32 problems

Let $C_1$ and $C_2$ be Cantor sets in $\mathbb R$ with Lebesgue measure zero. Palis hypothesis. The arithmetic sum or difference $C_1\mathbin{+}C_2$ or $C_1\mathbin{-}C_2$ is eithe…

Let $PM(d)$ be the space of primitive majors for degree $d$, and let $\Pi(s_1,\ldots,s_r)$ be a stratum other than $\Pi(d)$ and $\Pi(1,1,\ldots,1)$. Consider the set of primitive m…

Let $C$ be the classical Cantor ternary set, and let $x_1,x_2,x_3,x_4$ be elements of $C$. Athreya–Reznick–Tyson's four-squares conjecture. Every element of $[0,1]$ can be expresse…

Let $X$ and $X'$ be Cantor sets in $\mathbb R^N$, and let $\varepsilon>0$. An $\varepsilon$-homeomorphism is a homeomorphism $h:\mathbb R^N\to\mathbb R^N$ whose displacement is bou…

Existence conjecture. For each smooth manifold $M$ of dimension $n\ge 2$, there exists a diffeomorphism

For the Almost Mathieu operator with irrational frequency $\omega$ and coupling parameter $b$, the spectrum is denoted by $\sigma_b$. A spectral gap is a connected component of the…

Let $\bm{C}$ denote the middle-third Cantor set, and let $N$ be a positive integer. The notation $A\ll B$ means that $A$ is bounded above by a constant multiple of $B$, with the co…

Let $K\subset\mathbb{R}$ be a Cantor set satisfying $\tau(K)>1$, and let $f$ be a continuously differentiable function satisfying $f(0)=0$ and $f'(0)>0$. Non-degenerate smooth Roth…

Let $S(k)$ be the set of numbers in $[0,1]$ whose continued-fraction partial quotients satisfy $a_n(x)\geq k$ whenever they are defined, with $0$ included. Cusick's conjecture. For…

Let $a=(a_n)_{n\in\mathbb N}\in(0,1)^{\mathbb N}$, and let $C(a)$ denote the central Cantor set determined by the sequence $a$. Write $\operatorname{int}$ for interior in…

Let $d$ be a positive integer. A set is generically universal if every generic set—meaning a dense $G_{\delta}$ set whose complement has Lebesgue measure zero—contains a linear and…

Let $K\subset\mathbb R^N$ be a Krushkal Cantor set, with $N\geqslant4$. A sequence of Cantor sets $X_1,X_2,\ldots\subset\mathbb R^N\setminus K$ is said to converge homeomorphically…

Let $C_2$ denote the middle-$\frac12$ Cantor set, and let $m$ be an integer with $m\geq1$. Guo's conjecture. … This is the middle-half analogue of Guo's Waring problem for the Cant…

Let $C=\left\{\sum_{n=1}^\infty\frac{a_n}{3^n}:a_n\in\{0,2\}\right\}\subseteq[0,1]$ be the Cantor ternary set, and let $m$ be an integer with $m\geq4$. Guo's conjecture. … This is…

Let $K\subset\mathbb R$ be a perfect set. Write $K\in\mathcal{BLC}$ if there is an array of interpolating nodes from $K$ whose Lebesgue constants form a bounded sequence. Conjectur…

Uniform irrationality conjecture. The parameter $t$ may satisfy

Minimal uniqueness set conjecture. There is a minimal set of uniqueness of size $\left\lfloor\frac{N}{2}\right\rfloor$.

Let $m\geq 2$ and let $K_{\lambda}$ denote the uniform $\lambda$-Cantor set, with $\lambda\in[1/(m+1),1/m)$. Write $\mathcal{L}$ for Lebesgue measure and…

Let ${\mathcal{C}}$ be Cantor's middle-thirds set, equipped with its coin-tossing measure, which assigns equal probability to the digits $0$ and $2$ in the ternary expansion. A rat…

Let ${\mathcal{C}}$ be Cantor's middle-thirds set, and more generally let ${\mathcal{C}}$ be the set of numbers $x=\sum_{i=1}^{\infty}a_i b^{-i}$ with $a_i$ in a proper subset…

Cantor graph complex-dimension conjecture. One should have

Let $C$ denote the middle-thirds Cantor set. For a real number $u\in[0,1]$, consider representations of $u$ as a sum of four squares of elements of $C$. Four-square representation…

Let $K_0$ be the Cantor ternary set and let $\eta_{K_0}$ be its Cantor–Lebesgue measure. For each $n\in\mathbb{N}$, let $W_n(\eta_{K_0})$ denote the corresponding Widom factor. Wid…

Let $\{C_\lambda\}$ be a family of dynamically defined Cantor sets of class $C^{1+\alpha}$, with the dependence on the parameter possibly subject to extra conditions involving the…

Ultra-fat Cantor set construction conjecture. Such a set can be constructed by subtracting at every level $j\in\mathbb{N}_0$ an interval with radius proportional to $1/2^{2^{2^j}}$…