Let n_1<n_2<… be a lacunary sequence of integers and f∈ L^2([0,1]). Estimate the growth of, for almost all alpha, ∑_1≤ k≤ Nf( alpha n_k). For example, is it true that, for almost all alpha, ∑_1≤ k≤ Nf( alpha n_k)=o(N√loglog N)?

Erdős Problem #995 — Let $n_1<n_2<\cdots$ be a lacunary sequence of integers and $f\in L^2([0,1])$ .

About 62 years old · traced to 1964

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 $\alpha$ , $\sum_{1\leq k\leq N}f(\{ \alpha n_k\})=o(N\sqrt{\log\log N})?$