Rational approximation with digit-restricted denominators

TL;DR

This paper investigates how well real numbers can be approximated by rational numbers whose denominators are composed solely of digits 0 and 1 in a given base, providing new bounds and methods.

Contribution

It introduces new elementary estimates and improves approximation bounds by analyzing exponential sums for denominators with restricted digit sets.

Findings

Existence of good approximations with digit-restricted denominators

Enhanced bounds through exponential sum analysis

Elementary estimates for approximation quality

Abstract

We show the existence of ``good'' approximations to a real number $\gamma$ 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.