Research Problems in Function Theory — Problem 9.9

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 $(p_n)_{n\ge1}$ be positive integers with $\sum_{n\ge1}p_n^{-1}<\infty$. For a Jordan arc $\gamma\subset\mathbb C$, the space $C(\gamma)$ has its uniform norm; the powers span it when finite complex linear combinations of the restrictions $z^{p_n}|_\gamma$ are uniformly dense. No constant monomial is added unless it occurs in the specified family, and all $p_n$ are positive.

Statement recorded in the supplied source.

Does reciprocal summability imply that the powers fail to span $C(\gamma)$ for every planar Jordan arc $\gamma$? [S2]

Short English statement

Must reciprocal-summable positive powers fail to be dense in the continuous functions on every planar Jordan arc?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2309009 (AMR-022-9009). 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 9.9, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.