On the minimal Hamming weight of a multi-base representation

Given a finite set of bases $b_1$, $b_2$, \dots, $b_r$ (integers greater than $1$), a multi-base representation of an integer~$n$ is a sum with summands $db_1^{\alpha_1}b_2^{\alpha_2} \cdots b_r^{\alpha_r}$, where the $\alpha_j$ are nonnegative integers and the digits $d$ are taken from a fixed finite set. We consider multi-base representations with at least two bases that are multiplicatively independent. Our main result states that the order of magnitude of the minimal Hamming weight of an integer~$n$, i.e., the minimal number of nonzero summands in a representation of~$n$, is $\log n / (\log \log n)$. This is independent of the number of bases, the bases themselves, and the digit set. For the proof, the existing upper bound for prime bases is generalized to multiplicatively independent bases, for the required analysis of the natural greedy algorithm, an auxiliary result in Diophantine approximation is derived. The lower bound follows by a counting argument and alternatively by using communication complexity, thereby improving the existing bounds and closing the gap in the order of magnitude. This implies also that the greedy algorithm terminates after $\mathcal{O}(\log n/\log \log n)$ steps, and that this bound is sharp.