269 problems

Let $$ be the inhomogeneous Duffin–Schaeffer statement referred to as , and let be the additional divergence condition referred to as . Weaker inhomogeneous Duffin–Schaeffer conje…

Let $C(b,D)$ be a generalized Cantor set with Hausdorff dimension $cgamma$, with $D$ containing at least one of $0$ and $b-1$. Let $t\geq 2$ be an integer such that the sets of pri…

Let $C(b,D)$ be a generalized Cantor set with Hausdorff dimension $cgamma$, with $D$ containing at least one of $0$ and $b-1$. Let $t\geq 2$ be an integer such that the sets of pri…

Let $C(b,D)$ be a generalized Cantor set with Hausdorff dimension $cgamma$, where $b$ and $t$ have the same prime divisors, $D$ contains at least one of $0$ and $b-1$, and…

Effective Lang conjecture. There exists an absolute constant $C$ such that, for every real cubic irrational number $x$, all partial quotients satisfy

Throughout, let $K$ be the middle-third Cantor set, let $\mu$ be its natural probability measure, and for $\psi:\mathbb R\to[0,\infty)$ and $y\in\mathbb R$ define … Here $\|x\|$ de…

Let $a\geqslant 3$ be an odd integer, and let $\langle a,a+2\rangle$ denote the numerical semigroup generated by $a$ and $a+2$. Let $(u_n)_{n\geqslant 1}$ be the recursive sequence…

Polynomial lower-bound conjecture. For given $n\ge 1$ there exist constants $c(n)>0$ and $\lambda(n)>0$ such that

Let $(\alpha_1,\ldots,\alpha_n)$ be an $n$-tuple of real numbers in $\bigtriangleup$, and suppose that its triangle sequence is eventually periodic. Algebraicity conjecture for per…

Let $n$ be a positive integer and let $\bigtriangleup$ denote the region of ordered $n$-tuples used by the triangle algorithm. Let $0 \leq \alpha_n \leq \cdots \leq \alpha_1 < 1$ b…

Prime-denominator approximation conjecture. For every real $\xi$, there exists a constant $c(\xi)$ such that the inequality

Conjecture on cubic-order $a/b^2$-approximations. There exists a constant $c(\alpha)$ such that

Let $\epsilon>0$ be any small positive real number and let $N\geq 1$. Suppose that $\alpha$ has a rational approximation $a/q$ with integers $a,q$, $1\leq q\leq N^2$, $(a,q)=1$, an…

Let $\alpha$ be a real number, let $N\geq 1$ be real, and let $0\leq\beta\leq 1$. The integers $a_1,a_2$ and positive integers $q_1,q_2$ satisfy $1\leq q_1,q_2\leq N$. Interpolated…

Oppenheim's conjecture. The lattice

Let $\xi$ be an irrational real number whose base-$b$ expansion is generated by a finite automaton; equivalently, its digit sequence is $k$-automatic for some integer base…

Let $r,s\in\mathbb R\setminus\mathbb Q$ be irrational numbers, and let ${}^{\ast}\mathbb Z_r$ and ${}^{\ast}\mathbb Z_s$ denote their associated groups of nonstandard diophantine a…

Let $n\geqslant 3$, let $\Lambda\subset\mathbb{R}^n$ be an $n$-dimensional lattice, and let $\Pi$ be a sail generated by $\Lambda$. A sail's facets and edge stars are its facets an…

Let $n\geqslant 3$ and let $\Lambda\subset\mathbb{R}^n$ be an $n$-dimensional lattice with positive norm minimum $N(\Lambda)>0$. A lattice is called algebraic when it is similar, m…

Let $V$ be an algebraic variety defined over a field $k$, let $D$ be an ample divisor on $V$, and let $P\in V(k)$ be a $k$-rational point. Assume that there is a rational curve $C$…

Let $g(\theta)$ be the least exponent such that, for some $c_1,c_2>0$, every interval of the form … contains an integer $n=a_1b_1=a_2b_2$, where … and all four integers lie in … an…

Amoroso–Zannier's conjecture. There exists a constant $c(K)>0$ such that

Fix an integer $B\geq2$, and let $\mathcal{C}_B$ be the set of irrational numbers whose partial quotients are bounded in average by $B$. For each $n$, consider rational points…

Schneider–Lang–Ramachandra's exponential conjecture. Under these assumptions, the determinant of $M$ is nonzero.

The torus analogue of Mazur's conjecture. The connected component of the identity of the closure of the image of $F$ in $\Gamma\backslash S({\mathbb R})$ equals the connected compo…