We investigate the physics of a two-dimensional electron gas threaded by an accessible magnetic flux tube of strength φ (in units of Φ0=hc/e) and radius R→0. The Zeeman interaction of the magnetic field inside the flux tube with the magnetic moment of the electrons is taken explicitly into account. First we concentrate on the existence of bound states in an arbitrarily weak square-well attractive potential. When the spin σ=s/2 (s=±1) of the electrons is parallel to the flux, there is one bound state in the angular momentum quantum number m for which 1>m-φ=-m+φs>0. The binding energy approaches its maximal value (equal to the potential strength) as φ→m+1 from below. Second, we consider a two-dimensional electron gas confined in a disk (either clean or dirty). In this case the single-particle energy is a function of m-φ [and not of (m-φ)2 as in a one-dimensional ring geometry]. We discuss the pertinent consequence on the ground-state energy and its possible implications on the resulting persistent currents.