148 problems

Let $a\colon {\mathbb N}\to {\mathbb Z}$ be a strictly increasing sequence. For a system $(X,{\mathcal X},\mu,T)$, say that $a$ is good for seminorm control along $\ell$-term arith…

Let $a\colon {\mathbb N}\to {\mathbb Z}$ be a sequence. A sequence is good for $\ell$-step irrational equidistribution if it satisfies the corresponding irrational equidistribution…

Let $a\colon {\mathbb N}\to {\mathbb Z}$ be a sequence and let $\ell\in {\mathbb N}$. A sequence is good for mean convergence along $\ell$-term arithmetic progressions if the corre…

Let $X>1$, let $q$ be a positive integer, and let $a$ range over residue classes relatively prime to $q$. Define the error term $E(X,q,a)$ by … where … For $ell\geq 1$, define the…

Let $A$ be an increasing sequence of positive integers, allowing repetitions, and let $A(n)$ denote the number of elements of $A$, counted with multiplicity, that are at most $n$.…

Let $A$ be a strictly increasing sequence of positive integers, and write $A(n):=|A\cap[n]|$, where $[n]:=\{1,2,\dots,n\}$. Thus $A$ has asymptotic density at least $C\sqrt n$ when…

Let ${\bf f}=f_0f_1f_2\cdots$ be a paperfolding word over $\{1,4\}$, and define ${\bf v}=v_0v_1v_2\cdots$ by … for all $n\geq 0$, with the even-position symbols of ${\bf f}$ recode…

Let an infinite word be a sequence over a 4-letter alphabet. An $r$-power is a word consisting of repetitions of a nonempty block with exponent $r$, and a word avoids $r$-powers in…

Let $s,t$ be positive integers with $t\geqslant s+2$. An $s\times t$ integer matrix $\mathbf{A}$ is non-degenerate if it has rank $s$ and every non-zero vector in its row span over…

Folkman's conjecture. There is a constant $c$ such that every increasing sequence satisfying

Folkman's conjecture. There is a constant $C$ such that, if $A(n)\geq Cn$ for all sufficiently large $n$, then $A$ is subcomplete.

Structural conjecture. There exists a $K\in\mathbb N$ such that for every finite set $A\subseteq\mathbb N$ with $|A|>K$, if

Freiman's conjecture. There exists a natural number $K$ such that for any finite set of natural numbers $A$ with $|A|=k>K$ and

Sharpening of Szabó's conjecture. Equality holds in this lower bound for every $N\ge 1$, that is,

Let $A\geq 2$, and let $b$ and $l$ be coprime integers. Heath-Brown's conjecture. There exists a prime $p\equiv b\pmod l$ such that … This conjecture is used as a conditional input…

Let $\mathcal{A}$ be the set of positive integers $N$ for which the corresponding triple $N,N+d,N+2d$ is a three-term arithmetic progression of consecutive powerful numbers. Partit…

Let $\mathcal{A}$ be the set of positive integers $N$ for which some $d\in\mathbb{N}$ makes $N,N+d,N+2d$ a three-term arithmetic progression of consecutive powerful numbers. Let…

Let $M_k(n)$ be the packing number for the progressions $B_d=\{d,2d,\ldots,nd\}$ with $d=1,2,\ldots,k$, with $n$ fixed. Fixed-length progression-packing conjecture. For every fixed…

Let $M_k(n)$ denote the packing number with parameter $k$. Piecewise $M_k(n)$ conjecture. If $k\ll n$, then … while otherwise, … The source presents these as plausible asymptotics…

Let $m_k(n)$ denote the packing number for progressions with parameter $k$. Intermediate-length packing conjecture. For every $k$ satisfying $\Omega(\sqrt{n})=k<\sqrt{n}$, we have…

Let $\mathbb{P}(x)$ be the set of prime numbers not exceeding $x$, and let $M_{\mathbb{P}(n)}(n)$ denote the corresponding prime-restricted maximum packing number. Prime-restricted…

Let $M(n)$ denote the maximum packing number in the setting of the paper. Maximum-packing conjecture. … The conjecture predicts the leading constant for $M(n)$; the source presents…

Let $m(n)$ denote the maximum size of a packing of arithmetic progressions in the setting of the paper. Asymptotic packing conjecture. … This conjecture seeks the leading constant…

Let $K_1$ be a quadratic extension of $\mathbb{Q}(\sqrt{-2})$ and $K_2$ a quadratic extension of $\mathbb{Q}(\sqrt{2})$. An arithmetic progression of squares is non-constant if its…

Let $a,m\in\mathbb{N}$ satisfy … Let $\operatorname{GEM}(a,m)$ be the class of polynomials introduced in the paper, and let $\operatorname{GEM}_1(f;a,m)$ and…