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Interferometric measurements of the radio emission of extensive air showersallow reconstructing cosmic-ray properties. A recent simulation study with anidealised detector promised measurements of the depth of the shower maximum$X_\mathrm{max}$ with an accuracy better than 10$\,$g$\,$cm$^{-2}$. In thiscontribution, we evaluate the potential of interferometric $X_\mathrm{max}$measurements of (simulated) inclined air showers with realisticallydimensioned, sparse antenna arrays. We account for imperfect timesynchronisation between individual antennas and study its inter-dependency withthe antenna density in detail. We find a strong correlation between the antennamultiplicity (per event) and the maximum acceptable inaccuracy in the timesynchronisation of individual antennas. From this result, prerequisites for thedesign of antenna arrays for the application of interferometric measurementscan be concluded. For data recorded with a time synchronisation accurate to1$\,$ns within the commonly used frequency band of 30$\,$MHz to 80$\,$MHz, anantenna multiplicity of $\gtrsim 50$ is needed to achieve an $X_\mathrm{max}$reconstruction with an accuracy of 20$\,$g$\,$cm$^{-2}$. This multiplicity isachieved measuring inclined air showers with zenith angles $\theta \geq77.5^\circ$ with 1$\,$km spaced antenna arrays, while vertical air showers withzenith angles $\theta \leq 40^\circ$ require an antenna spacing below 100$\,$m.Furthermore, we find no improvement in $X_\mathrm{max}$ resolution applying theinterferometric reconstruction to measurements at higher frequencies, i.e., upto several hundred MHz.

Expected performance of interferometric air-shower measurements with radio antennas