19 problems

Let $f_1,\ldots,f_m$ be positive linear forms on $\mathbb{R}^d$, and let $s$ be the subspace on which they all vanish. For a permutation $\sigma$ of $[d]$, define the reverse domin…

Univariate inhomogeneous Duffin–Schaeffer conjecture for systems of linear forms. For almost every $\mathbf{x}\in \operatorname{Mat}_{n\times m}(\mathbb{R})$, there exist infinitel…

Oppenheim's conjecture. The lattice

Let $N,d,t,L$ be positive integers, let $\Psi=(\psi_1,\ldots,\psi_t)$ be a system of affine-linear forms with $\|\Psi\|_N\leqslant L$, and let $K\subset[-N,N]^d$ be a convex body.…

Arithmetic inclusion conjecture. There holds the inclusion

Fix $m,n\in\mathbb N$ and an inhomogeneous parameter $\mathbf y\in\mathbb R^m$. For every $\mathbf q\in\mathbb Z^n$, define … Let $W_{n,m}^P(\psi)$ be the corresponding limsup set…

Fix $m,n\in\mathbb N$. Let $\mathbf y\in\mathbb R^m$ and define, for $\mathbf q\in\mathbb Z^n$, … Here $W_{n,m}^P(\psi)$ is the corresponding limsup set and $\operatorname{m}$ deno…

Let $f_1(t)=a_1t+b_1$ and $f_2(t)=a_2t+b_2$ be linear forms with integers $a_1,a_2,b_1,b_2$ and $a_1,a_2>0$. Set $f=f_1f_2$, and suppose that $f$ is admissible, meaning that it doe…

Prime-tuple hypothesis. If no congruence obstruction exists, then there are infinitely many values of $x$ such that every $a_i x+b_i$ is prime.

Let $\Phi=(\phi_1,\dots,\phi_k)$ be a system of linear forms, with $\phi_i\colon\mathbb{F}_p^d\to\mathbb{F}_p$, and let $i_0\in[k]$. Let $s(\Phi,i_0)$ be the asymmetric true comple…

Harbourne–Schenck–Seceleanu conjecture. If $r+1\geq n\geq5$, then $R_{n,r,d}$ fails the WLP for $d\gg0$.

Let $m,d$ be natural numbers with $d\geqslant m+2$, and let $L:\mathbb{R}^d\to\mathbb{R}^m$ be a surjective linear map. Let $V^*_{\operatorname{degen}}(m,d)$ denote the degenerate…

Let $L$, $\mathbf{v}$, $F$, and $G$ be as in Theorem 1, without assuming that $L$ has algebraic coefficients. Let $W=\prod_{p\leq w}p$. Transcendental-case conjecture. There should…

Sárközy–Sós conjecture. There exists some infinite set of positive integers $\mathcal{A}$ such that $r_{\mathcal{A}}(n;k_1,\dots,k_d)$ is constant for $n$ large enough if and only…

Let $q$ be a prime, let $\mathbb F_q$ be the finite field with $q$ elements, let $d=\pi(q-1)$, and let $L_n$ be the linear forms from the paper on $\mathbb F_q^d$. For…

Let $N$, $d$, $t$, and $L$ be positive integers, and let … be a system of affine-linear forms, none constant and no two rational multiples of one another. Write each form as…

Gowers' conjecture. The two averages are close whenever $\\|f-g\\|_{U^k}$ is small.

Let $L_1,\dots,L_m$ be a system of linear forms in $d$ variables over $\mathbb{F}_p^n$. Its true complexity is the least integer $k$ such that, for every $\eta>0$, there exists…

Gowers–Wolf conjecture. The true complexity of $\mathcal{L}$ is equal to the smallest $k$ such that the functions $L_1^{k+1},\dots,L_m^{k+1}$ are linearly independent. This conject…