10 problems

Let $\mu$ be the unique normalized invariant measure of a cubic critical homeomorphism of the circle, and let $HD(\mu)$ denote the Hausdorff dimension of its measure-theoretical su…

Let $(\Gamma_n)$ be a recurrent Markov chain with origin $o$, let $\mu_\theta$ be the law of the coin-toss record obtained by using bias $\theta\geq 0$ whenever $\Gamma_n=o$ and fa…

Salem's conjecture. The Fourier–Stieltjes coefficients of the question mark function decay to zero at infinity with a polynomial rate: there exist constants $C>0$ and $\alpha>0$ su…

Let $q\geq 2$, and let $p_0,\ldots,p_{q-1}$ be non-negative numbers adding to $1$, with no $p_j=1$. Let $x_1,x_2,\ldots$ be independent, identically distributed random variables sa…

Let $\mu$ be a measure on $\mathbb T$ with $\mu'>0$ $m$-almost everywhere, and let $w$ be a positive function with $\log w\in L^1(m)$. Nevai's conjecture. One has … The paper state…

Translational absolute continuity conjecture. If $\mu$ is a frame-spectral measure, then $\mu$ must be translationally absolutely continuous.

Let Minkowski's question mark measure be the singular continuous measure associated with Minkowski's question mark function, and let Nevai's class denote the class of measures whos…

Minkowski's measure Nevai-class conjecture. The measure $\mu$ belongs to the Nevai class $N(\frac14,\frac12)$, equivalently its Jacobi coefficients have the limiting values…

Minkowski's measure regularity conjecture. Minkowski's measure $\mu$ is regular in the sense of Ullman--Stahl--Totik. This is a problem in logarithmic potential theory and orthogon…

Let $\nu$ be a finitely supported probability measure on $SL_2(\Bbb{R})$, and let its Furstenberg measure be the stationary measure induced by $\nu$ on the projective line…