  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 to
verify the inequality in several instances, in particular for small dimension.
Moreover, we provide counterexamples to generalizations, which contrasts 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 variants.
