399 problems

Let $W_{\beta,\lambda}(x)=\sum_{n=1}^{\infty}\lambda^{-\beta n}e^{2\pi i\lambda^n x}$, where $0<\beta\leq 1$ and $\lambda>1$. For a real-valued function $f$, write…

Let $\mathcal{W}^{\alpha,b}$ be the Banach space of $1$-periodic functions considered above. For a function $W_{\mathbf{t}}$ in this space, let … be the measure on its graph. For…

Mandelbrot's conjecture.

Minkowski dimension conjecture. Any such spiral trajectory has Minkowski dimension

Let an iterated function system (IFS) generate a self-similar set satisfying the relevant hypotheses, such as the open set condition. Call the IFS non-lattice when its contraction…

Let $\alpha\in{\mathbb R}\setminus{\mathbb Q}$, let $\Sigma_\alpha$ denote the spectrum of the critical almost Mathieu operator $H_{1,\alpha,\theta}$, and let $\dim_H$ denote Hausd…

Let $M_1=M(\{0,\pm1\})$ be the closure in $\mathbf{C}$ of the set of zeros of polynomials with coefficients in $\{0,\pm1\}$ and nonzero constant term. Write…

Let a Borel subgroup or Borel subring of the reals mean a subgroup or subring of $\mathbb{R}$ that is Borel as a subset, and let Hausdorff dimension refer to its dimension as a sub…

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 $N$.

Levesley–Salp–Velani conjecture.

Let $\Omega\subset\mathbb{R}^d$ be a domain with Dirichlet Laplacian eigenvalues $\{\lambda_k\}$, let $|\Omega|$ denote its volume, and let $\omega_d$ be the volume of the unit bal…

Modified Weyl–Berry conjecture. There is a positive constant $c_D$ depending only on $D$ such that

Quasi-Assouad dimension conjecture. The quasi-Assouad dimension of $F$, and therefore its Assouad spectrum, always coincides with the Hausdorff dimension of $F$.

Let $d\in(0,1)$, and suppose that $\mathcal{L}=(l_j)_{j=1}^{\infty}$ has Minkowski dimension $d$. Fractal-string packing conjecture. The following are equivalent: begin{enumerate}…

Let $X,Y\subset [0,1)$ be closed and invariant under $T_a,T_b$ respectively, where $T_m(x)=mx\bmod 1$ is multiplication by $m$ on the circle. Assume that $\log a/\log b$ is irratio…

Let $K$ be the attractor of a self-similar iterated function system (IFS) $\{\varphi_i:i\in\Lambda\}$ on $\mathbb{R}$ that contains no exact overlaps. Let $\lambda_i$ be the contra…

Let $E,F\subset\mathbb{R}^d$ be two totally disconnected non-trivial self-similar sets, generated by iterated function systems $\Phi=\{\varphi_i\}_{i\in I}$ and…

Exact overlap conjecture. If there is a dimension drop for $\Phi$, then $\Phi$ has exact overlaps.

Texan conjecture. The dimension spectrum is full:

Full dimension conjecture. For all $\lambda$ sufficiently close to $1$, one should have

Finite Hausdorff measure conjecture. If

Let $\mu_\gamma$ be Gaussian multiplicative chaos on the unit circle with parameter $\gamma$, and define the Fourier dimension $\operatorname{dim}_F(\mu)$ as the largest $s$ such t…

Let $\Omega\subset\mathbb R^n$ have a fractal boundary $\partial\Omega$ with Hausdorff dimension $H\in[n-1,n]$. Let ${\Greekmath 011A}(x)$ be the eigenvalue counting function for t…

Let $0<\alpha\le 1$, and let $p,q$ satisfy … Suppose that $E\subset [1,2]$ has bounded $\alpha$-Assouad characteristic. Fractal local smoothing conjecture. The estimate … holds wit…

Let $\Omega$ be a domain with fractal boundary, and let fractal curvatures be the limiting curvature quantities associated with parallel sets of the boundary, with scaling exponent…