Research Problems in Function Theory — Problem 9.17

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 $K\subset[0,1]$ be the middle-thirds Cantor set, equivalently \[K=\left\{\sum_{j=1}^{\infty}\frac{2\varepsilon_j}{3^j}:\varepsilon_j\in\{0,1\}\right\}.\] Identify $E=K\times K$ with $\{x+iy:x,y\in K\}\subset\mathbb C$, and let $\Omega=\widehat{\mathbb C}\setminus E$. The algebra $H^{\infty}(\Omega)$ consists of bounded analytic functions on this sphere domain, including analyticity at infinity.

Statement recorded in the supplied source.

Prove the corona theorem for $\Omega$: for every finite $m\ge1$ and $f_1,\ldots,f_m\in H^{\infty}(\Omega)$ such that \[\inf_{z\in\Omega}\sum_{j=1}^m|f_j(z)|>0,\] there must exist $g_1,\ldots,g_m\in H^{\infty}(\Omega)$ with $\sum_{j=1}^mf_jg_j=1$ on $\Omega$. [S2]

Short English statement

Prove the bounded-analytic corona theorem for the complement on the sphere of the Cartesian square of the middle-thirds Cantor set.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2309017 (AMR-022-9017). 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 9.17, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.