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 $\alpha$,$$\sum _{1\le k\le N}f(\{\alpha n_k\}).$$For example, is it true that, for almost all $\alpha$,$$\sum _{1\le k\le N}f(\{\alpha n_k\})=o(N\sqrt{\log \log N})?$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write [ S_N(\alpha):=\sum_{k=1}^N f({\alpha n_k}),\qquad \mu:=\int_0^1 f(x),dx,\qquad g:=f-\mu. ] Then [ S_N(\alpha)=N\mu+\sum_{k=1}^N g({\alpha n_k}), ] so the real issue is the *centered* sum (\sum_{k\le N} g({\alpha n_k})) [[nomath]](since $N\mu=o(N\sqrt{\log\log N})$ automatically)[[/nomath]].

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.