Square Functions and Rectifiability under Monotone Transformations of the Density

Chousionis–Garnett–Le–Tolsa (CGLT) showed that an n-AD-regular measure in ℝ^d is uniformly n-rectifiable if and only if a square function built from differences of its density across dyadic scales satisfies a Carleson condition. This paper proves that the characterization still holds after the density is composed with any function F that is bi-Lipschitz on the range fixed by the AD-regularity constant.

The main example is F = log. It gives the scale-invariant square function Δ^log_μ(x, r) = log(μ(B(x, r)) / μ(B(x, 2r))) + n log 2, which compares mass at two scales without normalizing by rⁿ. This is the log-density statistic behind the UR–JEPA regularizer and its anomaly diagnostic.

Main results

• Uniform rectifiability is invariant under bi-Lipschitz transformations of the density. The CGLT Carleson characterization holds for the square function of F(density), with a complete proof.
• Smooth square functions. The result extends to the smooth CGLT square functions, where the density is replaced by the convolution of μ with a Gaussian or a more general radial kernel.
• Beyond bi-Lipschitz. The paper discusses what happens when F is not bi-Lipschitz. For finite measures, the behavior of F near zero matters in only one of the two implications.
• Pointwise version. The Tolsa–Toro characterization of n-rectifiable measures, via the same square function at μ-almost every point, holds after composition with any locally bi-Lipschitz F.