Research Problems in Function Theory — Problem 9.5

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $K_1,K_2,K_3$ be pairwise disjoint closed subsets of the Riemann sphere $\widehat{\mathbb C}$ and $C_1,C_2,C_3\in\mathbb C$. On $K=\bigcup_iK_i$ put $f|_{K_i}=C_i$. For $n\ge0$ let $R_n$ be the rational functions of degree at most $n$: $p/q$ in reduced form with $\max(\deg p,\deg q)\le n$. Only functions finite on $K$, including at infinity when it belongs to $K$, can have finite approximation error. Set \[\rho_n(f)=\inf_{r\in R_n}\sup_{z\in K}|f(z)-r(z)|.\]

Statement recorded in the supplied source.

Find a geometric characterization of the exponential approximation rate \[\lim_{n\to\infty}\rho_n(f)^{1/n},\] including the existence and degenerate cases of this limit, in terms of the three sets and their assigned constants. [S2]

Short English statement

Describe geometrically the exponential rate of best rational approximation to a function constant on three disjoint closed sphere sets.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2309005 (AMR-022-9005). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 9.5, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.