OPEN
This is open, and cannot be resolved with a finite computation.
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 $\alpha$,\[\sum_{1\leq k\leq N}f(\{ \alpha n_k\})=o(N\sqrt{\log\log N})?\]
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.
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #995, https://www.erdosproblems.com/995, accessed 2026-01-14
