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.

中文翻译：

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关于多基表示的最小汉明权

给定一组有限的基数 $b_1$、$b_2$、\dots、$b_r$（大于 $1$ 的整数），整数的多基数表示~$n$ 是一个加数为 $db_1^{\ alpha_1}b_2^{\alpha_2} \cdots b_r^{\alpha_r}$，其中 $\alpha_j$ 是非负整数，数字 $d$ 取自一个固定的有限集合。我们考虑具有至少两个乘法独立的基的多基表示。我们的主要结果表明，整数~$n$ 的最小汉明权的数量级，即~$n$ 表示中非零被加数的最小数量，是 $\log n / (\log \log n)$。这与碱基数量、碱基本身和数字集无关。为了证明，素数基的现有上界被推广到乘法独立的基，对于自然贪婪算法所需的分析，导出了丢番图逼近的辅助结果。下限后跟一个计数参数，或者通过使用通信复杂性，从而改进现有的界限并在数量级上缩小差距。这也意味着贪心算法在 $\mathcal{O}(\log n/\log \log n)$ 步之后终止，并且这个界限是尖锐的。