两个进制下的数位和函数之比
Ratio of sum of digits functions in two bases
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中文总结 AI 辅助
该研究证明，对任意一对乘法独立的进制，两个进制的数位和函数之比可取到每个有理数无穷多次，还刻画了乘法相关的两进制情形。
中文摘要 AI 辅助
2019年，拉布雷泰什（La Bretèche）、斯托尔（Stoll）和滕南鲍姆（Tennenbaum）证明，两个乘法独立的进制$q_1$和$q_2$的数位和函数之比$s_{q_1}(n)/s_{q_2}(n)$在正有理数集$\boldsymbol{Q}^+$中稠密。近期，施皮格尔霍费尔（Spiegelhofer）证明，在$s_2(n)/s_3(n)=1$的特殊情形下存在无穷多解；他与德莫塔（Drmota）合作将结果推广，证明对几乎所有正整数对$\boldsymbol{N}^2$，数对$(s_2(n),s_3(n))$均可取到，因此特别地，每个有理数比值均可取到无穷多次。本文中，我们证明，对任意一对乘法独立的进制$p$和$q$，该比值均可取到每个有理数无穷多次；我们还研究了乘法相关情形下的该问题，从而给出了两个进制情形的完整刻画。
英文摘要
In 2019 La Bretèche, Stoll and Tennenbaum showed that the ratio of the sum of digits function $s_{q_1}(n)/s_{q_2}(n)$ of two multiplicatively independent bases $q_1$ and $q_2$ is dense in $\mathbb{Q}^+$. Recently Spiegelhofer proved that in the special case $s_2(n)/s_3(n)=1$ we have infinitely many solutions. Spiegelhofer extended this jointly with Drmota to show that the pair $(s_2(n),s_3(n))$ attains almost every value of $\mathbb{N}^2$ and hence, in particular, that every rational ratio is attained infinitely many times.\\ In this paper we show that, indeed, for any pair of multiplicatively independent bases $p$ and $q$, that the ratio attains every rational number infinitely many times. We also study this problem in the multiplicatively dependent case, hence giving a complete characterisation in the case of 2 bases.
