# Metric results on summatory arithmetic functions on Beatty sets

## TL;DR

This paper proves an improved almost-everywhere asymptotic formula for summatory arithmetic functions over Beatty sets, using Fourier analysis and probabilistic methods, and demonstrates the optimality of the error term for certain functions.

## Contribution

It significantly refines previous results on the distribution of arithmetic functions over Beatty sets, employing advanced Fourier-analytic techniques and probabilistic arguments.

## Key findings

- The asymptotic formula holds for almost all  with respect to Lebesgue measure.
- The error term is shown to be optimal up to logarithmic factors for some functions.
- The proof leverages recent Fourier-analytic results based on Carleson--Hunt inequality.

## Abstract

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.