On simply normal numbers with digit dependencies

Given an integer $b⩾2$ and a set $P$ of prime numbers, the set $T_P$ of Toeplitz numbers comprises all elements of $[0,b[$ whose digits $(a_n{)}_{n⩾1}$ in the base-$b$ expansion satisfy $a_n=a_{pn}$ for all $p\in P$ and $n⩾1$. Using a completely additive arithmetical function, we construct a number in~$T_P$ that is simply Borel normal if, and only if, ${\sum }_{p\in ̸P}1\mathrm{/}p=\mathrm{\infty }$. We then provide an effective bound for the discrepancy.