Metric results on summatory arithmetic functions on Beatty sets and beyond

Abstract: The \emph{Beatty set} $\mathcal{B}(\alpha) = \lbrace\, \lfloor n\alpha \rfloor : n\in\mathbb{N} \,\rbrace$ associated to a real number $\alpha>1$ may be viewed as a generalised arithmetic progression (consecutive elements differ by either $\lfloor \alpha \rfloor$ or $\lfloor \alpha \rfloor+1$) and there are numerous results in the literature on averages of arithmetically interesting function $f\colon\mathbb{N}\to\mathbb{C}$ along such Beatty sets. (Here $\lfloor\xi\rfloor$ denotes the integer part of a real number $\xi$.) For fixed $\alpha$, the quality of such results is usually intricately linked to Diophantine properties of $\alpha$. However, it turns out that the metric theory is much cleaner: in this talk I will discuss recent joint work with A.\ Zafeiropoulos showing that

\[ \Bigl\lvert \sum_{\substack{ 1\leq m\leq x \\ m\in \mathcal{B}(\alpha) }} f(m) - \frac{1}{\alpha} \sum_{1\leq m\leq x} f(m) \Bigr\rvert^2 \ll_{f,\alpha,\varepsilon} (\log x) (\log\log x)^{3+\varepsilon} \sum_{1\leq m\leq x} \lvert f(m) \rvert^2 \]

holds for almost all $\alpha>1$ with respect to the Lebesgue measure. This significantly improves a beautiful 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, it can be shown that the above result is optimal (up to logarithmic factors) in a suitable sense. If time permits, I shall also discuss ongoing work on Piatetski-Shaprio sequences $\lbrace\, \lfloor n^c \rfloor : n\in\mathbb{N} \,\rbrace$ ($c>1$) of a related spirit.

number theory

Audience: researchers in the topic

Number theory during lockdown

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