A note on quadratic approximation for Liouville numbers

TL;DR

This paper investigates the limitations of extending known approximation exponents from strong Liouville numbers to all Liouville numbers, highlighting the boundaries of current approximation techniques.

Contribution

The paper demonstrates that Schleischitz's results on approximation exponents do not generalize to all Liouville numbers, clarifying the scope of existing approximation methods.

Findings

Schleischitz's exponents apply to strong Liouville numbers

The results cannot be extended to arbitrary Liouville numbers

Limits of current approximation techniques are identified

Abstract

Schleischitz [arXiv:1701.01129] determined exponents of best approximations to a strong Liouville number by integer polynomials and algebraic numbers of precribed degree. In this note, we show that we cannot extend his result to arbitrary Liouville numbers.