Research Problems in Function Theory — Problem 7.68

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 $n_k$ be strictly increasing positive integers such that $n_{k+1}/n_k\ge q>1$ for one fixed $q$. Set $S_m(x)=\sum_{k=1}^m\cos(n_kx)$. Measure on $[0,2\pi)$ is Lebesgue length. Define \[E=\{x\in[0,2\pi):\text{for every }\xi\in\mathbb R, \ \liminf_{m\to\infty}|S_m(x)-\xi|=0\}. \] Thus every real number must be a subsequential limit of these partial sums; the exceptional set is common to all targets.

Statement recorded in the supplied source.

For every such Hadamard gap sequence, must $|E|=2\pi$? [S2]

Short English statement

Do the partial sums of every lacunary cosine series approach every real target along subsequences at almost every point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307068 (AMR-022-7068). 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 7.68, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.