I have been recently reading X-G Wen's book on gauge theories. In chapter 6 he gave two examples of gauge theories showing confinement. The first one is a compact U(1) gauge theory in 2+1D, and the second one is a lattice U(1) theory in 3+1D. The treatments he gave for the two cases are different, for the 2+1D case he relied on duality a lot.
In both cases, in the confined phase, the gauge theories are also gapped. However, I do not immediately see the relation between the two concepts.
The main reason for my puzzle is, if a gauge theory is gapped, it should become short-ranged, and naively one should NOT expect the charges to strongly interact (in analogy with screening effect), let alone confinement. Hence the question in the title.
Update: I think I have figured out what is going on (see my comments below), but more comments/discussions are welcome!