Research Problems in Function Theory — Problem 7.62

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 $\omega_1$ be the first uncountable ordinal; $\alpha<\omega_1$ means that $\alpha$ is countable. For each $\alpha$ seek a nonnegative nonincreasing sequence $(x_{n,\alpha})_{n\ge1}$ with finite sum. Every asserted quotient below is required to be defined for all sufficiently large $n$, so that a denominator in that assertion is eventually positive. No monotonicity is imposed on the test sequence $(x_n)$ in the second condition.

Statement recorded in the supplied source.

Without assuming the Continuum Hypothesis, can one construct such a family satisfying both \[\alpha<\beta<\omega_1\quad\Longrightarrow\quad \lim_{n\to\infty}\frac{x_{n,\alpha}}{x_{n,\beta}}=0, \] and \[x_n>0,\quad\sum_{n=1}^\infty x_n<\infty \quad\Longrightarrow\quad \exists\alpha<\omega_1:\lim_{n\to\infty}\frac{x_n}{x_{n,\alpha}}=0? \] The construction is required in the usual axioms of set theory with choice, without the extra continuum assumption. [S2]

Short English statement

Can a countable-ordinal hierarchy of convergent decreasing series grow successively more slowly and eventually dominate every positive summable sequence, without assuming the Continuum Hypothesis?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307062 (AMR-022-7062). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.62, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.