20 problems

Let $n$, $a$, and $d$ be positive integers, and set … Let $\varphi(n)$ denote Euler's totient function and let $\zeta_k$ be a primitive $k$-th root of unity. Arithmetic conjecture.…

Arithmetic removal conjecture. If $A\subseteq[N]$ and the number of vectors $x\in A^m$ satisfying $Lx=0$ is $o(N^{m-r})$, then

Polynomial-scale density conjecture. The full counting function satisfies

Fix a prime $p$ and let $\pi_p:\mathbb{Z}\to\mathbb{F}_p$ be the canonical projection. For $U\subset\mathbb{Z}$, a $U$-pattern in $\mathbb{F}_p^n$ is a set of the form…

Fix a positive integer $n$. Let $\mathcal{f}^M_n(k)$ be the infimum of the Minkowski dimension of a set $B_k\subset\mathbb{R}^n$ containing a $k$-term arithmetic progression with e…

For $A\subset\mathbb{Z}\times\mathbb{Z}$ and $r\in\mathbb{Q}\cup\\{\infty\\}$, define … with $x+\infty y=y$. Projection formulation of the arithmetic Kakeya conjecture. For every…

Let $(U_k)_{k\in\mathbb{N}}$ be an increasing chain of subsets of $\mathbb{N}^+$ with $|U_k|=k$. Let $G_{U_k}(N)$ be the size of the smallest set of integers containing a $U_k$-pat…

Let $(U_k)_{k\in\mathbb{N}}$ be an increasing chain of subsets of $\mathbb{Q}^+$ with $|U_k|=k$. A $U_k$-pattern is a set obtained from $U_k$ by a common scaling and translation;…

Let $k$ and $N$ be positive integers. A $k$-term arithmetic progression has a basepoint $x$ and a common difference. Let $G_k(N)$ be the size of the smallest set of integers contai…

Let $S\subseteq \{0,\ldots,N\}$ satisfy $|S|\geq N^{\varepsilon}$ and occupy fewer than $\alpha p$ residue classes modulo every prime $p$, where $0<\alpha<1$. Helfgott–Venkatesh an…

Let $T_{\mathbb N^*}$ be the infinite triangle generated by iterating the $Z$-rule on the positive integers, and call its first column the left edge. A number is square-free when n…

Let $p$ be a prime and let $g_1,g_2\geq 2$. For each $g$, let $\mathfrak S(p,g)$ be the connected collection of cells that starts at $p^g$ and contains only powers of $p$ larger th…

Let $X\subset [N]$ be a subset and let $\epsilon>0$ be real. Assume that, for every prime $p\geq\epsilon^{-1}$, the reduction $X\pmod p$ occupies at most $0.99p$ residue classes. F…

Let $X\subset\mathbb{Z}$ be a subset and let $\epsilon>0$ be real. Assume that, for every prime $p\geq\epsilon^{-1}$, the reduction $X\pmod p$ occupies at most $0.99p$ residue clas…

Inverse sieve conjecture. At least one of the following holds: $|X|\ll_{\epsilon}N^{\epsilon}$, or there exists a polynomial $f(x)\in\mathbb{Q}[x]$ of degree $d\in[2,C]$ and height…

Let $X\subset [N]$ be a subset. Suppose that, for every prime $p$, the reduction $X\pmod p$ occupies at most $0.9p$ residue classes. Inverse sieve conjecture, rough form. Either…

Let $g$ be a polynomial over the integers. For each integer $a$, define … A geometric progression of length $b$ is a sequence of $b$ terms with a common ratio. Geometric-progressio…

Let $g(x)$ be a polynomial over the rationals, and write $g(\mathbb{Q})$ for its range on rational inputs. An arithmetic progression is a finite sequence of the form $a+id$ with fi…

Green–Tao conjecture for function fields. For every finite field $\mathbf{F}$, the monic irreducible polynomials in $\mathbf{F}[t]$ contain affine spaces of arbitrarily high dimens…

Let $p$ be an odd prime, and let $\Sigma_{p+1}$ be the Kerov character polynomial on a cycle of length $p+1$, with $R_i$ denoting the free cumulants. Gal's conjecture. The expressi…