5 problems

Totik's conjecture. For almost every $x\in\mathbb{T}$,

Let $E\subset(0,1)$ be a measurable set, and define $f(x)=\chi_E(\{x\})$, where $\{x\}$ denotes the fractional part of $x$. Khinchin's question. Is it true that for almost every…

Let $(n_k)$ be a lacunary sequence of natural numbers, and let $f$ belong to the Orlicz class $L\log^+\log^+ L[0,2\pi)$. Almost-everywhere convergence along $(n_k)$ means that the…

In the setting of the preceding result, let $f$ be a function of bounded variation satisfying the source's Fourier-coefficient conditions, and let $(n_\ell)_{\ell\geq1}$ be an incr…

Rubio de Francia's conjecture. Carleson's theorem should hold for all UMD Banach spaces: the Fourier series of every suitable $\mathrm{X}$-valued function should converge almost ev…