28 problems

Let $\phi_1,\dots,\phi_m\in\mathbf{Z}[x_1,\dots,x_d]$ be a system of linear forms in $d$ variables, and let $s$ denote its true complexity: the least natural number for which small…

Gowers–Wolf conjecture. These averages may be controlled by the Gowers $U^{s+1}$-norm of $f$.

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…

Let $L_1,\dots,L_m$ be a system of linear forms, and let $\operatorname{true\ complexity}(L_1,\dots,L_m)$ denote the least integer $k$ such that the $U^{k+1}$ norm controls the ass…

Let $n\geq 3$, let $\langle\mathbf{L}_1,\,\cdot\,\rangle,\dots,\langle\mathbf{L}_n,\,\cdot\,\rangle$ be linearly independent irrational linear forms on $\mathbb{R}^n$, and define ……

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.…

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…

Arithmetic inclusion conjecture. There holds the inclusion

Let $A$ and $B$ be the varieties appearing above, and consider the mixed motive $H^1(A,A\cap B)$ together with its exact sequence … Here $\mathbb{Q}(0)$ denotes the Tate motive of…

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 $m,n$ be positive integers with $\min\{m,n\}>1$, let $\Vert\cdot\Vert_1$ and $\Vert\cdot\Vert_2$ be any pair of norms, and let $FS_{m,n}(\Vert\cdot\Vert_1,\Vert\cdot\Vert_2)$ d…

For $x\in\mathbb{R}$, let $|x|_+=\max(|x|,1)$. For $\boldsymbol{x}=(x_1,\dots,x_d)\in\mathbb{R}^{d}$, define … in particular, for $\boldsymbol{x}\in\mathbb{Z}^{d}$,…

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 $\Phi$ be a linear datum with $W_i^\Phi=\mathbb{F}_p$ for every $i\in I^\Phi$, so that $\Phi$ is a system of linear forms. Let $s\geq1$, and suppose there is a function…

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 $\mathbb{L}$ be a lattice in $n$-dimensional Euclidean space $\mathbb{R}^n$ with a Korkine–Zolotareff-reduced basis … where the $A_i$ are positive. Woods' conjecture. If…

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…

Let $L$ be a surjective linear map as in Theorem 1, and let $w$ be the auxiliary function occurring there. Write $V^*_{\operatorname{degen}}(m,d)$ for the degenerate locus of linea…

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…

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

Let $F=(f_1(x),\ldots,f_m(x))$ be a linear polynomial map on $\mathbb{Z}^n$, where the $f_i$ are non-constant affine-linear forms. Call $F$ admissible if, for every positive intege…

Let $A$ be an integral $k\times k$ matrix defining a linear system $L_3=A X^T$, where $X=(x_1,\ldots,x_k)^T$ and the component forms are $f_i(X)$. Say that $A$ has the good propert…