We study Piatetski-Shapiro sequences $(\lfloor n^{c}\rfloor )_{n}$ modulo $m$, for non-integer $c>1$ and positive $m$, and we are particularly interested in subword occurrences in those sequences. We prove that each block $\in \{0,1\}^{k}$ of length $k<c+1$ occurs as a subword with the frequency $2^{-k}$, while there are always blocks that do not occur. In particular, those sequences are not normal. For $1<c<2$, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi–Kátai criterion, we prove that the sequence $\lfloor n^{c}\rfloor$ modulo $m$ is asymptotically orthogonal to multiplicative functions bounded by 1 and with mean value 0.

Benford's law (to base $B$) for an infinite sequence $\{x_k: k \ge 1\}$ of positive quantities $x_k$ is the assertion that $\{ \log_B x_k : k \ge 1\}$ is uniformly distributed $(\bmod\ 1)$. The $3x+1$ function $T(n)$ is given by $T(n)=(3n+1)/{2}$ if $n$ is odd, and $T(n)= n/2$ if $n$ is even. This paper studies the initial iterates $x_k= T^{(k)}(x_0)$ for $1 \le k \le N$ of the $3x+1$ function, where $N$ is fixed. It shows that for most initial values $x_0$, such sequences approximately satisfy Benford's law, in the sense that the discrepancy of the finite sequence $\{\log_B x_k: 1 \le k \le N \}$ is small.