22 problems

Generalized Binomial Biroot Conjecture. The approximations converge according to

Contour-integral bound conjecture. The remaining contour integral satisfies

Let $\alpha$ be the exponent, let $\beta\in[0,2)$, and consider the V-shaped domain $[0,1]e^{\pm i\beta\pi/2}$. Let $r(z)$ be a lightning + polynomial approximation with $N_1$ ligh…

Let $\alpha$ be the exponent in the approximation of $x^\alpha$ on $[0,1]$, and let $r(x)$ be a lightning + polynomial approximation with $N_1$ lightning terms,…

Random Runge approximation conjecture. There is a sequence $R_k(\omega,z)$ of random rational functions such that, for each $\omega$, $R_k(\omega,\cdot)\rightarrow f(\omega,\cdot)$…

Conjecture on cubic-order $a/b^2$-approximations. There exists a constant $c(\alpha)$ such that

Let $\epsilon>0$ be any small positive real number and let $N\geq 1$. Suppose that $\alpha$ has a rational approximation $a/q$ with integers $a,q$, $1\leq q\leq N^2$, $(a,q)=1$, an…

Let $\alpha$ be a real number, let $N\geq 1$ be real, and let $0\leq\beta\leq 1$. The integers $a_1,a_2$ and positive integers $q_1,q_2$ satisfy $1\leq q_1,q_2\leq N$. Interpolated…

Higher-power modulus spacing conjecture. The local multiplicity of $S_{k,Q}$ at scale $Q^{-k-1}$ should satisfy

Spacing conjecture for square moduli.

For a positive expansion parameter $c$, let $\beta_m^2(1,c)$ denote the factored square biroot evaluated at $1$. Factored square-biroot identity. The value is … The source presents…

Let $P_{[p/q]}^{(c)}(x)$ denote the $[p/q]$ Padé approximant associated with the square-root expansion at $c$, and let $\beta_m^2(x,\sqrt{c})$ be the square biroot approximation. S…

Herremans, Huybrechs and Trefethen's conjecture. There exist coefficients $\{a_j\}_{j=1}^{N_1}$ and a polynomial $P_{N_2}$ with $N_2=\mathcal{O}(\sqrt{N_1})$ such that

Let $K_1$ and $K_2$ be compact subsets of the plane, and let $\alpha$ denote continuous analytic capacity. Vitushkin's quasiadditivity conjecture. There exists a constant $C$ such…

Positivity and convex-hull conjecture. The real weights are positive, and the convex hull of the original matrix weighted rational Bézier curve can be computed as the convex hull o…

Let $K_1$, $K_2$, and $k$ be random compact subsets of $\mathbb C$ with $K_1(\omega)\cap K_2(\omega)=\emptyset$ for every $\omega$. Assume … Let…

Let $\Phi:\mathbb N\to\mathbb R_{\geqslant 0}$ satisfy the hypothesis denoted by in the source. Duffin--Schaeffer conjecture. For almost all ${\beta}\in\mathbb R$, the inequality ……

Let $f$ belong to the Laguerre class $\mathscr L$, represented by … where $\alpha_j\in\mathbb R\setminus\mathbb Z$, $e_j\in\mathbb R$, and $e_1<\dots<e_{2q}$. Assume that $1,f,f^2$…

Let $a,b,r,s$ be the fixed parameters of the $(a,b,r,s)$-approximation setting, and let $\operatorname{Bad}$ denote the set of badly approximable real numbers. Consider the set of…

Let $a,b,r,s$ be integers as in the $(a,b,r,s)$-approximation setting, and let $\theta$ denote the constant on the right-hand side of the relevant approximation inequality. The con…

Let $s\geq 2$, and let $\widehat{R}_{s/s,3}$ denote the maximal radius of absolute monotonicity in the restricted rational-function class used by the paper. Third-order diagonal ra…

Extended Equal Ripple conjecture. There are infinitely many arrays $bbda_1,bbda_2,bbda_3$ for which there exists $l$ such that