Distributions of weights and a question of Wilf

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.