Metric results on summatory arithmetic functions on Beatty sets
2019
Let $f\colon\mathbb{N}\rightarrow\mathbb{C}$ be an arithmetic function and consider the Beatty set $\mathcal{B}(\alpha) = \lbrace\, \lfloor n\alpha \rfloor : n\in\mathbb{N} \,\rbrace$ associated to a real number $\alpha$, where $\lfloor\xi\rfloor$ denotes the integer part of a real number $\xi$. We show that the asymptotic formula \[
\Bigl\lvert
\sum_{\substack{ 1\leq m\leq x \\ m\in \mathcal{B}(\alpha) }} f(m)
- \frac{1}{\alpha} \sum_{1\leq m\leq x} f(m)
\Bigr\rvert^2
\ll_{f,\alpha,\varepsilon}
(\log x)
(\log\log x)^{3+\varepsilon}
\sum_{1\leq m\leq x} \lvert f(m) \rvert^2 \] holds for almost all $\alpha>1$ with respect to the Lebesgue measure. This significantly improves an earlier result due to Abercrombie, Banks, and Shparlinski. The proof uses a recent Fourier-analytic result of Lewko and Radziwi{\l}{\l} based on the classical Carleson--Hunt inequality. Moreover, using a probabilistic argument, we establish the existence of functions $f\colon\mathbb{N}\to\lbrace\,\pm 1\,\rbrace$ for which the above error term is optimal up to logarithmic factors.
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