The Heinis spectrum \(\varOmega \) is the set of all pairs \((\alpha _u,\beta _u)\) such that \(\alpha _u=\liminf _{n\rightarrow \infty}\frac{p_u(n)}{n}\) and \(\beta _u=\limsup _{n\rightarrow \infty}\frac{p_u(n)}{n}\) for some infinite word u. In this paper, we demonstrate that there exists a closed connected set with non-empty interior contained in \(\varOmega \) . Furthermore, every point in this set can be represented as the pair \((\alpha _u,\beta _u)\) for some recurrent word u. The construction is explicit, algorithmic in nature and is based on constructing certain “Cantor sets of integers”, whose “gaps” correspond to blocks of zeros.