  Let $S$ be a numerical semigroup of embedding dimension $e$ and conductor
$c$. The question of Wilf is, if $\#(\mathbb N\setminus S)/c\leq e-1/e$.
  \noindent In (An asymptotic result concerning a question of Wilf,
arXiv:1111.2779v1 [math.CO], 2011, Lemma 3), Zhai has shown an analogous
inequality for the distribution of weights $x\cdot\gamma$, $x\in\mathbb N^d$,
w.\,r. to a positive weight vector $\gamma$:
  \noindent Let $B\subseteq\mathbb N^d$ be finite and the complement of an
$\mathbb N^d$-ideal. Denote by $\operatorname{mean}(B\cdot\gamma)$ the average
weight of $B$. Then \[\operatorname{mean}(B\cdot\gamma)/\max(B\cdot\gamma)\leq
d/d+1.\]
  $\bullet$ For the family $\Delta_n:=\{x\in\mathbb N^d|x\cdot\gamma<n+1\}$ of
such sets we are able to show, that
$\operatorname{mean}(\Delta_n\cdot\gamma)/\max(\Delta_n\cdot\gamma)$ converges
to $d/d+1$, as $n$ goes to infinity.
  $\bullet$ Applying Zhai's Lemma 3 to the Hilbert function of a positively
graded Artinian algebra yields a new class of numerical semigroups satisfying
Wilf's inequality.
