74 problems

Bloom–Walker conjecture. Under this hypothesis, the sequence $(a_n)$ has metric Poissonian pair correlation; that is, for every $s>0$,

Full Hausdorff dimension conjecture. The set $\boldsymbol{FS}_m$ has full Hausdorff dimension, namely

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 $C$) be the middle-third Cantor set, let $\mu$ be its natural Cantor probability measure, and write … For a positive function $\psi$, define … Equivalently, $x\in W_2(\psi)$ if…

Inhomogeneous Duffin--Schaeffer conjecture.

Baker's conjecture. For almost every real number $x$, the inequality

Khintchine-type threshold conjecture. For almost all $\alpha$, the sequence $(\{a_n\alpha\})_{n\geq 1}$ has PPC if and only if

Inhomogeneous Gallagher divergence conjecture. If

Let $n\geq 1$ and let $\psi:\mathbb{N}\to\mathbb{R}^{\geq 0}$ be a non-negative function. Define … Here $\varphi(q)$ denotes Euler's phi function. Multiplicative Duffin–Schaeffer c…

Let $I$ be an open interval in $\mathbb{R}$ and let $f\in C^{(3)}(I)$ satisfy, for every $x\in I$, constants $c_1>c_2>0$ with $c_1>f'(x)>c_2$ and $f^{\prime\prime}(x)\neq 0$. Defin…

Baker's conjecture. The Veronese curve $\mathcal{V}_n$ is of Groshev type for convergence.

Baker's conjecture. In the assertion that almost all points on $C_n$ are not $psi$-approximable, Baker's condition could be replaced by the standard convergence condition. This con…

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…

Cai--Hambrook's conjecture. If

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 $\mathcal M$ 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 $F$ be the function field and let $mathcal M$ be an analytic nondegenerate submanifold of $F^n$, identified with either columns $F^{n \times 1}$ in simultaneous Diophantine app…

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…

The Generalised Baker–Schmidt Problem concerns the Hausdorff $f$-measure of the set of $\boldsymbol{\Psi}$-approximable points on a nondegenerate submanifold. The constructions in…

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 $1$…

Conjecture on coincidence of the bounds. For any $k\le n$, these two bounds coincide, so both displayed inequalities are equalities. The corresponding right-hand sides are known to…

Let $(s_n)_{n\geq 1}$ be any sequence of distinct integers, and let $x>1$. Consider the subsequence $(\{x^{s_n}\})_{n\geq 1}$. Knuth's complete uniform distribution conjecture. For…

Fix $x>1$ and consider the sequence $(\{x^n\})_{n\geq 1}$ of fractional parts. Knuth's pseudorandomness conjecture. For almost all $x>1$, this sequence should be a good candidate t…

Let $f$ be a 1-periodic Lebesgue integrable function. The averages … should converge almost everywhere to the integral of $f$ over one period. Khinchin's conjecture. The convergenc…

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…