177 problems

Fix $n,m\in\mathbb{N}$ and let … for $A,B\subseteq\mathbb{N}_0$. Let $r,s\in\mathbb{N}$ be multiplicatively independent, and let $A,B\subseteq\mathbb{N}_0$ be $\times r$- and…

Density conjecture. The real parts of the complex dimensions of $\mathcal{L}$ form a set that is dense in the connected interval $[\sigma_l,D]$.

Let $\mathbb{A}$ be an annulus in $\mathbb{R}$, and let $0<\delta\ll1$. Let $A\subset\mathbb{A}$ be a $(\delta,1/2)_1$ set with measure $\approx\delta^{1/2}$. Discretized Erdős rin…

Let $0<\delta\ll1$, let $\Omega$ be a $\delta$-separated set of directions, and for each $\omega\in\Omega$ let $R_\omega$ be a $(\delta,1/2)_2$ set contained in a rectangle of dime…

For a compact set $K\subset\mathbb{R}^2$, let $\operatorname{dist}(K)=\{|x-y|:x,y\in K\}$ be its distance set, and let ${\text{\roman dim}}$ denote Hausdorff dimension. Distance-se…

Let $n\ge 3$ and let $A\subseteq \mathbb R$ be a Borel set. Write $\dim_H A$ for the Hausdorff dimension of $A$. Shifted-product nonempty-interior conjecture. For every $n\ge 3$, t…

Let $K \subset \mathbb{R}^{2}$ be compact, let $\alpha \in[0,1]$, and let $\operatorname{vis}_{\sigma}(K)$ be the visible part of $K$ in direction $\sigma$. Exceptional-set estimat…

Let $K \subset \mathbb{R}^{2}$ be compact. For $\sigma \in[-1,1]$, let $\operatorname{vis}_{\sigma}(K)$ denote the points of $K$ that are not followed by another point of $K$ along…

Let $E$) be a moiety of a bounded primitive Eisenstein circle packing. Let $N$ be a positive real number, and let $c_E$ denote a constant depending on $E$. Let…

Let $Z_F$ be constructed from the non-trivial zeta zeros, and consider orderings of those zeros, including the standard ordering by increasing imaginary part. Canonical Ordering co…

Let $Z_F$ be the fractal zero set constructed from the positive imaginary parts of the non-trivial zeros of $\zeta(s)$ by the recursive four-subinterval construction described in t…

Let $P_{\text{ess}}$ be the essential fractal prime set, with information measure $\iota(P_{\text{ess}})=-\zeta(1/2)$, and let $Z_F$ be the fractal zero set constructed from the po…

Finite Hausdorff measure conjecture. If

Hausdorff measure conjecture. 1. If $h_{\mathcal{F}}<2\theta_{\mathcal{F}}$, then

Let $K\subset\mathbb{R}^d$ be a regular Cantor set, and write $\operatorname{\dim_{H}}(K)$ for its Hausdorff dimension. A pair of sets has a $C^r$-stable intersection if this inters…

Let $u$ be a compactly supported probability measure on $\mathbb{R}^d$ satisfying … and … Assume that $\alpha>2d/3$. Three-term progression distance conjecture. Under appropriate q…

The averaged Zaremba conjecture. For $N\in\mathbb{N}$,

Hensley's conjecture. One has $\delta_A>1/2$ if and only if $D_A\supset\mathbb{N}_{\gg 1}$, meaning that Zaremba's conjecture holds for all sufficiently large $N$.

Let $b$ be the base and let $D_1,D_2\subseteq\{0,1,\ldots,b-1\}$ be digit sets, with $\#D_1$ and $\#D_2$ denoting their cardinalities. A pair $(D_1,D_2)$ is called degenerate or Ke…

Bugeaud–Durand conjecture.

Levesley–Salp–Velani conjecture.

Let $Q$ be a polygon, and let its singularity set be the set of points where the outer length billiard is singular. Let an orbit be a sequence of points generated by the billiard,…

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…

Super-exponential condensation conjecture. Any analytic IFS which has super-exponential condensation but no exact overlaps must be sub-conjugated to a self-similar IFS.

Minkowski dimension conjecture. Any such spiral trajectory has Minkowski dimension