Wirsing’s conjecture for approximation by real algebraic numbers

Prove that every real number \(\xi\) that is not algebraic of degree at most \(n\) satisfies the optimal lower bound \(\omega_n^*(\xi)\geq n\) for approximation by infinitely many real algebraic numbers of degree at most \(n\), for every integer \(n\geq3\).

Background

For a real number $\xi$, the exponent $\omega_n^*(\xi)$ measures the quality of approximation to $\xi$ by infinitely many real algebraic numbers of degree at most $n$, normalized by their naive heights. Wirsing established the lower bound $\omega_n^*(\xi)\geq (n+1)/2$ when $\xi$ is not algebraic of degree at most $n$. He conjectured that the optimal uniform lower bound is $n$.

The conjecture was proved by Davenport and Schmidt for $n=2$, but the paper explicitly states that it remains unresolved for every $n\geq3$. The paper’s main result concerns a $p$-adic analogue of a more recent improvement to Wirsing’s bound and does not resolve this real approximation conjecture.