Problem 996

Erdős Problem #996: declared status 'open'. Formalized: yes. Let $n_1<n_2<\cdots$ be a lacunary sequence of integers, and let $f\in L^2([0,1])$. Let $f_n$ be the $n$th partial sum of the Fourier series of $f(x)$. Is there an absolute constant $C>0$ such that, if$$\| f-f_n\|_2 \ll \frac{1}{(\log\log\log n)^{C}}$$then$$\lim_{N\to\infty}\frac{1}{N}\sum_{k\leq N}f(\{\alpha n_k\})=\int_0^1 f(x)\mathrm{d}x$$for almost every $\alpha$? Current best: Raikov proved the conclusion always holds (for every $f\in L^2([0,1])$, with no assumption on $\| f-f_n\|_2$) if $n_k=a^k$ for some integer $a\geq 2$. Prize: no. Tags: analysis.