A note on exact approximations

TL;DR

The paper investigates exact Diophantine approximations for real numbers by rationals, focusing on when the inequality $|\beta-p/q|<\gamma/q^2$ has infinitely many solutions while a slightly stronger bound fails to do so. Building on Hall’s theorem and the structure of the Lagrange spectrum, it sharpens prior Baker–Ward formulations for the specific case $\Psi(q)=\gamma/q^2$ by constructing a real $\beta$ with infinitely many good approximations at rate $\gamma/q^2$ but only finitely many at the slightly reduced rate $\gamma/q^2\left(1-\varpi(q)/q^2\right)$, where $\varpi$ grows to infinity. The proof uses a tailored continued-fraction construction with partial quotients bounded by 4, Hall’s decomposition of $1/\gamma$, Perron’s formula, and Legendre-type accuracy bounds to control convergents and non-convergents. The results sharpen our understanding of the boundary between abundant and scarce rational approximants for a fixed quadratic-irrational convergence rate and have implications for the optimality of exact-approximation phenomena.

Abstract

Based on M. Hall's theorem we prove a simple result dealing with real numbers which admit exact approximations by rationals.