Minkowski’s conjecture is true in dimension n=6

You need to know: Euclidean space ${\mathbb R}^n$ , basis for ${\mathbb R}^n$ , inner product $(x,y)=\sum\limits_{i=1}^n x_i y_i$ in ${\mathbb R}^n$ , matrix, determinant of a matrix.

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 ${\mathbb R}^n$ . A lattice $L$ is called unimodular if the determinant of the $n \times n$ matrix with entries $(v_i,v_j)$ is $1$ or $-1$ . This property does not depend on the choice of basis for L.

For $x=(x_1, \dots, x_n) \in {\mathbb R}^n$ , denote $N(x)=|x_1\cdot x_2 \cdot \dots \cdot x_n|$ .

The Theorem: On 27th August 2004, Curtis McMullen submitted to the journal of the AMS a paper in which he proved that for any unimodular lattice $L \subset {\mathbb R}^6$ , and any $x\in {\mathbb R}^6$ , there is a $y\in L$ such that $N(x - y) \leq 2^{-6}=\frac{1}{64}$ .

Short context: Minkowski’s conjecture states that for any unimodular lattice $L \subset {\mathbb R}^n$ , we have $\sup\limits_{x\in {\mathbb R}^n} \inf\limits_{y\in L} N(x - y) \leq 2^{-n}$ . The conjecture is interesting in its own, and also has number-theoretic consequences. Before 2004, it was known for $n \leq 5$ . The Theorem proves it for $n=6$ .

Links: The original paper is available here.

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Linear growth of the number of quartic fields with bounded discriminant

You need to know: Prime numbers, infinite product, field, isomorphic and non-isomorphic fields. Also, see this previous theorem description for the concepts of number field, degree of a number field, and discriminant of a number field.

Background: Number fields of degree $n=4$ are called quartic. 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. Also, let ${\cal P}$ be the set of prime numbers.

The Theorem: On 7th June 2004, Manjul Bhargava submitted to the Annals of Mathematics a paper in which he proved that the limit $\lim\limits_{X\to\infty}\frac{N_4(X)}{X}$ exists and is equal to $c_4 = \frac{5}{24}\prod\limits_{p\in {\cal P}}\left(1+p^{-2}-p^{-3}-p^{-4}\right) = 0.253...$ .

Short context: Counting number fields up to isomorphism is a basic and important open problem in the area. There is and old folklore conjecture that $\lim\limits_{n\to\infty}\frac{N_n(X)}{X} = c_n>0$ for every fixed n, but, before 2004, this was known only for $n\leq 3$ (see here for the best upper bounds for $N_n(X)$ available for general n). The Theorem proves this conjecture for $n=4$ (quartic fields). In a later work, Bhargava also proved it for $n=5$ .

Links: The original paper is available here. See also Section 5.9 of this book for an accessible description of the Theorem.

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The Dufﬁn–Schaeffer conjecture implies its Hausdorff measure version

You need to know: Set ${\mathbb N}$ of natural numbers, set ${\mathbb R}^+$ of positive real numbers, co-prime integers, monotonic function, infimum, Lebesgue measure, infinite sum.

Background: For a function $g:{\mathbb N}\to {\mathbb R}^+$ , let $S(g)$ be the set of all real numbers $x \in [0,1]$ such that the inequality $\left|x-\frac{a}{b}\right| < \frac{g(b)}{b}$ has infinitely many co-prime solutions $a,b$ . The Duffin–Schaeffer conjecture states that set $S(g) \subset [0,1]$ has Lebesgue measure 1 if and only if $\sum\limits_{n=1}^{\infty} g(n)\frac{\phi(n)}{n}=\infty$ , where $\phi(n)$ is the number of positive integers which are less than n and co-prime with it.

Let $f:[0,\infty) \to [0,\infty)$ be a continuous, non-decreasing function, with $f(0)=0$ . For $\delta>0$ , a $\delta$ -cover of set $S \subset {\mathbb R}$ is a sequence of intervals $I_1, I_2, \dots, I_n, \dots$ of lengths $r_i = |I_i|\leq \delta$ for all i such that $S \subset \bigcup\limits_{i=1}^\infty I_i$ . Let $H_\delta^f(S) = \inf \sum\limits_{i=1}^{\infty}f(r_i/2)$ , where the infimum is taken over all $\delta$ -covers of S. The number $H^f(S)=\lim\limits_{\delta\to 0} H_\delta^f(S)$ is called the Hausdorff f-measure of S. The Hausdorff measure version of the Duffin–Schaeffer conjecture states that if $f(r)/r$ is monotonic, then $H^f(S(g))=H^f([0,1])$ if and only if $\sum\limits_{n=1}^{\infty} f(\frac{g(n)}{n})\phi(n)=\infty$ .

The Theorem: On 2nd June 2004, Victor Beresnevich and Sanju Velani submitted to the Annals of Mathematics a paper in which they proved that the Duffin–Schaeffer conjecture implies its Hausdorff measure version.

Short context: The Duffin–Schaeffer conjecture is a fundamental conjecture in the theory of rational approximation. Its Hausdorff measure version looks much more general and difficult, but the Theorem states that it reduces to the original conjecture. In a later work, Koukoulopoulos and Maynard proved the Duffin–Schaeffer conjecture. By The Theorem, this implies that its Hausdorff measure version is also true.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 6.7 of this book for an accessible description of the Theorem.

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No arithmetic sequence is very well-distributed

You need to know: Prime numbers, logarithm, $O(1)$ notation,

Background: Let ${\cal P}$ be the set of primes. For $x>0$ , let ${\cal P}(x)=\{p\in {\cal P}: p<x\}$ be the set of primes less than x, $\theta(x)=\sum\limits_{p\in{\cal P}(x)} \log p$ , and let $\Delta(x,y) = \frac{\theta(x+y) - \theta(x) - y}{y}$ .

The Theorem: On 1st June 2004, Andrew Granville and Kannan Soundararajan submitted to arxiv and the Annals of Mathematics a paper in which they proved, among other results, the following theorem. Let x be large and y be such that $\log x \leq y \leq \exp\left(\frac{\beta \sqrt{\log x}}{2\sqrt{\log\log x}}\right),$ where $\beta>0$ is an absolute constant. Then there exist numbers $x_+$ and $x_-$ in $(x,2x)$ such that $\Delta(x_+,y) \geq y^{-\delta(x,y)}$ and $\Delta(x_-,y) \leq -y^{-\delta(x,y)},$ where $\delta(x,y) = \frac{1}{\log\log x}\left( \log\left(\frac{\log y}{\log \log x}\right) + \log\log \left(\frac{\log y}{\log \log x}\right) + O(1) \right)$ .

Short context: Function $\theta(x)$ counts primes up to x, each prime p with weight $\log p$ . Famous prime number theorem states that that $\theta(x) \approx x$ for large x, hence the number of weighted primes on interval $(x, x+y]$ is about $\theta(x+y) - \theta(x) \approx y$ . Function $\Delta(x,y)$ measures the quality of this approximation, and the Theorem states that there are intervals with much more and much less primes than average. In author’s words, primes are not very well distributed. In fact, the authors proved a much more general (and surprising) result that the same holds for any “arithmetic sequence”, but the exact definition of this is too difficult to be presented here.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.3 of this book for an accessible description of the Theorem.

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The complexity of every irrational algebraic number has superlinear growth

You need to know: Rational and irrational numbers, decimal expansion of a real number, limit inferior $\liminf$ . You also need to know b-ary expansion of real numbers to understand the context.

Background: By complexity function (or just “complexity”) of a real number $\alpha$ we mean the number $p(n)$ of distinct blocks of digits of length n occurring in its decimal expansion. A real number $\alpha$ is called algebraic number if there exists a polynomial P with integer coefficients such that $P(\alpha)=0$ .

The Theorem: On 30th May 2004, Boris Adamczewski and Yann Bugeaud submitted to the Annals of Mathematics a paper in which they proved that the complexity function $p(n)$ of every irrational algebraic number satisfies $\liminf\limits_{n\to\infty}\frac{p(n)}{n} =+\infty$ .

Short context: It is natural to conjecture that the decimal expansion of irrational algebraic numbers such as $\sqrt{2}$ contains all possible digit patterns, that is, $p(n)=2^n$ . However, before 2004, it was only known that $\liminf\limits_{n\to\infty}(p(n)-n) =+\infty$ . The Theorem is a considerable improvement in comparison with this result. It remains true in any b-ary expansion with any base $b\geq 2$ .

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.2 of this book for an accessible description of the Theorem.

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The primes contain arbitrarily long arithmetic progressions

You need to know: Prime numbers.

Background: A (non-trivial) arithmetic progression of length n is a sequence $a_1, a_2, \dots, a_n$ of real numbers such that $a_i = a_1 + (i-1)d, \, i=1,2,\dots,n$ for some $d\neq 0$ .

The Theorem: On 8th April 2004, Ben Green and Terence Tao submitted to arxiv a paper in which they proved that for every positive integer n there exist an arithmetic progression of length n consisting of only prime numbers.

Short context: In 1939, Van der Corput proved that the set of primes contains infinitely many arithmetic progressions of length $n=3$ . The Theorem proves this for all n. Before 2004, this was open even for $n=4$ . Moreover, Green and Tao proved that even any positive proportion of primes contains infinitely many arithmetic progressions of length n for all n. This generalises an earlier theorem of Green, which proves this for $n=3$ .

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 8.2 of this book for an accessible description of the Theorem.

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Quartic rings can be explicitly parametrized

You need to know: Matrix, invertible matrix, determinant of a matrix, group, abelian group, group ${\mathbb Z}^n$ , ring, commutative ring, isomorphic groups and rings.

Background: Each commutative ring R is an abelian group with respect to addition. We say that R has rank n if this group is isomorphic to ${\mathbb Z}^n$ . Rings of rank $n=4$ are called quartic rings. An integral ternary quadratic form is an expression of the form $ax^4+bx^3y+cx^2y^2+dxy^3+ey^4$ with $a,b,c,d,e \in {\mathbb Z}$ . We say that two such forms A and B are linearly independent over ${\mathbb Q}$ if $uA+vB=0$ with rational $u,v$ is possible only if $u=v=0$ . Let $GL_n({\mathbb Z})$ be the set of invertible $n \times n$ matrices with integer entries, and let $GL_2^{\pm 1}({\mathbb Q})$ be the set of $2 \times 2$ matrices with rational entries and determinant $\pm 1$ .

The Theorem: On 29th February 2004, Manjul Bhargava submitted to the Annals of Mathematics a paper in which he proved the existence of a canonical bijection between isomorphism classes of nontrivial quartic rings and $GL_3({\mathbb Z}) \times GL^{\pm 1}_2({\mathbb Q})$ -equivalence classes of pairs $(A,B)$ of integral ternary quadratic forms where A and B are linearly independent over ${\mathbb Q}$ .

Short context: Explicit parametrizations for rings of ranks $n=2$ (known as quadratic rings) and $n=3$ (cubic rings) are relatively easy and well-known. The Theorem provides such a parametrization for quartic rings. In a later work, Bhargava also obtained explicit parametrization for rings of rank $n=5$ (quintic rings).

Links: The original paper is available here. See also Section 4.7 of this book for an accessible description of the Theorem.

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The number of distinct products in N by N multiplication table

You need to know: Basic arithmetic, logarithm.

Background: For positive integer N, let $M(N)$ denote the number of distinct integers n which can be written as $n=a\cdot b$ , where a and b are positive integers not exceeding N.

The Theorem: On 18th January 2004, Kevin Ford submitted to arxiv a paper in which he proved, among other results, the existence of positive constants $c_1$ and $c_2$ such that inequality $c_1\frac{N^2}{(\log N)^\delta(\log\log N)^{3/2}} \leq M(N) \leq c_2\frac{N^2}{(\log N)^\delta(\log\log N)^{3/2}}$ holds for all N, where $\delta=1-\frac{1+\log\log 2}{\log 2}=0.08607...$ .

Short context: The number $M(N)$ of distinct products in $N\times N$ multiplication table has been studied starting since 1955, when Erdős proved that $\lim\limits_{N\to\infty}\frac{M(N)}{N^2}=0$ . However, exactly how fast $M(N)$ grows was an open question, answered by the Theorem. In fact, this result is only one out of many corollaries of a deep theory developed by Ford for estimating the number $H(x,y,z)$ of positive integers $n \leq x$ having a divisor in $(y,z]$ and the number $H_r(x,y,z)$ of positive integers $n \leq x$ having exactly r such divisors.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 8.10 of this book for an accessible description of the Theorem.

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The only perfect powers in the Fibonacci sequence are 0, 1, 8, and 144

You need to know: Basic arithmetic only.

Background: Let $F_0, F_1, \dots, F_n, \dots$ be the Fibonacci sequence defined by $F_0 = 0, F_1 = 1,$ and $F_{n+2} = F_{n+1} + F_n$ for $n\geq 0$ . Let $L_0, L_1, \dots, L_n, \dots$ be the Lucas sequence defined by $L_0 = 2, L_1 = 1,$ and $L_{n+2} = L_{n+1} + L_n$ for $n\geq 0$ . A perfect power is an integer of the form $m^p$ for integers $m$ and $p\geq 2$ .

The Theorem: On 24th November 2003, Yann Bugeaud, Maurice Mignotte, and Samir Siksek submitted to the Annals of Mathematics a paper in which they proved that (a) the only perfect powers in the Fibonacci sequence are $F_0 = 0$ , $F_1 = 1$ , $F_2 = 1$ , $F_6 = 8$ , and $F_{12} = 144$ , and (b) the only perfect powers in the Lucas sequence are $L_1=1$ and $L_3=4$ .

Short context: The Fibonacci sequence is perhaps the most famous and well-studied sequence of integers in mathematics. In 1951, Ljunggren proved that the only perfect squares in this sequence are $0=0^2, 1=1^2$ , and $144=12^2$ . In 1969, London and Finkelstein proved that the only perfect cubes are $0,1$ and $8=2^3$ . Are there any other perfect powers in the sequence? By 2003, it was known that there are no more p-th powers with $p\leq 17$ or with $p\geq 5.1\cdot 10^{17}$ . The Theorem proves this for all p, and resolves this question also for the Lucas sequence.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 6.3 of this book for an accessible description of the Theorem.

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Lehmer’s conjecture is true for polynomials with odd coefficients

You need to know: Polynomials, degree of a polynomial, roots of a polynomial, divisor of a polynomial, irreducible polynomial, complex numbers, notation ${\mathbb Z}[x]$ for polynomials in variable x with integer coefficients, notation $a \equiv b\,(\text{mod}\, m)$ if $a-b$ is divisible by m.

Background: An irreducible polynomial $P \in {\mathbb Z}[x]$ is cyclotomic if $P$ is a divisor of $x^n-1$ for some integer $n \geq 1$ . Every polynomial $P \in {\mathbb Z}[x]$ of degree n can be written as $P(x)=a\prod\limits_{i=1}^n(x-\alpha_i)$ , where $a\in {\mathbb Z}$ and $\alpha_i$ are (possibly complex) roots of P. Mahler ’s measure of P is $M(P):=|a|\prod\limits_{i=1}^n\max\{1,|\alpha_i|\}$ .

For integer $m\geq 2$ , let $D_m:=\left\{\sum\limits_{i=0}^n a_ix^i \in {\mathbb Z}[x]\,|\,a_i \equiv 1 (\text{mod}\, m), \, 0\leq i \leq n\right\}$ .

The Theorem: On 2nd October 2003, Peter Borwein, Edward Dobrowolski, and Michael Mossinghoff submitted to the Annals of Mathematics a paper in which they proved that inequality $\log M(P) \geq c_m\left(1-\frac{1}{n+1}\right)$ holds for every $P \in D_m$ of degree n and no cyclotomic divisors, where $c_2=(\log 5)/4$ and $c_m=\log(\sqrt{m^2+1}/2)$ for $m>2$ .

Short context: In 1933, Lehmer asked if for every $\epsilon>0$ there exists a polynomial $P \in {\mathbb Z}[x]$ satisfying $1<M(P)<1+\epsilon$ . It is conjectured that the answer to this question is negative, and this is known as Lehmer’s conjecture. The Theorem implies that this conjecture holds for polynomials $P \in D_m$ . In particular, case $m=2$ of the Theorem implies that Lehmer’s conjecture holds for polynomials with odd coefficients.

Links: The original paper is available here. See also Section 7.7 of this book for an accessible description of the Theorem.

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