Let \(\Delta =\{| z| <1\}\) and \(C=\{| z| =1\}\) be the unit disk and its circumference in the complex plane. For \(\omega\neq 0\) a continuous modulus of continuity on [0,2\(\pi\) ], define the generalized BCH (Beurling-Carleson-Hayman) sets or \(\omega\)-sets as the closed subsets K of C of linar measure \(| K| =0\) with \(\sum \omega (| I_ k|)<\infty\), where \((I_ k)\) is an enumeration of the component arcs of \(C\setminus K\). Let \(\Lambda_{\omega}\) denote the continuous functions f on \({\bar \Delta}=\{| z| \leq 1\}\) analytic on \(\Delta\) having modulus of continuity \(\omega_ f\leq c\omega\) \((c>0)\). Shirokov recently generalized to arbitrary moduli of continuity \(\omega\) the characterization due to Beurling and Carleson of the boundary zero sets of f in \(\Lambda_{\omega}\) for \(\omega (t)=t^{\alpha}\), \(0<\alpha \leq 1\), as BCH sets. The characterization of the entire zero sets was actually given, with the boundary zero sets being the \(\Phi_{\omega}\)-sets where \(\Phi_{\omega}(t)=\int^{t}_{0}\log [1/\omega (s)]ds\). Using this fact, it is shown that a necessary and sufficient condition for the zero sets of functions of \(\Lambda_{\omega}\) to be the same as those of \(\Lambda_{\nu}\) is that \(\omega^{\alpha}\leq \nu \leq \omega^{\beta}\) for some \(\alpha,\beta >0\). Results of Ahern and Shapiro dealing with the relationship between Hausdorff measure and BCH sets are also generalized to arbitrary \(\omega\)-sets. Questions concerning uniform and asymptotic boundary descent to 0 are also considered.