of positive constants c and $\epsilon$ such that inequality $\max(|A+A|, |A \cdot A|) \geq c|A|^{1+\epsilon}$ holds for any finite non-empty set A of real numbers. The Erdős–Szemerédi theorem can be viewed as an assertion that it is not possible for a large set to behave like an arithmetic progression and as a geometric progression simultaneously. The Theorem establishes a finite field analogue of this fundamental result.

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

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The tau_n=n conjecture in approximation by algebraic integers is false

You need to know: Polynomial, degree of a polynomial, leading coefficient, irreducible polynomial over the integers, supremum.

Background: A real number $\alpha$ is called algebraic number if there exists a polynomial P with integer coefficients such that $P(\alpha)=0$ . Real numbers which are not algebraic are called transcendental. Below, we may and will assume that P is irreducible over the integers. The largest absolute value of the coefficients of P is called the height of $\alpha$ and denoted $H(\alpha)$ . If the leading coefficient of P is 1, $\alpha$ is called algebraic integer. The degree of an algebraic integer $\alpha$ is the degree of P. Also, let $\gamma=\frac{1+\sqrt{5}}{2}$ denote the golden ratio.

The Theorem: On 11th October 2002, Damien Roy submitted to arxiv and Annals of Mathematics a paper in which he proved the existence of a transcendental real number $\xi$ and constant $c>0$ such that, for any algebraic integer $\alpha$ of degree at most 3, we have $|\xi-\alpha| \geq c H(\alpha)^{-\gamma^2}$ .

Short context: For a positive integer n, let $\tau_n$ be the supremum of all $\tau \in {\mathbb R}$ such that for any transcendental $\xi \in {\mathbb R}$ there exist infinitely many algebraic integers $\alpha$ of degree at most n such that $|\xi-\alpha| \leq H(\alpha)^{-\tau}$ . $\tau_n$ measures the quality of approximation of real numbers by algebraic integers. It is known that $\tau_2=2$ and has been conjectured that $\tau_n=n$ for all $n\geq 2$ . In 1969, Davenport and Schmidt proved that $\tau_3\geq \gamma^2$ . The Theorem implies that $\tau_3\leq \gamma^2$ . Hence $\tau_3=\gamma^2=\frac{3+\sqrt{5}}{2}<3$ , and the $\tau_n=n$ conjecture is false.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 3.8 of this book for an accessible description of the Theorem.

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For partition function p(.), no prime q other than 5,7,11 divides p(qn+b) for all n

You need to know: Integers, divisibility, prime numbers.

Background: Let $p(n)$ be the partition function, that is, the number of representations of positive integer n as a sum of positive integers (representations which differ by order of terms only are considered the same). By convention, we also put $p(0) = 1$ and $p(n) = 0$ for $n < 0$ .

The Theorem: On 22nd September 2002, Scott Ahlgren and Matthew Boylan submitted to Inventiones mathematicae a paper in which they proved that there are no prime $q \neq 5, 7, 11$ , and integer b such that $p(qn+b)$ is divisible by q for all integers n.

Short context: In 1920, Ramanujan proved that $p(5n+4)$ always divisible by 5, $p(7n+5)$ – by 7, $p(11n+6)$ – by 11, and conjectured that “there are no equally simple properties for any moduli involving primes other than these three”. The Theorem confirms (a natural formalisation of) this conjecture.

Links: The original paper is available here.

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