Rational approximation with digit-restricted denominators

TL;DR

This work studies rational approximation using denominators restricted to digit-strings in base $b$, by focusing on the digit-set $\mathfrak{D}_{b}$. It first shows $\mathfrak{D}_{b}$ is an $\,\mathcal{H}$-set with an elementary decay bound and then strengthens the estimate using exponential-sum techniques, achieving a bound of $\min_{1\le n\le N,\ n\in \mathfrak{D}_{b}} \|\gamma n\| \ll 1/(\log N)^{2}$ for all $\gamma$. A complementary analysis proves that $\mathfrak{D}_{b}$ is not strong-approximating for $b\ge3$, providing explicit lower bounds on $\min_{1\le n\le N,\ n\in \mathfrak{D}_{b}} \|\gamma n\|$ for some $\gamma$, namely $\gg N^{-{\frac{\log_{2} b}{b-1}}}/b^{4}$. The paper concludes with conjectures suggesting the true decay could be a positive power $N^{-c}$ and discusses extensions to related digit-sets, highlighting the interplay between combinatorial digit restrictions and Diophantine approximation via exponential-sum methods.

Abstract

We show the existence of ``good'' approximations to a real number $γ$ using rationals with denominators formed by digits $0$ and $1$ in base $b$. We derive an elementary estimate and enhance this result by managing exponential sums.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• Lemma 3.4
• proof
• Lemma 4.1
• proof