4 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…

Van de Lune's conjecture. The records of $S_n(\sqrt 2)$ satisfy a Pell-like recurrence. The source states that this specific conjecture was confirmed in prior work, so it is includ…

For an irrational number $\xi$, define the deterministic random walk by … Call $r$ a record if $S_r(\xi)$ is different from every preceding partial sum, including $S_0(\xi)=0$. Van…

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…