In this text the fractional part of a real $x$ shall be denoted $\{x\}$, such that $x = \lfloor x \rfloor + \{x\}$.

Let the set of functions $f^n_b : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}^n$ be defined through the $i$th $\left(1 \leq i \leq n \right)$ component of $f^n_b$:

$$f^n_b(x) \cdot \hat{e}_i = \sum_{k \in \mathbb{Z}}{b^k \left\lfloor b \left\{ \frac{x}{b^{kn+i}} \right\} \right\rfloor} $$

Examples of what $f^n_b$ does:

$$f^2_{10}(12345.67890) = (135.79, 24.680) \\ f^3_{10}(321321.321321) = (11.11, 22.22, 33.33) $$

It may be useful to note that for all nonnegative real $x$ and positive integer $b$:

$$ \left\lfloor x \right\rfloor \equiv \left\lfloor b \left\{ \frac{x}{b} \right\} \right\rfloor \pmod{b} $$

Is $f^n_b$ surjective for all positive integer $n$ and integer $b>1$? If so, how can this be proven?


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