19 problems

Let ${\mathcal{A}}$ be a primitive Apollonian circle packing. Let $S_{\mathcal{A}}$ be the set of missing curvatures that do not lie in any of the quadratic or quartic obstruction…

Let ${\mathcal{A}}$ be a primitive integral Apollonian circle packing. A positive curvature $c$ is missing in ${\mathcal{A}}$ if curvatures congruent to $c$ modulo $24$ occur in…

Let $G$ be one of the remaining hypergeometric monodromy groups in the cases listed in the paper's table of open cases. The groups are being considered with the amalgamated free-pr…

The Dwork family consists of the hypergeometric groups associated with the Dwork family of parameters; in particular, the septic case has…

For $n\geq 2$, let $R_n,U_n,T_n\in\operatorname{GL}_n(\mathbb{C})$ be the matrices defined above, let $H_n=\langle R_n,U_n,T_n\rangle$, and write $\langle R_n\rangle$ and…

The local-global multiplicity conjecture. If $t\asymp X$ is admissible, then

Let $\Gamma(2)'$ be the commutator subgroup of the level-2 principal congruence subgroup $\Gamma(2)\subset\operatorname{SL}_2(\mathbb{Z})$, and let an integer $t$ be admissible if,…

Consider the periodic continued fractions $[\overline{a_1,\ldots,a_k}]$ with entries in $\mathbb{Q}(\sqrt{5})$ satisfying $a_i\leq 2$. McMullen's exponential-growth conjecture. The…

Let $\mathcal A$ be a finite alphabet, let $\Gamma_{\mathcal A}$ be the associated semigroup in $\operatorname{SL}_2(\mathbb{Z})$, let $\delta_{\mathcal A}$ be its Hausdorff dimens…

Cohen's refined Zaremba conjecture. The multiplicity satisfies

The Local-Global Conjecture. Assume that $A\ge2$, so that $\Gamma_A$ is Zariski dense in $\operatorname{SL}_2$, and that $F(\Gamma_A)$ is infinite; equivalently, the Zariski closur…

For $N>0$, let $\|\cdot\|$ be the matrix norm and count quadruples of matrices $\g_1,\dots,\g_4\in\operatorname{SL}_{2}(\mathbb Z)$ with $\|\g_j\|<N$ satisfying an additive relatio…

Local-global conjecture. If $\gd_{\cA}>1/2$, then $\cT_{\cA}$ contains every sufficiently large admissible integer. Moreover, for an admissible $t\in[N,2N)$,

For $A\geq 1$, let $\Gamma_A$ be the semigroup generated by products … The Bourgain–Kontorovich theorem gives an asymptotic formula for the number of integers $n\leq N$ that occur…

Let $H(\alpha,\beta)$ denote a hyperbolic hypergeometric monodromy group, where $\alpha$ and $\beta$ are the parameter tuples defining the corresponding hypergeometric equation. A…

For hyperbolic hypergeometrics $(\alpha,\beta)$, the group $G(\alpha,\beta)$ is an orthogonal group $O_f$ defined over $\mathbb{Q}$, where $f$ is a rational quadratic form in an od…

Let $\Or$ be a thin orbit of primitive integral Pythagorean triples ${\bx}=(x,y,z)$, without affine injections, and define … Here $\cP(k)$ denotes the integers having at most $k$ p…

Let $\Or$ be an orbit of primitive integral Pythagorean triples ${\bx}=(x,y,z)$, and define … Here $\cP(5)$ denotes the integers having at most five prime factors. Coordinate-produ…

Let $\Or$ be a thin orbit of primitive integral Pythagorean triples, and let $\G$ be the acting group. For a triple ${\bx}=(x,y,z)$, define … Here $\cP(3)$ denotes the integers hav…