The joint distribution of binary and ternary digits sums
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929654399500288 |
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| author | Drmota, Michael Spiegelhofer, Lukas |
| author_facet | Drmota, Michael Spiegelhofer, Lukas |
| contents | We consider the sum-of-digits functions $s_2$ and $s_3$ in bases $2$ and $3$. These functions just return the minimal numbers of powers of two (resp. three) needed in order to represent a nonnegative integer as their sum. A result of the second author states that there are infinitely many \emph{collisions} of $s_2$ and $s_3$, that is, positive integers $n$ such that \[s_2(n)=s_3(n).\] This resolved a long-standing folklore conjecture.
In the present paper, we prove a strong generalization of this statement, stating that $(s_2(n),s_3(n))$ attains almost all values in $\mathbb N^2$, in the sense of asymptotic density. In particular, this yields \emph{generalized collisions}: for any pair $(a,b)$ of positive integers, the equation \[as_2(n)=bs_3(n)\] admits infinitely many solutions in $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00850 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The joint distribution of binary and ternary digits sums Drmota, Michael Spiegelhofer, Lukas Number Theory Primary: 11A63, Secondary: 11N60 We consider the sum-of-digits functions $s_2$ and $s_3$ in bases $2$ and $3$. These functions just return the minimal numbers of powers of two (resp. three) needed in order to represent a nonnegative integer as their sum. A result of the second author states that there are infinitely many \emph{collisions} of $s_2$ and $s_3$, that is, positive integers $n$ such that \[s_2(n)=s_3(n).\] This resolved a long-standing folklore conjecture. In the present paper, we prove a strong generalization of this statement, stating that $(s_2(n),s_3(n))$ attains almost all values in $\mathbb N^2$, in the sense of asymptotic density. In particular, this yields \emph{generalized collisions}: for any pair $(a,b)$ of positive integers, the equation \[as_2(n)=bs_3(n)\] admits infinitely many solutions in $n$. |
| title | The joint distribution of binary and ternary digits sums |
| topic | Number Theory Primary: 11A63, Secondary: 11N60 |
| url | https://arxiv.org/abs/2501.00850 |