The circle method applies to rational points on Perazzo-type cubic fourfolds and exceeds the square-root barrier.
Swarbrick Jones, Weak approximation for cubic hypersurfaces of large dimension.Algebra & Number Theory7(2013), 1353–1363
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Beyond the square-root barrier: cubic forms of Perazzo type
The circle method applies to rational points on Perazzo-type cubic fourfolds and exceeds the square-root barrier.