Gaussian rigidity and width of measures
Abstract
Given 0≤ m<n and a positive Radon measure μ on ℝn with at most polynomial growth, assume that it has finite m-dimensional Gaussian discrepancy, defined as 𝒟(μ):=limsupR→∞[R2supx∈sptμ|GRμ(x)−1|], where GRμ(x):=(4πR2)−m/2∫ℝn e−|z−x|2/(4R2) dμ(z). Provided that μ admits a flat blow-down ℋmllcorner V (an assumption that cannot be dropped entirely), we prove that there exists a finite (smallest) 0≤δ<∞, the so-called width of μ, such that sptμ⊆ B̅δ(V+x0), δ2≤ C(m,n)𝒟(μ), where B̅δ(V+x0) denotes the closed δ-neighborhood of a suitable translate of V. Note that the exponent is optimal because of scale invariance. In particular, when 𝒟(μ)=0, the support of μ is included in an m-dimensional affine plane. We use this fact to give a simpler proof of Preiss's rectifiability theorem: while still based on Gaussian averages of μ, this new proof avoids showing uniqueness (and hence conicality) of blow-downs of m-uniform measures, as well as studying Gaussian-weighted moments beyond second order. The strategy of proof consists of an anisotropic blow-down of μ, followed by an analysis of the resulting Young measure, using a rigidity lemma involving the heat semigroup.
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BibTeX
@misc{pigati2026gaussian,
title = {{Gaussian rigidity and width of measures}},
author = {Alessandro Pigati},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20385},
eprint = {2610.20385},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20385},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gaussian rigidity and width of measures
AU - Alessandro Pigati
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.20385
UR - https://arxiv.org/abs/2610.20385
AB - Given 0≤ m<n and a positive Radon measure μ on ℝ^n with at most polynomial growth, assume that it has finite m-dimensional Gaussian discrepancy, defined as 𝒟(μ):=limsup_R→∞[R^2sup_x∈sptμ|G_R^μ(x)−1|], where G_R^μ(x):=(4πR^2)^−m/2∫_ℝ^n e^−|z−x|^2/(4R^2) dμ(z). Provided that μ admits a flat blow-down ℋ^mllcorner V (an assumption that cannot be dropped entirely), we prove that there exists a finite (smallest) 0≤δ<∞, the so-called width of μ, such that sptμ⊆ B̅_δ(V+x_0), δ^2≤ C(m,n)𝒟(μ), where B̅_δ(V+x_0) denotes the closed δ-neighborhood of a suitable translate of V. Note that the exponent is optimal because of scale invariance. In particular, when 𝒟(μ)=0, the support of μ is included in an m-dimensional affine plane. We use this fact to give a simpler proof of Preiss's rectifiability theorem: while still based on Gaussian averages of μ, this new proof avoids showing uniqueness (and hence conicality) of blow-downs of m-uniform measures, as well as studying Gaussian-weighted moments beyond second order. The strategy of proof consists of an anisotropic blow-down of μ, followed by an analysis of the resulting Young measure, using a rigidity lemma involving the heat semigroup.
ER -