36 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…

Almost-everywhere Weyl-sum exponent conjecture. For $\alpha\in[0,1]$,

Let $C$ be the middle-third Cantor set and let … W2:=left{xin[0,1]: |2^nx|<(n)text{ for infinitely many }ninright}. … is monotonic, then … This is a Khintchine-type zeroone law for…

González Robert–et al. conjecture. For almost every sequence ${\bm \omega}$ in $\mathbb{R}^{d}$, for every $d$-tuple of non-increasing functions ${\bf r}=(r_1,\ldots,r_d)$,

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…

For positive integers $n,m$, a function $\psi: \mathbb{N} \to \mathbb{R}_{\geq 0}$, and an inhomogeneous parameter $\mathbf{y} \in \mathbb{R}^m$, let…

Let $\mathcal M$ be a non-degenerate submanifold of $\mathbb R^k$, and let $\psi:\mathbb N\to[0,1)$ be non-increasing. Let $W_k^\times(\psi)$ denote the set of points in $[0,1)^k$…

Let $l\geq 2$ and $n\geq 2l$. A pair $(l,n)$ is called good when the conditions (I) and (II) defined earlier in the paper hold for the associated matrix construction. The induced m…

Conjectural refinement. These inequalities should hold for all $n\ge2$ and $w\in[n,\infty]$. This would improve the established lower bounds, whose corresponding numerators are $1$…

Let $\Delta_1,\Delta_2,\dots\geqslant0$, define … and let $CA$ be the set of $x\in[0,1]$ for which $|x-a/q|<\Delta_q$ holds for infinitely many pairs…

Let $n,m$ be positive integers with $nm=2$, and let $\Psi=(B_q)_{q\in\mathbb{N}}$ be any sequence of balls in $\mathbb{R}^m/\mathbb{Z}^m$. Let $\mathcal A_{n,m}(\Psi)$ denote the c…

Let $n,m$ be positive integers with $nm=2$, let $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$, and let $\mathbf{y}\in\mathbb{R}^m$. Let $\mathcal A_{n,m}^{\mathbf{y}}(\psi)$ denote the s…

For a nondegenerate curve in $\mathbb{R}^n$, let $\tau_{n,1}$ denote the threshold governing the dimension of the set of simultaneously Diophantine-approximable points on such curv…

Let $\tau_{n,d}$ denote the threshold governing the dimension of the set of simultaneously Diophantine-approximable points on nondegenerate $d$-dimensional manifolds in…

Dual Gallagher problem. For almost all $\alpha_k\in\mathbb R$, there exist infinitely many $(n_1,\ldots,n_k)\in\mathbb Z^k$ such that

Let $\psi:{\mathbb{N}}\to{\mathbb{R}}^+$ be monotonic, and let … where $\mathcal A(2)=\{2^n:n=0,1,2,\ldots\}$ and $\mu$ denotes Lebesgue measure. Velani's conjecture. … This is the…

For $\lambda\in[2,\infty]$, let $\mathscr{W}_\lambda$ denote the set of real numbers with the corresponding irrationality exponent, and let $n\geq 1$. The irrationality-exponent pr…

For $m\geq 1$, let $U_m$ be the set of real $U_m$-numbers in Mahler's classification, and for $n\geq 1$ let $U_m^n$ denote their $n$-fold Cartesian product. The $U_m$ product-dimen…

Define … Let $\mathcal X=\{(t,t^2):t\in[0,1]\}$ and let $\mu$ be the natural probability measure on $\mathcal X$. Parabolic natural-measure conjecture. … This asserts that the larg…

For an integer $d\geqslant2$ and $\alpha\in(0,1)$, define … Let $\mathcal X=\{(t,t^2):t\in[0,1]\}\subset\mathsf{T}_2$. Parabolic restriction dimension conjecture. … This is a conje…

For an integer $d\geqslant2$ and $\mathbf x\in\mathsf{T}_d$, let … be the orbit of the Weyl sums. Weyl-sum orbit density conjecture. For almost all $\mathbf x\in\mathsf{T}_d$ in th…

Let $U$ be an open subset of $\mathbb{R}^k$, and for each $u\in U$ let $\Phi_u=\{\phi_{i,u}(x)=A_i(u)x+t_i(u)\}_{i=1}^l$ be an iterated function system with attractor $X_u$. Let…

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…

Let $\psi:\mathbb{N}\to\mathbb{R}^+$ be a decreasing approximation function, and define … Here $\|\cdot\|$ denotes distance to the nearest integer. Let $k$ be the codimension of an…

Let $m,d\geq1$, let $\Psi:\mathbb Z^m\to(0,\infty)$ be arbitrary, and let $f$ be a dimension function such that the source's series … diverges and $x\mapsto x^{-md+1}f(x)$ is monot…