11 problems

Maximal-projection conjecture. One has

Let $\mu$ be a measure on the unit circle, and consider para-orthogonal polynomials whose zeros include a fixed point $w$ lying in $\operatorname{supp}\mu$. The strong limit points…

For each $n$, let $\{\xi_{n,k}\}$ be the set of spurious roots of the orthogonal polynomial $\varphi_n$ on the unit circle, and let $\{s_{n,\ell}\}$ be the roots of its electrostat…

Let … be a polynomial potential on the unit circle, and suppose that minimizing over the stated class of equally distributed measures reduces to choosing a minimal sub-sum of its c…

Erdős's conjecture. The sequence $(A_n)_{n=1}^{∞}$ is unbounded. Erdős also asked how fast

Brilleslyper and Schaubroeck's conjecture. The number of roots of $x^n+x^m-1$ in the interior of the unit circle is

Let $Q(z)=1+z^n$, where $n$ is a positive integer, and let $\|\cdot\|_s$ denote the corresponding quasi-norm on the unit circle. Baernstein's conjecture. For every…

Energy lower-limit conjecture. The following inequalities hold:

Modified higher-order Szegő conjecture. The weighted logarithmic integrability condition

Let ${\Bbb S}^1$ be the unit circle, let $p>0$, and let … be the set of $n$-th roots of unity. For a configuration $\omega_n\subset {\Bbb S}^1$, write $M^p(\omega_n,{\Bbb S}^1)$ fo…

Let $T=\{z\in\mathbb C:|z|=1\}$, let $\xi_k=e^{2\pi i k/n}$ for $k=1,\dots,n$, and let $M_n^p(T)$ denote the maximal Chebyshev constant for $n$ points on $T$ associated with the ke…