Problem Statement
Let $n_1<n_2<\cdots$ be a lacunary sequence of integers and $f\in L^2([0,1])$. Estimate the growth of, for almost all $\alpha$,\[\sum_{1\leq k\leq N}f(\{ \alpha n_k\}).\]For example, is it true that, for almost all $\alpha$,\[\sum_{1\leq k\leq N}f(\{ \alpha n_k\})=o(N\sqrt{\log\log N})?\]
Categories:
Analysis Discrepancy
Progress
Erdős [Er49d] constructed a lacunary sequence and $f\in L^2([0,1])$ such that, for every $\epsilon>0$, for almost all $\alpha$\[\limsup_{N\to \infty}\frac{1}{N(\log\log N)^{\frac{1}{2}-\epsilon}}\sum_{1\leq k\leq N}f(\{\alpha n_k\})=\infty.\]Erdős also proved that, for every lacunary sequence and $f\in L^2$, for every $\epsilon>0$, for almost all $\alpha$,\[\sum_{1\leq k\leq N}\sum_{1\leq k\leq N}f(\{\alpha n_k\})=o( N(\log N)^{\frac{1}{2}+\epsilon}).\]Erdős [Er64b] thought that his lower bound was closer to the truth.Source: erdosproblems.com/995 | Last verified: January 19, 2026