Erdős Problem #995

declared status 'open'. Formalized: no. 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 $\alpha$,$$\sum_{1\leq k\leq N}f(\{ \alpha n_k\})=o(N\sqrt{\log\log N})?$$ Current best: Erd\H{o}s [Er49d] constructed a lacunary sequence and $f\in L^2([0,1])$ such that, for every $\epsilon>0$, for almost all $\alpha$$$\limsup_{N\to \infty}\frac{1}{N(\log\log N)^{\frac{1}{2}-\epsilon}}\sum_{1\leq k\leq N}f(\{\alpha n_k\})=\infty.$$Erd\H{o}s also proved that, for every lacunary sequence and $f\in L^2$, for every $\epsilon>0$, for almost all $\alpha$,$$\sum_{1\leq k\leq N}\sum_{1\leq k\leq N}f(\{\alpha n_k\})=o( N(\log N)^{\frac{1}{2}+\epsilon}).$$Erd\H{o}s [Er64b] thought that his lower bound was closer to the truth. Prize: no. Tags: analysis, discrepancy.

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