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We study the electronic and local structural properties of pure andIn-substituted $\beta$-Ga$_2$O$_3$ using density functional theory (DFT). Ourmain result is that the structural energetics of In in Ga$_2$O$_3$ causes mostsites to be essentially inaccessible to In substitution, thus reducing themaximum In content in thi to somewhere between 12 and 25 \% in this phase. Wealso find that the gap variation with doping is essentially due to "chemicalpressure", i.e. volume variations with doping.

Structure and gap of low-$x$ (Ga$_{1-x}$In$_x$)$_2$O$_3$ alloys