On two uniform exponents of approximation related to Wirsing’s problem

We aim to fill a gap in the proof of an inequality relating two exponents of uniform Diophantine approximation stated in a paper by Bugeaud. We succeed in verifying the inequality in several instances, in particular for small dimension. Moreover, we provide counterexamples to generalizations, which contrasts with the case of ordinary approximation where such phenomena do not occur. Our results contribute to the understanding of the discrepancy between small absolute values of a polynomial at a given real number and approximation to the number by algebraic numbers of absolutely bounded degree, a fundamental issue in the famous problem of Wirsing and its variants.