281 problems

Let $\mathcal{M}$ be a compact submanifold of $\mathbb{R}^n$ of dimension $m$, with codimension $k=n-m$, and let $N_{\mathcal{M}}(Q,\delta)$ denote the number of rational points of…

Let $\varphi=(1+\sqrt{5})/2$ and, for each integer $n\ge 1$, define the row $R_n=\{\lfloor k\varphi^n\rfloor:k\ge 1\}$. Kimberling's conjectures are: (1) for every $n\ge 1$,…

Determine whether every irrational real number $\alpha$ has infinitely many primes $\ell$ such that $\|\alpha\ell\|<\ell^{-1}$, where $\|x\|$ denotes the distance from $x$ to the n…

For every sextuple of distinct positive integers $v_1,\ldots,v_6$, define $\operatorname{ML}(v_1,\ldots,v_6)=\max_{t\in\mathbb{R}}\min_{1\leq i\leq 6}\|t v_i\|$, where…

For every pair $(x,y)\in\mathbb{R}^{2}$, where $\langle t\rangle$ denotes the distance from $t$ to the nearest integer, one has…

For a finite alphabet $\mathcal A$, determine the spectrum $\operatorname{Spec}_{\mathrm{rD}}(\mathcal A)=\{\operatorname{rD}(w):w\in\mathcal A^{\mathbb N}\}$ of the refined Diopha…

Let $(\alpha,\beta)$ be a pair of real numbers, and let $\epsilon>0$. The system … uses solutions $(p,r,q)\in\mathbb{Z}\times\mathbb{Z}\times\mathbb{N}$ for an unbounded set of…

Let $x,y$ be positive integers with $x^3\ne y^2$. Weak Hall's conjecture. There exists an absolute constant $\kappa>0$ such that … This weaker form replaces Hall's exponent $1/6$ b…

For any $0<\alpha<1$, let $f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}-\{ \alpha k\}).$ Does $f(\alpha,n)$ have an asymptotic distribution function? In other wo…

Let $S(N,A,c)$ be the measure of $α∈(0,1)$ for which $|α-x/y|<A/y^2$ has a coprime solution with $N≤y≤cN$. Does $S(N,A,c)$ have a limit as $N→∞$, and what is its explicit form?

For a strictly increasing sequence of positive integers $A=(n_k)_{k\geq0}$, define $\phi_A(k)$ to be the number of integers $m$ with $1\leq m\leq n_k$ such that … for every $j<k$.…

For nonnegative numbers $ψ(q)$, is divergence of $Σ_{q≥1}φ(q)ψ(q)/q$ necessary and sufficient for $|α-p/q|<ψ(q)/q$ to have infinitely many coprime solutions $(p,q)$ for almost ever…

Let $α$ be irrational. If the number of $1≤m≤n$ whose fractional part $mα$ lies in $[u,v)$ equals $n(v-u)+O(1)$, must $u$ and $v$ each be fractional parts of integer multiples of…

For every irrational real $α$ and every $ε>0$, do there exist integers $x,y,z$ such that $|x^2+y^2-αz^2|<ε$?

Let $\alpha,\beta \in \mathbb{R}$. Is it true that $\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0$ where $\|x\|$ is the distance from $x$ to the nearest integer?

Let $\mathbf{x}=(x_1,\ldots,x_k)\in\mathbb{R}^k$, where $k\geqslant2$. Littlewood's conjecture in $k$ dimensions. … This is the multiplicative form of simultaneous Diophantine appr…

Let $f\colon \mathbb{N}\rightarrow [1,+\infty)$, and let $B_f$ be the set of real numbers $\alpha\in [1,+\infty)$ such that … for infinitely many $r,k_1,\ldots,k_r$ with…

Let $K$ be a field complete with respect to a non-archimedean absolute value, let $A/K$ be a semiabelian variety, and let $X \subset A$ be a closed subvariety. Tate–Voloch conjectu…

Let $p$ be a prime and let $b1$ be an irrational real number. Let $\lvert q\rvert_p=p^{-v_p(q)}$ denote the $p$-adic absolute value of the integer $q$, where…

Let $\mathcal{M}\subset\mathbb{R}^n$ be a $d$-dimensional non-degenerate submanifold, let $\mu_d$ be the normalised $d$-dimensional Lebesgue measure induced on $\mathcal{M}$, and l…

Bugeaud–Durand conjecture.

Given pairwise distinct positive velocities $v_1,\dotsc,v_n \in \mathbb{R}$ and arbitrary starting points $s_1,\dotsc,s_n \in \mathbb{R}$, the distance from $s_j+tv_j$ to the neare…

Linnik's conjecture. For each such $n$, there exists a representation with

Let $X$ be a smooth projective variety defined over a number field $k$, let $P\in X(k)$, and let $A$ be an ample $\mathbb{Q}$-Cartier divisor on $X$. For an algebraic point $P$, wr…

Oppenheim's conjecture. The lattice