For non-involutive distributions, a tangency set of positive measure is possible exactly when its fractional Sobolev regularity is low enough relative to the surface gradient regularity.
Pauls rectifiable and purely Pauls un- rectifiable smooth hypersurfaces
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Frobenius theorem and fine structure of tangency sets to non-involutive distributions
For non-involutive distributions, a tangency set of positive measure is possible exactly when its fractional Sobolev regularity is low enough relative to the surface gradient regularity.