16 problems

Let $C(b,D)$ be a generalized Cantor set with Hausdorff dimension $cgamma$, with $D$ containing at least one of $0$ and $b-1$. Let $t\geq 2$ be an integer such that the sets of pri…

Let $C(b,D)$ be a generalized Cantor set with Hausdorff dimension $cgamma$, with $D$ containing at least one of $0$ and $b-1$. Let $t\geq 2$ be an integer such that the sets of pri…

Let $C(b,D)$ be a generalized Cantor set with Hausdorff dimension $cgamma$, where $b$ and $t$ have the same prime divisors, $D$ contains at least one of $0$ and $b-1$, and…

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 $K\subset\mathbb{R}$ 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…

Waring conjecture for $C_\alpha$.

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…

Exceptional set conjecture. The $3$-adic exceptional set has Hausdorff dimension zero:

A regular Cantor set $K$ is called affine if its defining map has constant derivative on every interval $K_i$. For two affine Cantor sets $K$ and $K'$, consider their arithmetic di…

Uniqueness conjecture. It is possible to characterize base-$N$ repunit primes as precisely the primes whose reciprocals belong to $\mathcal{C}_N$ only when $N=3$; for every…