Problem Statement
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 $\alpha$?
Categories:
Analysis
Progress
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 $a\geq 2$. Erdős [Er64b] also asked whether this is true for $n_k=\lfloor a^k\rfloor$ for some $a>1$.Kac, Salem, and Zygmund [KSZ48] proved that the conclusion holds if\[\| f-f_n\|_2 \ll \frac{1}{(\log n)^{c}}\]for some constant $c>1$. Erdős [Er49d] proved that the conclusion holds if\[\| f-f_n\|_2 \ll \frac{1}{(\log\log n)^{c}}\]for some constant $c>1$. Matsuyama [Ma66] improved this to $c>1/2$.
In [Er64b] Erdős asked whether the conclusion holds for all bounded functions $f$ and lacunary sequences $n_k$.
Source: erdosproblems.com/996 | Last verified: January 19, 2026