On uniform polynomial approximation (Q6584974)

Let \(\xi\) be a non-zero real number and let \(n\) be a positive integer. The uniform exponent of approximation, \(\widehat{\omega}_{n}(\xi)\), is the supremum of the real numbers \(\omega>0\) such that the system\N\[\N\| P \| \leq H \text{ and } 0 < \left| P(\xi) \right| \leq H^{-\omega}\N\]\Nhas a non-zero solution \(P \in {\mathbb Z}[X]\) of degree at most \(n\) for each sufficiently large \(H\), where \(\| P \|\) is the naive height of \(P\).\N\NIf \(\xi\) is an algebraic number of degree \(d\), then we know \(\widehat{\omega}_{n}(\xi)=\min(n,d-1)\), so the outstanding case is when \(\xi\) is transcendental. In 1969, \textit{H. Davenport} and \textit{W. M. Schmidt} [Acta Arith. 15, 393--416 (1969; Zbl 0186.08603)] showed that for any transcendental real number \(\xi\) and any integer \(n \geq 2\), we have \(\widehat{\omega}_{n}(\xi) \leq 2n-1\).\N\NIn 2017, \textit{J. Schleischitz} [JP J. Algebra Number Theory Appl. 39, No. 2, 115--150 (2017; Zbl 1420.11097)]. showed that \(\widehat{\omega}_{n}(\xi) \leq 2n-2\) holds for \(n \geq 10\) follows from a conjecture of Schmidt and Summerer. This conjecture was proven afterwards by \textit{A. Marnat} and \textit{N. G. Moshchevitin} [Mathematika 66, No. 3, 818--854 (2020; Zbl 1503.11100)].\N\NHere the author improves Schleischitz' upper bound showing that for any transcendental real number, \(\xi\), \(\widehat{\omega}_{n}(\xi) \leq 2n-n^{1/3}/3\) for \(n\) sufficiently large (effectively computable from the proof). This is Theorem~1.2 in this paper. He also shows that Schleischitz' upper bound, \(\widehat{\omega}_{n}(\xi) \leq 2n-2\), holds for all \(n \geq 4\) and that \(\widehat{\omega}_{3}(\xi) \leq 2+\sqrt{5}\) (see his Theorem~1.1). Arguably, Theorem~1.2 is the first significant advance on this problem since the aforementioned work of Davenport and Schmidt over 50 years ago.\N\NThe key idea in the proof is to work with a large number of good linearly independent polynomial approximations \(Q_{0},\ldots, Q_{j+1}\), whereas Davenport and Schmidt used just two polynomials \(P\) and \(Q\) in their proof.\N\NThe author also notes that if we were able to work directly with the sequence of what he calls minimal polynomials, then it may be possible to improve the upper bound in his Theorem~1.2 to \(2n-O \left( n^{1/2} \right)\).