9 problems

Let $\mathcal C_k$ be Cantor sets, and let a dilation mean scaling these sets by a factor. Their fractal uncertainty exponent is denoted by $\beta$. Dyatlov's conjecture. If the Ca…

Let $M$ tend to infinity, and consider Cantor sets of dimension $\frac12<\delta<1$. Their fractal uncertainty exponent is denoted by $\beta$. Dyatlov's conjecture. There exists a s…

Let $\mathcal A,\mathcal B\subset\{0,\ldots,M-1\}^2$ be digit sets, let $\mathcal C_{k,\mathcal A}$ and $\mathcal C_{k,\mathcal B}$ be the associated discrete product Cantor sets,…

Higher-dimensional logarithmic-phase FUP conjecture. For each $\nu>0$ there exists $\beta=\beta(\nu)>0$ such that

Patterson–Sullivan additive energy conjecture. For every $\varepsilon>0$ there exists $C_\varepsilon$ such that, for all $y\in\Lambda_\Gamma$ and $h\in(0,1]$,

Generic dilation conjecture. There exists $\beta>\max(0,\frac12-\delta)$ depending only on $\delta$ such that, for a generic choice of $\alpha\in[1,M]$,

Exponential-smallness conjecture. Fix $\delta\in(1/2,1)$. There exists a sequence of pairs $(M_j,\mathcal A_j)$ such that

Zworski's conjecture. Whenever the trapped set is hyperbolic, there is an essential spectral gap.

Let $(M,g)$ be a convex co-compact hyperbolic surface with $\delta=\frac{1}{2}$. The estimate referred to as is a polynomial resolvent bound of the form … for sufficiently large…