37 problems

Let $f\in\mathbb{R}[x,y]$ be a fixed bivariate polynomial, and let $A,B\subset\mathbb{R}$ be finite sets with $|A|=|B|=n$. Call $f$ non-special if it is neither additive, meaning…

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

Arithmetic spherical maximal-function conjecture. The arithmetic spherical maximal function $M_\Lambda$ is bounded on $\ell^p(\mathbb{Z}^d)$ for all $p>\eta(\Lambda)$. The precedin…

Near-maximal spectrum conjecture.

Let $G$ be the filtered Lie group containing a polynomial sequence $g$, let $\Psi=(\psi_1,\ldots,\psi_t)$ be the system of linear forms under consideration, and let $g^\Psi$ denote…

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.…

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…

Fix a positive integer $n$. Let $F'_{n,k}(N)$ be the minimum cardinality of a set containing a $k$-term arithmetic progression in $\mathbb{R}^n$ with $N^n$ distinct common differen…

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…

Bhattacharyya–Grigorescu–Shapira conjecture. Every linear-invariant, semi subspace-hereditary property is testable.

Induced arithmetic removal conjecture. A statement analogous to the induced arithmetic removal theorem for finite collections of $r$-colored complexity $1$ patterns over finite-dim…

Let $d\geq 3$. The two-dimensional inverse sieve theorem considers subsets of $\{0,\ldots,N\}^{2}$ occupying at most $\alpha p$ residue classes modulo every prime, and concludes th…

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 $k$ be a positive integer. For $N$ different integer common differences, let $F'_k(N)$ be the cardinality of the smallest set $A\subset\mathbf{Z}$ containing a $k$-term arithme…

Let $k$ be a positive integer and let $p$ be prime. Write $f_{k,n}(p)$ for the size of the smallest set containing, for every nonzero $d\in\mathbf{F}_p^n$, a $k$-term progression w…

Katz–Tao projection conjecture. For every $\varepsilon>0$, there are $r_1,\dots,r_k\in\mathbf{Q}\cup\\{\infty\\}$, none equal to $-1$, such that

Entropy arithmetic Kakeya conjecture. For every $\varepsilon>0$, there are $r_1,\dots,r_k\in\mathbf{Q}$, none equal to $-1$, such that

Let $k, N$ be positive integers. Write $F_k(N)$ for the size of the smallest set of integers containing, for each $d \in \\{1,\dots, N\\}$, a $k$-term arithmetic progression with c…

Power-monoid sets-of-lengths conjecture. The systems of sets of lengths of $\mathcal P_{\rm fin}(H)$ and $\mathcal P_{{\rm fin},\times}(H)$ contain every non-empty finite subset of…

Let $p$ be a prime and let $\epsilon,\delta>0$ be parameters in the arithmetic triangle removal lemma over a vector space. The lemma asserts that if a set of triangles has density…