Quantitative Erdős–Kac conjecture for Beatty sequences

Let $\alpha>0$ and $\beta\in\mathbb{R}$. For $n\sim U[N]$, let $d_K$ denote the Kolmogorov distance, and define

$$X_N=\frac{\omega(\lfloor\alpha n+\beta\rfloor)-\log\log N}{\sqrt{\log\log N}}.$$

Quantitative Erdős–Kac conjecture for Beatty sequences. The distance between $X_N$ and the standard Gaussian $\mathcal{N}(0,1)$ is bounded above by

$$O_{\alpha,\beta}\left(\frac{1}{\sqrt{\log\log N}}\right)$$

as $N\rightarrow\infty$, with the implied constant allowed to depend on $\alpha$ and $\beta$.

The paper proves the weaker bound $O_{\alpha,\beta}(\log\log\log N/\sqrt{\log\log N})$ using probabilistic methods. The conjectured estimate is the optimal order suggested by the lattice spacing of the normalized random variable, but no proof is given.