Digit Function
Idea
The digit function,
$$D_b^n(x)=\left\lfloor\frac{x}{b^n}\right\rfloor-b\left\lfloor\frac{x}{b^{n+1}}\right\rfloor,$$
returns the $n^\text{th}$ digit from the base $b$ expansion of $x$. I.e;
$$\sum_{n\in\mathbb{Z}}b^n D_b^n(x)=x.$$
So by $n^\text{th}$ digit, we mean the digit coresponding to the $n^\text{th}$ power of $b$. It is the subject of a lot of my research.
p-adic connection
Without any modification, the digit function yields the
$p$-adic expansion for negative integers in base $p$. For example, the 2-adic expansion of $-1$ is
$$\dots1111111_2=-1,$$
which corresponds to the fact that
$$D_2^n(2)=1,$$
for all $n\in\mathbb{N}$.
Articles referencing here
