Poissonian Pair Correlation for Bernoulli Orbits

Let $\alpha ,\beta \in \mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$, and let $({\xi }_n{)}_{n\ge 1}$ be independent random variables taking the values $\alpha$ and $\beta$ with equal probability. We study the random walk $$X_n={\xi }_1+\cdots +{\xi }_n(\mathrm{m}\mathrm{o}\mathrm{d}1)$$ and present a deterministic Diophantine condition on $(\alpha ,\beta )$ that implies almost sure Poisson pair correlation. To the best of our knowledge, this is the first result providing explicit examples in this framework where the increment measure is purely discrete. The classical Fourier contraction method fails in this setting; a different approach is required. Our proof uses the parabolic geometry of returns to $1$ and an exact covariance formula for overlapping blocks.