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I am interested in the distributedness or "mixing" behavior of certain linear recurrences modulo powers of $2$. In particular, consider the Pell sequence (https://oeis.org/A000129), modulo $...

$\alpha \in \mathbb{R} \backslash \mathbb{Q}$, given $\varepsilon>0 \quad \lambda>0$. given $n_{1}, \cdots, n_{k} \in N^{*}$ satisfied $(1-\varepsilon) \lambda<n_{1}<\cdots<n_{k}<(1+...