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on Erdős #995 · analysis / discrepancy

The 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})?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.