Integers with small digits in multiple bases
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918134916579328 |
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| author | Bloom, Thomas F. Croot, Ernie |
| author_facet | Bloom, Thomas F. Croot, Ernie |
| contents | We show that, for any $r\geq 1$, if $g_1,\ldots,g_r$ are distinct coprime integers, sufficiently large depending only on $r$, then for any $ε>0$ there are infinitely many integers $n$ such that all but $ε\log n$ of the digits of $n$ are $\leq g_i/2$ in base $g_i$ for all $1\leq i\leq r$. In other words, for any fixed large bases, there are infinitely many $n$ such that almost all of the digits of $n$ are small in all bases simultaneously. This is both a quantitative and qualitative improvement over previous work of Croot, Mousavi, and Schmidt. As a consequence, we obtain a weak answer to a conjecture of Graham concerning divisibility of $\binom{2n}{n}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_02835 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integers with small digits in multiple bases Bloom, Thomas F. Croot, Ernie Number Theory Combinatorics We show that, for any $r\geq 1$, if $g_1,\ldots,g_r$ are distinct coprime integers, sufficiently large depending only on $r$, then for any $ε>0$ there are infinitely many integers $n$ such that all but $ε\log n$ of the digits of $n$ are $\leq g_i/2$ in base $g_i$ for all $1\leq i\leq r$. In other words, for any fixed large bases, there are infinitely many $n$ such that almost all of the digits of $n$ are small in all bases simultaneously. This is both a quantitative and qualitative improvement over previous work of Croot, Mousavi, and Schmidt. As a consequence, we obtain a weak answer to a conjecture of Graham concerning divisibility of $\binom{2n}{n}$. |
| title | Integers with small digits in multiple bases |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2509.02835 |