Research Problems in Function Theory — Problem 7.32

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.

For a continuous real function $\mu$ on $[0,1]$, define \[\omega_1(h,\mu)=\sup_{|s-t|\le h,\ s,t\in[0,1]}|\mu(s)-\mu(t)|,\] \[\omega_2(h,\mu)=\sup_{0\le s\le h,\ t,t+s,t+2s\in[0,1]}|\mu(t+2s)-2\mu(t+s)+\mu(t)|.\] A singular increasing function here is nonconstant, continuous and nondecreasing, with derivative zero Lebesgue-almost everywhere. Big-$O$ constants may depend on the chosen functions but not on $h$.

Statement recorded in the supplied source.

For every function $\phi:(0,1]\to(0,\infty)$ increasing to $+\infty$ as its argument decreases to zero, does there exist a singular increasing $\mu$ for which both \[\omega_1(h,\mu)=O(h\phi(h)),\qquad \omega_2(h,\mu)=O\!\left(h(\log(1/h))^{-1/2}\right)\quad(h\downarrow0) \] hold? [S2] The update reports an affirmative answer by Anderson, Fernández and Shields. [S2]

Short English statement

Can the two sharp borderline bounds on continuity and second differences be attained simultaneously by a singular increasing function, for every prescribed slowly diverging factor?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307032 (AMR-022-7032). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.32 and its complete update. https://arxiv.org/abs/1809.07200.