29 problems

Bourgain–Gamburd–Sarnak connectivity conjecture. The graph ${\mathcal{G}(p)}$ is connected for every prime $p>3$. This is the strong-approximation conjecture for the Markoff equati…

Baragar–Bourgain–Gamburd–Sarnak conjecture. The Markoff equation $\mathbb{X}$ satisfies strong approximation at every prime $p$. Equivalently, $\operatorname{Out}(\Pi)$ acts transi…

Optimal strong approximation conjecture. For every $\epsilon>0$, as $q\to\infty$, there exists a set $Y\subset G_q$ with

Baragar's transitivity conjecture. Just as $\Gamma$ acts transitively on $\mathcal{M}(\mathbb{Z}_{>0})$, it should act transitively on $\mathcal{M}^*(\mathbb{F}_p)$. This would imp…

Let $p$ be a prime and let $\mathbf{M}_k$ be the Markoff surface … Let $\mathcal{E}_k$ denote the union of the exceptional orbits in $\mathbf{M}_k(\mathbb{F}_p)$, and let…

Consider the stochastic differential equation and its equidistant Euler approximation $X^{\mathrm{E}}_{n,1}$ from the paper, with bounded drift coefficient $\mu\in W^{s,p}$, where…

Let $\mathcal{M}$ be the set of Markoff triples over $\mathbb{Z}$, let $\mathcal{M}_p$ be the set of their reductions modulo a prime $p$, and let $\mathcal{G}^\times_p$ be the Mark…

Let $\mathcal{X}$ be the affine surface in $\mathbb{A}^3$ defined by … and let $\Gamma$ be the group of affine integral morphisms generated by coordinate permutations and the Vieta…

For an integer $m$, let $U_m$ be the affine Markoff surface … For $m=0$, let $U_0^*(\mathbb{Z}/p\mathbb{Z})=U_0(\mathbb{Z}/p\mathbb{Z})\setminus\{(0,0,0)\}$. Let $\Gamma$ be the gr…

Let $W$ be a Brownian motion, and consider the scalar SDE from Theorem 1 with drift coefficient $\mu$ and diffusion coefficient $\sigma$. Assume that $\mu$ satisfies $(\mu1)$ and…

Strong Approximation Conjecture. All non-zero solutions of the Markov equation modulo $p$ come from Markov triples.

Let $k$ be a number field, let $P_1(t),\cdots, P_n(t)$ be pairwise distinct irreducible polynomials in $k[t]$. Set $k_i=k[t]/(P_i(t))$, let $a_i$ be the class of $t$ in $k_i$, let…

Let $q$ be a prime power of odd characteristic. Let $F$ be a quadratic form over $\mathbb{F}_q[t]$ in $4$ variables with discriminant $\Delta\ne0$. Let $f,g\in\mathbb{F}_q[t]$ be n…

Let $N,m,\lambda_1,\dots,\lambda_4\in\mathbb{Z}$ satisfy … Four-square strong approximation conjecture. If $N\gg m^{4+\varepsilon}$, then there exists…

Let $G$ range over LPS Ramanujan graphs, and let $|G|$ denote the number of vertices. Sardari's diameter conjecture. The diameter is asymptotically … The paper states that the lowe…

Let $f\colon X\to X$ be a nonuniformly expanding map with reference measure $m$, return time $\tau$, and physical invariant measure $\mu$. Let $\varphi\colon X\to\mathbb{R}$ be Höl…

Let $(X_n)_{n\geq 0}$ be a random process satisfying an almost sure invariance principle, meaning that, possibly after enlarging the probability space, there is a Brownian motion…

The paper considers the condition denoted by … should hold. The authors state that this is suggested by assertion (iv) of the preceding theorem and by the discussion cited below, b…

Convergence-order conjecture. In the full parameter range $a,6sigma\typo2(0,6infty)$, the convergence order in the estimate is equal to

Let $\mathbb{X}$ be the affine surface defined by … Let $\Gamma$ be the group generated by coordinate permutations and the Vieta involutions preserving this equation. For a prime…

Non-polynomial lower-bound conjecture. A similar negative result should hold for such approximations: there exist stochastic differential equations for which no adaptive approximat…

Strong Approximation Conjecture. For any prime $p$, $\mathbb X(\mathbb Z/p\mathbb Z)$ consists of exactly two $\Gamma$-orbits, namely

Harari–Wittenberg's conjecture. The set

Optimal strong approximation conjecture. The same strong approximation result as for quadratic forms in five or more variables should hold for $F$ provided that

Let … where $\alpha_{ij},\beta_j,\gamma,\delta$ are integers, and let $\Gamma_Y$ be generated by the corresponding Vieta involutions. The paper shows that the finite…