Accessing the homogeneity scale with 21 cm intensity mapping surveys

The homogeneity scale, $R_{\rm H}$, offers a fundamental test of the Cosmological Principle, yet it has not yet been measured with 21cm intensity mapping surveys. A key limitation for such a measurement is the telescope beam, which artificially smooths the observed signal. We quantify this effect using the two-point correlation function and the correlation dimension, $\mathcal{D}_2(r)$, to model how beam convolution suppresses intrinsic clustering. For any given redshift $z$, we identify a maximum beam width, $\sigma_{\rm max}(z)$, beyond which the homogeneity scale cannot be recovered. This limit defines an inaccessible region in the $\sigma \times z$ parameter space, where $R_{\rm H}$ is erased by beam smoothing. Applying this framework to several current and upcoming radio telescopes, we assess their ability to probe $R_{\rm H}$. Our results provide the first quantitative forecast of the instrumental requirements for measuring the cosmic homogeneity scale with 21cm IM, and establish a theoretical basis for future observational applications.