570 problems

Let $\alpha$ and $\beta$ be real numbers, and let $\|x\|$ denote the distance from $x$ to the nearest integer. Littlewood's conjecture. For every pair of real numbers $\alpha$ and…

Every rational number $a/q\in(0,1)$ has a continued-fraction expansion … where the $a_i\geq 1$ are its partial quotients. Zaremba's conjecture. There exists a constant…

Let $d=2$. A weight vector is a vector $\mathbf{w}=(w_1,w_2)\in\mathbb{R}^2$ with $w_i\geq 0$ and $w_1+w_2=1$. For a weight vector $\mathbf{w}$, let … where $|\cdot|_{\mathbb{Z}}$…

Let $Q$ be an indefinite quadratic form in $n\ge3$ variables that is not proportional to a quadratic form with rational coefficients. Oppenheim conjecture. The set … is dense in…

Let $\mathcal{M}\subseteq\mathbb{R}^n$ be an analytic manifold. It is nondegenerate if it is not contained locally in any proper affine subspace of $\mathbb{R}^n$. Sprindžuk's conj…

Let $n$ be a positive integer and let $\alpha_1<\dots<\alpha_n$ be nonzero real numbers. For $x\in\mathbb{R}$, write $\lVert x\rVert_{\mathbb{R}/\mathbb{Z}}$ for the distance from…

Let $p$ be a prime, let $x\in\mathbb{R}$, and let $||\cdot||$ denote the distance to the nearest integer. The $p$-adic Littlewood conjecture. For every prime $p$ and every real num…

Lonely runner conjecture. There exists $x\in\mathbb{T}$ such that

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 $x,y$ range over integers, and let $\epsilon>0$. Hall's conjecture. For any $\epsilon>0$, the inequality … has only finitely many solutions. This conjecture concerns the possib…

Let $K_{1/3}$ be the middle-third Cantor set, let $\tau>1$, and let $W_1(\psi_\tau,0)$ denote the set of points satisfying the corresponding one-dimensional Diophantine approximati…

An automatic number is a real number generated by an automatic sequence; an $S$-number is a transcendental number with finite, nonzero Diophantine exponent in the relevant classifi…

Let $\alpha$ be a real number whose Diophantine type is defined by the existence of a constant $c_\alpha>0$ such that … for all coprime integers $p$ and $q$. Consider the sequence…

Let $\mathcal{alpha}$ be a real number, let $p$ be a prime, and write $|n|_p=p^{-\nu_p(n)}$ for the $p$-adic norm of a positive integer $n$. Let $|\langle x\rangle|$ denote the dis…

Let $n\ge 3$ be an integer, let $\mathbf{u}\in\mathbb{R}^n$ have coordinates linearly independent over $\mathbb{Q}$, and let $\gamma=(1+\sqrt{5})/2$. For…

The mixed Littlewood conjecture. For every real number $\alpha\in\mathbb{R}$ and every pseudo-absolute sequence $\mathcal{D}$, we have

Let $L\subset {\mathbb R}^n$ be a unimodular lattice, and define the multiplicative norm of ${\boldsymbol x}=(x_1,\ldots,x_n)$ by $N({\boldsymbol x})=|x_1\cdots x_n|$. Minkowski's…

Let $n,Re\mathbb{N}$ and $d=n+R$, and let $\mathcal{M}\subseteq\mathbb{R}^d$ be a bounded, immersed, $n$-dimensional smooth submanifold with boundary. For $Q\in\mathbb{N}$ and…

Hensley's conjecture. One has $\delta_A>1/2$ if and only if $D_A\supset\mathbb{N}_{\gg 1}$, meaning that Zaremba's conjecture holds for all sufficiently large $N$.

Levesley–Salp–Velani conjecture.

Let $a_1=(a_{11},a_{12})$ and $a_2=(a_{21},a_{22})$ be elements of $(\mathbb{Q}^{\times})^2$, and let $\mathcal{L}=\langle a_1,a_2\rangle$. Define … Assume that the rows and column…

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…

Lubinsky's conjecture. The set $S$ is empty. This conjecture was disproved: Verschueren, and independently Grepstad, Kaltenböck and Neumüller, proved that the Golden Ratio…

Let $M$ be an analytic nondegenerate manifold, and consider the set of points on $M$ that are very well approximable in the dual Diophantine sense. Sprindzhuk's conjecture. The set…

Let $\mathcal{M} \subset \mathbb{R}^n$ be a connected analytic manifold. For $\mathbf{x}\in\mathcal{M}$ and $\theta\in\mathbb{R}$, let $\omega(\mathbf{x},\theta)$ denote the inhomo…