Conjecture regarding primal representaion of vectors I have a conjecture which I cannot prove or disprove.
Denote the $i$'th digit of $x$ in binary expansion by $d_i(x)$, where for $i=1$ the MSB is taken. Example: $d_3(110.1)=0$ and $d_4(110.1)=1$ (so it just ignores the dot).
Denote by $p_i$ the $i$'th prime number, i.e. $p_1=2, p_2=3, p_3=5$ and so on.
Let $$h_i^j(x)=\frac {d_i(x)} {p_i^{p_{2j}}}$$ if the $i$'th digit of $x$ is left to the decimal point, and $$h_i^j(x)=\frac {d_i(x)} {p_i^{p_{2j+1}}}$$ otherwise.
Define $f(x):\mathbb R^n\rightarrow\mathbb R$ as: 
$$f(x)=\sum_{j=1}^n \sum_{i=1}^\infty  h_i^{j+in}(x_j) $$
Conjecture: $f$ is injective and finitely integrable over any finite measure balls on $\mathbb R^n$.
Caution: some explanations might work well for rational numbers only.
Please, if you think if it's incorrect, try to see if you can slightly modify $f$ to make the conjecture correct.
Thanks!!!

I managed to find an injection as requested, pretty much like the one proposed before.
I'll post a link to the full paper when ready
