Multivariate and quantitative Erd\H{o}s-Kac laws for Beatty sequences

Abstract

The classical Erd\H{o}s-Kac theorem states that for $n$ chosen uniformly at random from $1, \dots, N$, the random variable $(\omega(n) - \log\log N)/\sqrt{\log\log N}$ converges in distribution to the standard Gaussian as $N$ tends to infinity. Banks and Shparlinski showed that this Gaussian convergence holds for any Beatty sequence $\lfloor\alpha n + \beta\rfloor$ in place of $n$. Continuing in this spirit, Crn\v{c}evi\'c, Hern\'andez, Rizk, Sereesuchart and Tao considered the joint distribution of $\omega(n)$ and $\omega(\lfloor\alpha n\rfloor)$, which they showed to be asymptotically independent for irrational values of $\alpha$. Generalising both results, we show that for any positive integer $k$, real numbers $\alpha_1, \dots, \alpha_k > 0$ and $\beta_1, \dots, \beta_k$, where $\alpha_i/\alpha_j$ is irrational for $i\neq j$, the joint distribution of $(\omega(\lfloor\alpha_in + \beta_i\rfloor) - \log\log N)/\sqrt{\log\log N}$ converges to the $k$-dimensional standard Gaussian. We next discuss quantitative bounds on the rate of convergence which do not depend on the values taken by the relevant parameters. Banks and Shparlinski remarked that such quantitative bounds may be given for a single Beatty sequence $\lfloor\alpha n + \beta\rfloor$ under Diophantine type assumptions on $\alpha$. We show that such assumptions are in fact unnecessary. Specifically, for any real numbers $\alpha > 0, \beta$, we show that the Kolmogorov distance between the random variable $(\omega(\lfloor\alpha n + \beta\rfloor) - \log\log N)/\sqrt{\log\log N}$ and the standard Gaussian is bounded above by $O(\log\log\log N/\sqrt{\log\log N})$ as $N$ tends to infinity. On the other hand, we show that universal quantitative bounds of this kind do not exist for higher-degree generalised polynomials or for the joint convergence for multiple Beatty sequences.