19 problems

Fourier decay conjecture.

Let $s\in[0,1]$, and let $\sigma$ be an $(s,1)$-Frostman measure on the curve $\Gamma$. The sharp exponent in the $L^6$ Fourier-decay estimate … was conjectured to be the sharp Fou…

Bell–Ho–Strichartz conjecture. As $m\to\infty$, the law of $r\circ c^{(w)}$ under $\lambda_m$ converges to the Dirac measure at $\sqrt{8/75}$, while the law of…

Let the irrigated measure be the measure describing the deviation of the boundary magnetization from the uniform state in the branched transport model associated with the reduced t…

Let $\widetilde\nu$ be the Cantor measure pushed forward to $(0,1)$, let $H_{M-1}=\sum_{n=1}^{M-1}n^{-1}$ be the harmonic number, and set $M=\lfloor\sqrt t\rfloor$. Cantor-weighted…

Let $\nu$ be the Cantor measure used in the paper and let $\widetilde\nu$ be its pushforward to $(0,1)$. Let $\zeta(s,\alpha)$ denote the Hurwitz zeta function. Cantor-weighted Hur…

Affirmative Beurling density conjecture. For every $(t,s)\in(0,\frac{1}{2}]\times[0,\infty)$, there exists a spectrum $Lambda:= Lambda{t,s}$ of $mu{4,{0,2}}$ such that

Suppose that $\mu$ is an Ahlfors--David regular probability measure supported on $[1,2]$ and satisfying … for $r<\operatorname{diam}(\operatorname{supp}\mu)$. Rough local smoothing…

Let $c8\colon \mathbb{N}\to[0,1/2]$, let $\gamma\in[0,1]$, and let $\mu$ be the natural measure on the middle third Cantor set. For…

For $b3\beta b5b4b1$?

Kleinbock–Tomanov's friendliness of pushforwards conjecture. The $p$-adic analogues of the main results in Kleinbock, Lindenstrauss, and Weiss should hold.

Let $d=2$ and $1<\alpha\le 3$. For $1<\alpha\le 2$, define … and for $2<\alpha\le 3$, define … Here $\mathfrak C^3(\alpha)$ denotes the class of $\alpha$-dimensional measures, and…

Let $\mu$ be an absolutely continuous probability measure supported on $[0,1]$ with density $f$. Assume there exist $C>0$ and $p>1$ such that, for all $0<r<R<1$ and $x$ in the supp…

Let $\mu$ be an $L^1$ density on $[0,1]$, and let $\operatorname{dim}_{\mathrm{L}}\mu$ denote its lower dimension. Lower-dimension integrability conjecture. If … then $\mu\in L^p$…

For $<\lambda<1$, let $\nu_\lambda$ denote the corresponding Bernoulli convolution, namely the distribution of the random series … where the $\epsilon_n$ are independent random sig…

Maximum-value conjecture. The absolute maximum value taken by $g_E$ is $\frac{11}{225}$, and

Local-dimension conjecture. With the exception of weak Perron parameters $\beta$, the only value of the local dimension larger than one for a Bernoulli convolution is

Let $(N_j,B_j)$ be the almost-Parseval-frame tower defined in Theorem 0.1, and let $bc$ be its associated measure. The measure $bc$ is spectral if there exists a set…

Let $\mu$ be a finite Borel measure on $\mathbb{R}^d$. Say that $\mu$ is translationally absolutely continuous if, for every Borel set $F$ in the support of $\mu$ with $\mu(F)>0$ a…