The size of subset of {1,…,N} without distinct degree polynomial progressions is O(N/(log log N)^c)

You need to know: Polynomial, degree of a polynomial, notation ${\mathbb Z}[y]$ for the set of polynomials in 1 variable with integer coefficients, notation $[N]$ for the set $\{1,2,\dots,N\}$ .

Background: Let $P=(P_1, \dots, P_m)$ be a finite set of polynomials $P_i \in {\mathbb Z}[y]$ . For integer $N>0$ , let $r_P(N)$ be the size of the largest subset of $[N]$ containing no subset of the form $x, x+P_1(y), \dots, x+P_m(y)$ with $y\neq 0$ .

The Theorem: On 1st September 2019, Sarah Peluse submitted to arxiv a paper in which she proved that if all polynomials $P_i$ in $P$ have distinct degrees and zero constant terms, then there exists a constant $c$ depending on $P_1, \dots, P_m$ such that
$r_P(N) \leq \frac{N}{(\log\log N)^c}$ .

Short context: Let $r_k(N)$ be the cardinality of the largest subset of $[N]$ which contains no nontrivial k-term arithmetic progressions. Famous Szemerédi’s theorem states that $\lim\limits_{N\to\infty}\frac{r_k(N)}{N}=0$ . In 2000, Gowers proved an explicit bound $r_k(N) < N(\log\log N)^{-c}$ for some constant $c=c(k)>0$ . In 1996, Bergelson and Leibman extended Szemerédi’s theorem to polynomial progressions and proved that $\lim\limits_{N\to\infty} \frac{r_P(N)}{N} = 0$ , if polynomials $P_i$ all have zero constant terms. The Theorem establishes an explicit upper bound on $r_P(N)$ , provided that all polynomials $P_i$ have distinct degrees.

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

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Kesten’s theorem holds for random Kronecker sequences on the torus T^d

You need to know: Matrix, determinant of a matrix, notation $AC$ for the image of set $C$ under linear transformation defined by matrix $A$ , notation $\times$ for direct product of sets, notation “ $x \mod 1$ ” for the real number $y\in[0,1)$ such that $x-y$ is an integer, notation ${\mathbb P}$ for the probability, selection uniformly at random.

Background: Let $d>0$ be an integer, $I=\{1,2,\dots,d\}$ , $T=[0,1)$ , and $T^d$ be set of vectors $x=(x_1, \dots, x_d)$ with $x_i \in T$ for every $i \in I$ . Let $U$ be the set of vectors $u=(u_1, \dots, u_d)$ such that $v_i\leq u_i \leq w_i$ for every $i \in I$ , where $v_i, w_i$ are fixed such that $0<v_i<w_i<1/2$ for every $i \in I$ . For each $u\in U$ , let $C_u$ be the set of vectors $y=(y_1, \dots, y_d)$ such that $|y_i|\leq u_i$ for every $i \in I$ . For a (small) $\eta>0$ , let $G_\eta$ be the set of $d\times d$ matrices with determinant $1$ and real entries $a_{ij}$ , such that $|a_{ii}-1|<\eta$ for all $i$ and $|a_{ij}|<\eta$ for all $i\neq j$ . Let $X=T^d \times T^d \times U \times G_\eta$ . For $\nu = (\alpha, x, u, A) \in X$ and integer $N>0$ , let $M(\nu, N)$ be the number of integers $1\leq m \leq N$ such that $(x+m\alpha) \mod 1 \in A C_u$ , and let $D(\nu, N)=M(\nu, N) - 2^d (\prod_{i=1}^d u_i) N$ .

The Theorem: On 19th November 2012 Dmitry Dolgopyat and Bassam Fayad submitted to arxiv a paper in which they proved that if $\nu$ is selected in $X$ uniformly at random, then, for all real $z$ , $\lim\limits_{N\to\infty}{\mathbb P}\left(\frac{D(\nu,N)}{(\ln N)^d}\leq z\right)=F(\rho_d z)$ , where $\rho_d$ is a constant depending only on $d$ , and $F(z)=\frac{\arctan(z)}{\pi}+\frac{1}{2}$ .

Short context: For every irrational $\alpha$ , it is known that sequence $\alpha, 2\alpha, \dots, m\alpha, \dots$ (called the Kronecker sequence) is uniformly distributed on $[0,1)$ , and the same is true for shifted sequence $x+\alpha, \dots, x+m\alpha, \dots$ . More formally, if $M(N)$ is the number of terms of this sequence (with $1\leq m\leq N$ ) belonging to some interval $[a,b) \subset [0,1)$ , then $\lim\limits_{N\to\infty}\frac{M(N)}{N}=b-a$ . Quantity $D(N)=M(N)-(b-a)N$ measures how fast this convergence happens. In 1962, Kesten established the limiting distribution of $D(N)$ , after appropriate scaling, provided that $x$ and $\alpha$ are selected at random. The Theorem establishes a multidimensional version of this result.

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

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There are at most B_n X^(C log^3 n) degree n number fields with discriminant at most X

You need to know: 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: 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 31st July 2019, Jean-Marc Couveignes submitted to arxiv a paper in which he proved the existence of constant $C>0$ such that inequality $N_n(X) \leq n^{Cn\log^3 n} X^{C\log^3 n}$ holds for all integers $n\geq C$ and $X\geq 1$ .

Short context: Counting number fields up to isomorphism is a basic and important open problem in the area. There is a folklore conjecture that $N_n(X)$ grows as linear function of X for every fixed n, but this is known only for $n\leq 5$ , see here and here. For general n, the best upper bound was $N_n(X) \leq B_n X^{\exp(C\sqrt{\log n})}$ , see here. The Theorem significantly improves the last result and provides the first upper bound with exponent polynomial in $\log n$ . In a later work (submitted 28th May 2020), Robert Lemke Oliver and Frank Thorne improved the bound further to $N_n(X) \leq B_n X^{c \log^2 n}$ for $n\geq 6$ , where one can take $c=1.564$ .

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

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The Weyl-type upper bound holds for Dirichlet L-functions of cube-free conductor

You need to know: Set ${\mathbb Z}$ of integers, greatest common divisor (gcd) of 2 integers, coprime integers, cubefree integer (integer $n$ not divisible by $m^3$ for any integer $m\geq 2$ ), set ${\mathbb C}$ of complex numbers, notation $i$ for $\sqrt{-1}$ , absolute value $|z|$ and real part $\text{Re}(z)$ of complex number z, function of complex variable, meromorphic function, analytic continuation.

Background: Function $\chi:{\mathbb Z}\to{\mathbb C}$ is called a Dirichlet character modulo integer $q>0$ if (i) $\chi(n)=\chi(n+q)$ for all n, (ii) if $\text{gcd}(n,q) > 1$ then $\chi(n)=0$ ; if $\text{gcd}(n,q) = 1$ then $\chi(n) \neq 0$ , and (iii) $\chi(mn)=\chi(m)\chi(n)$ for all integers m and n. The conductor of $\chi$ is the least positive integer $q_1$ dividing $q$ for which $\chi(n+q_1)=\chi(n)$ for all $n$ coprime to $q$ . For any Dirichlet character $\chi$ , and complex number $z$ with $\text{Re}(z)>1$ , let $L(z,\chi)=\sum\limits_{n=1}^\infty \frac{\chi(n)}{n^z}$ . By analytic continuation, function $L(z,\chi)$ can be extended to a meromorphic function on the whole complex plane, and it is called a Dirichlet L-function.

The Theorem: On 6th November 2018, Ian Petrow and Matthew Young submitted to the Annals of Mathematics a paper in which they proved that for any $\epsilon>0$ there exist a constant $C_\epsilon$ such that inequality $|L(1/2+it,\chi)| \leq C_\epsilon q^{1/6 + \epsilon}(1+|t|)^{1/6 + \epsilon}$ holds for every Dirichlet character $\chi$ with a cubefree conductor $q$ , and for all real $t$ .

Short context: Dirichlet L-functions $L(z,\chi)$ are generalisations of famous Riemann zeta function (which corresponds to the case $\chi(n)=1$ for all $n$ ), and finding upper bounds for their values at line $z=1/2+it$ (which is called the critical line), is an important problem in number theory with many applications. See here for a related problem. For the general case, Burgess proved in 1963 the bound with exponent $3/16 + \epsilon$ , which remains unimproved for over 55 years. The Theorem improves the exponent to $1/6 + \epsilon$ in the case when the conductor $q$ is cubefree. Earlier, bound with this exponent was derived (by Weyl) only for the Riemann zeta function, and is known as the Weyl bound.

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

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The size of subset of {1,…,N} without 3-term arithmetic progressions is O(N/(log N)^(1+c)) for some c>0

You need to know: A (nontrivial) 3-term arithmetic progression, big O notation, small o notation.

Background: For integer $N>0$ , let $r_3(N)$ be the cardinality of the largest subset of $\{1, 2, \dots , N\}$ which contains no nontrivial 3-term arithmetic progressions.

The Theorem: On 7th July 2020, Thomas Bloom and Olof Sisask submitted to arxiv a paper in which they proved that $r_3(N) = O\left(\frac{N}{\log^{1+c} N}\right)$ for some absolute constant $c>0$ .

Short context: In 1936, Erdős and Turán conjectured that any set containing a positive proportion of integers must contain a 3-term arithmetic progression (3APs). This is equivalent to $\lim\limits_{N\to\infty}\frac{r_3(N)}{N}=0$ . In 1953, Roth confirmed this conjecture by proving that $r_3(N) = O\left(\frac{N}{\log\log N}\right)$ . Another famous conjecture of Erdős states that if A is a set of positive integers such that $\sum\limits_{n \in A}\frac{1}{n}$ diverges then A contains arithmetic progressions of length k for all k. The $k=3$ case of this conjecture was known to follow from $r_3(N) = o\left(\frac{N}{\log N}\right)$ . In 2011, Sanders came close to this by proving that $r_3(N) = O\left(\frac{N}{\log^{1-o(1)} N}\right)$ . The Theorem finally achieves the bound better than $\frac{N}{\log N}$ , and thus implies the $k=3$ case of the Erdős conjecture. Because there are $O\left(\frac{N}{\log N}\right)$ primes up to N, the Theorem also implies the 1939 Van der Corput theorem that the set of primes contains infinitely many 3APs, as well as this generalisation by Green.

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

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The Schinzel-Zassenhaus conjecture on integer polynomials is true

You need to know: Polynomials, degree of a polynomial, constant polynomial (the one of degree $0$ ), leading coefficient of a polynomial (the nonzero coefficient of highest degree), monic polynomial (polynomial with leading coefficient $1$ ), roots of a polynomial, set ${\mathbb C}$ of complex numbers, absolute value $|z|$ of complex numbers $z\in{\mathbb C}$ .

Background: Let ${\mathbb Z}[x]$ be the set of polynomials in one variable x with integer coefficients. Polynomial $Q(x) \in {\mathbb Z}[x]$ is called a divisor of $P(x) \in {\mathbb Z}[x]$ if $P(x)=Q(x)R(x)$ for some $R(x)\in {\mathbb Z}[x]$ . If $P(x) \in {\mathbb Z}[x]$ cannot be written as $P(x)=Q(x)R(x)$ for non-constant $Q(x),R(x) \in {\mathbb Z}[x]$ , we say that $P(x)$ is irreducible. An irreducible polynomial $P \in {\mathbb Z}[x]$ is called cyclotomic if $P$ is a divisor of $x^n-1$ for some integer $n \geq 1$ .

The Theorem: On 28th December 2019, Vesselin Dimitrov submitted to arxiv a paper in which he proved that every non-cyclotomic monic irreducible polynomial $P(x) \in {\mathbb Z}[x]$ of degree $n>1$ has at least one root $\alpha\in{\mathbb C}$ satisfying $|\alpha|\geq 2^{1/4n}$ .

Short context: It is easy to see that all roots of cyclotomic polynomials have absolute value 1. In 1965, Schinzel and Zassenhaus conjectured that any other monic irreducible polynomial of degree n must have a root $\alpha$ satisfying $|\alpha|\geq 1+\frac{c}{n}$ , where $c>0$ is a universal constant. The conjecture attracted a lot of attension, but, before 2019, was proved only in special cases, e.g. for polynomials with odd coefficients, as a corollary of this Theorem. Because $2^{1/4n} \geq 1+\frac{\log 2}{4n}$ , the Theorem confirms this conjecture in full generality, with $c=\frac{\log 2}{4}$ .

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

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The Duffin–Schaeffer conjecture is true

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 $a,b$ . Also, let $\phi(n)$ denote the number of positive integers which are less than n and co-prime with it.

The Theorem: On 10th July 2019, Dimitris Koukoulopoulos and James Maynard submitted to arxiv a paper in which they proved that set $S(g) \subset [0,1]$ has Lebesgue measure 1 if and only if $\sum\limits_{n=1}^{\infty} g(n)\frac{\phi(n)}{n}=\infty$ .

Short context: The Theorem confirms a conjecture of Duffin and Schaeffer made in 1941, which was the fundamental conjecture in the theory of rational approximation. It has many important consequences. For example, this earlier theorem states that it implies an even more general conjecture of Beresnevich and Velani.

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

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All but finitely many of the fixed degree Jensen polynomials for the Riemann zeta function are hyperbolic

You need to know: Set ${\mathbb C}$ of complex numbers, real part $\text{Re}(z)$ of complex number z, function of complex variable, infinite series, integration, meromorphic function, analytic continuation, notation ${{d}\choose{j}}=\frac{d!}{j!(n-j)!}$ .

Background: For $z \in{\mathbb C}$ with $\text{Re}(z)>1$ , let $\zeta(z)=\sum\limits_{n=1}^\infty n^{-z}$ . By analytic continuation, function $\zeta(z)$ can be extended to a meromorphic function on the whole ${\mathbb C}$ , and it is called the Riemann zeta function. Similarly, let $\Gamma(z)$ be the analytic continuation of integral $\Gamma(z)=\int\limits_0^\infty x^{z-1}e^{-x}dx$ , defined for $\text{Re}(z)>0$ . Let $\Lambda(z)=\pi^{-z/2}\Gamma(z/2)\zeta(z)$ , and let sequence $\{\gamma(n)\}_{n=0}^\infty$ be defined by $(-1+4z^2)\Lambda\left(\frac{1}{2}+z\right)=\sum\limits_{n=0}^\infty\frac{\gamma(n)}{n!}z^{2n}$ . The Jensen polynomial for the Riemann zeta function of degree d and shift n is the polynomial $J_{\gamma}^{d,n}(x)=\sum\limits_{j=0}^n {{d}\choose{j}}\gamma(n+j)x^j$ . We say that a polynomial with real coefficients is hyperbolic if all of its zeros are real.

The Theorem: On 12th February 2019, Michael Griffin, Ken Ono, Larry Rolen, and Don Zagier submitted to the Proceedings of the National Academy of Sciences a paper in which they proved that for any $d\geq 1$ there is a constant $N=N(d)$ , such that the polynomial $J_{\gamma}^{d,n}(x)$ is hyperbolic for all $n\geq N$ .

Short context: The Riemann hypothesis (RH) states that if $\zeta(z)=0$ then either $z=-2k$ for some integer $k>0$ or $\text{Re}(z)=\frac{1}{2}$ . It is one of the most important open problems in the whole mathematics, and has many equivalent formulations. One of the equivalent formulations of RH, established by Pólya in 1927, states that polynomials $J_{\gamma}^{d,n}(x)$ are hyperbolic for all integers $d\geq 0$ and $n\geq 0$ . Before 2019, this statement was known to hold only for $d\leq 3$ . The Theorem proves the hyperbolicity of $J_{\gamma}^{d,n}(x)$ for all d, assuming that n is sufficiently large (depending on d). As a corollary, the authors also proved this for all n if $d\leq 8$ .

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

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Local Fourier Uniformity conjecture is true for s = 1 at scale X^θ

You need to know: Prime factorisation of positive integers, complex numbers, integration, supremum $\sup$ , small o notation,

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. Also, denote $e(t)=\exp(2\pi i t)$ .

The Theorem: On 4th December 2018, Kaisa Matomäki, Maksym Radziwiłł, and Terence Tao submitted to arxiv a paper in which they proved the following result. Let $\theta\in(0,1)$ be given and set $H = X^\theta$ . Then $\int\limits_X^{2X}\sup\limits_{\alpha} \left|\sum\limits_{x<n\leq x+H} \lambda(n)e(-\alpha n)\right|dx = o(XH)$ as $X\to\infty$ .

Short context: The Theorem confirms the first non-trivial case of a conjecture which is called the Local Fourier Uniformity conjecture. In turn, this conjecture is needed to prove the general case of the logarithmically averaged version of the Chowla conjecture, see here for its formulation and partial progress.

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

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The logarithmically averaged Chowla conjecture for two-point correlations is true

You need to know: Prime factorisation of positive integers, small o notation.

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 Theorem: On 17th September 2015, Terence Tao submitted to arxiv and the Forum of mathematics, Pi a paper in which he proved the following result. Let $a_1, a_2$ be natural numbers, and let $b_1, b_2$ be integers such that $a_1b_2 - a_2b_1 \neq 0$ . Let $1 \leq w(x) \leq x$ be a quantity depending on x that goes to infinity as $x \to \infty$ . Then one has $\sum\limits_{x/w(x)<n\leq x} \frac{\lambda(a_1n+b_1)\lambda(a_2n+b_2)}{n}=o(\log w(x))$ as $x\to\infty$ .

Short context: Famous Chowla conjecture predicts that for any sequence of distinct integers $h_1, h_2, \dots, h_k$ , one has $\sum\limits_{n \leq x}\lambda(n+h_1)\dots\lambda(n+h_k)=o(x)$ as $x\to\infty$ . It is open for all $k\geq 2$ . The $k=2$ case (also called Chowla conjecture for two-point correlations) states that $\sum\limits_{n \leq x}\lambda(n)\lambda(n+h)=o(x)$ as $x\to\infty$ . With $w(x)=x$ , $a_1=a_2=1$ , $b_1=0$ and $b_2=h$ , the Theorem implies that $\sum\limits_{n \leq x}\frac{\lambda(n)\lambda(n+h)}{n}=o(\log x)$ . The author called the Theorem “logarithmically averaged version” of the $k=2$ case of the Chowla conjecture.