One of the nice things about the base $b$ representation of numbers for any integer $b\ge 2$ is that the representation is unique. However, what if we replaced the powers of $b$ in the 'computation' of the numbers with something else? If the representation (which won't exist for all integers) is still unique, does this say something about how fast those new numbers grow?
Question: Let $b\ge 2$ and $n\ge 1$ be integers. Define $$B_n:=\{(b_i)_{i=1}^n:0\le b_i<b\}$$ and, for all positive integer sequences $x=(x_i)_{i=1}^n$, define $f_x:B\to\mathbb{Z}_{\ge 0}$ by $$f_x(b_1,\ldots,b_n):=\sum_{i=1}^nx_ib_i.$$ If $f_x$ is injective, does that imply that $\max_ix_i\ge b^{i-1}$?
Note: For $b=2$ this is equivalent to: Let $S$ be any set of positive integers such that all subsets have distinct sum, then $\max S\ge 2^{|S|-1}$.
The claim is false. Knowing that, find a counterexample.
Hint: There is a counterexample with $b=2$, $ |S| = 4$ and $ \max S = 7 < 2^3$, so it's not that bad of a search.
There's quite a bit of literature on "subsets with distinct sum". I encourage you to read up on that.
E.g. The minimum of $\max S_n$ is given by the Conway-Guy Sequence, defined as $a(n + 1) = 2a(n) - a(n - \lfloor 1/2 + \sqrt{2n} \rfloor). $ The initial terms are $0, 1, 2, 4, 7, 13, 24, 44 \ldots $