Let us now take a quick look at the class $N$, with the purpose of determining how much of Theorems 1 and 2 is true here.
This inspired us to take a fresh look at all the results in [BG].
A careful look at the proofs reveals that ......
The preceding observation, when looked at from a more general point of view, leads to ......
Rather than discuss this in full generality, let us look at a particular situation of this kind.
Suppose, to look at a more specific situation, that ......
The next lemma shows how such a semilattice looks when embedded in a larger compact semilattice.
Let us see what such a formula might look like, by analogy with Fourier series.
The proof is nonconstructive and gives no indication of what the exceptional set may look like.
This observation prompted the author to look for a more constructive solution.
Here, of course, the set $A$ produced is rather thin and certainly nowhere near the densities we are looking for.