2 problems

For an irrational number $\xi$, define the deterministic random walk by … Call $r$ a record if $S_r(2\xi)$ has not occurred among the preceding partial sums, including…

Let $S_n(\xi)=\sum_{j=1}^n(-1)^{\lfloor j\xi\rfloor}$ be a deterministic random walk, and call an integer $n$ a record when $S_n(\xi)$ has not occurred among the preceding partial…