Metric results on summatory arithmetic functions on Beatty sets
Volume 197 / 2021
Abstract
Let $f\colon \mathbb N \rightarrow \mathbb C $ be an arithmetic function and consider the Beatty set $\mathcal{B} (\alpha ) = \{ \lfloor {n\alpha }\rfloor : n\in \mathbb N \}$ 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 \[\biggl| \sum _{\substack { 1\leq m\leq x \\ m\in \mathcal{B} (\alpha ) }} f(m) - \frac {1}{\alpha } \sum _{1\leq m\leq x} f(m) \biggr|^2 \ll _{f,\alpha ,\varepsilon } (\log x) (\log \log x)^{3+\varepsilon } \sum _{1\leq m\leq x} | {f(m)}|^2 \] holds for almost all $\alpha \gt 1$ 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 Radziwiłł based on the classical Carleson–Hunt inequality.
Moreover, using a probabilistic argument, we establish the existence of functions $f\colon \mathbb N \to \{\pm 1\}$ for which the above error term is optimal up to logarithmic factors.
