2025/09/02 by Thomas F. Bloom, Ernie Croot, Bloom, Thomas F. +1 · 1 citation
Mathematics · #Limits and Structures in Graph Theory #Analytic Number Theory Research #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2509.02835
We show that, for any r≥ 1, if g1,…,gr 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 εlog n of the digits of n are ≤ gi/2 in base gi for all 1≤ i≤ 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 \binom2nn.