Rectifiability of measures and the β<inf>p</inf> coefficients

In some former works of Azzam and Tolsa it was shown that n-rectifiability can be characterized in terms of a square function involving the David–Semmes β2 coefficients. In the present paper we construct some counterexamples which show that a similar characterization does not hold for the βp coefficients with p 6= 2. This is in strong contrast with what happens in the case of uniform n-rectifiability. In the second part of this paper we provide an alternative argument for a recent result of Edelen, Naber, and Valtorta about the n-rectifiability of measures with bounded lower n-dimensional density. Our alternative proof follows from a slight variant of the corona decomposition in one of the aforementioned works of Azzam and Tolsa and a suitable approximation argument.