Digital shift randomization in base $b$

Fix an integer $b\ge 2$, a dimension $d\ge 1$, and points $a_1,\dots ,a_n\in [0,1{)}^d$. Write each coordinate in its terminating base-$b$ expansion when it has two expansions, and otherwise in its unique expansion, $a_{i,j}={\sum }_{k\ge 1}{a}_{i,j,k}{b}^{-k}$. Fix a bijection $q:\{1,\dots ,d\}\times {\mathbb{N}}_{>0}\to {\mathbb{N}}_{>0}$. From the independent uniform permutations ${\pi }_q$ supplied by A single uniform variable supplies countably many independent base-$b$ digits, set $S_{j,k}={\pi }_{q(j,k)}(0)$. These are mutually independent uniform digits in $\{0,\dots ,b-1\}$. A digital shift applies the same digit shifts to every point: $x_{i,j,k}=(a_{i,j,k}+S_{j,k})<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>b,{\tilde{X}}_{i,j}={\sum }_{k\ge 1}{x}_{i,j,k}{b}^{-k},X_{i,j}=\{\begin{array}{ll}{\tilde{X}}_{i,j},& {\tilde{X}}_{i,j}<1,\\ 0,& {\tilde{X}}_{i,j}=1.\end{array}$ The value $0$ is the half-open representative when the digit series equals $1$, so the output always lies in $[0,1{)}^d$. For each fixed input point and coordinate this exceptional event is null, since it requires every shifted digit to be $b-1$. The shift is global per coordinate and digit: when $n\ge 2$, the first shifted digits of any two points in one coordinate differ by their fixed input-digit difference modulo $b$, so the point variables are dependent. This operation differs from the prefix-dependent permutations in Nested uniform scrambling of a finite point set.