Year: 2025
Type: preprint
Abstract: 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 $\binom{2n}{n}$.
Source: arXiv (Cornell University)
Authors: Bloom, Thomas F., Croot, Ernie
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4417409493
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