250 Boolean Analysis — Rotation

250.1 Overview

Gaussian rotations preserve the joint Gaussian law. Their disagreement probabilities satisfy continuity, subadditivity, and equal-step lower bounds.

250.2 Declarations

Theorem 250.1 Rotation invariance of paired Gaussian vectors

✓

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). For any real \(t\), the map \((g,h)\mapsto (\cos t\, g+\sin t\, h,-\sin t\, g+\cos t\, h)\) preserves \(\gamma _n\otimes \gamma _n\).

Theorem 250.2 Subadditivity of Gaussian disagreement

✓

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). If \(A\subseteq \mathbb R^n\) is measurable, then \(s_A(s+t)\le s_A(s)+s_A(t)\) for all real \(s,t\).

Theorem 250.3 Disagreement at a sum of angles

✓

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). For any measurable \(A\subseteq \mathbb R^n\), integer \(\ell \ge 0\), and real angles \(\delta _1,\ldots ,\delta _\ell \), \(s_A(\sum _{j=1}^\ell \delta _j)\le \sum _{j=1}^\ell s_A(\delta _j)\).

Theorem 250.4 Equal-step rotation bound

✓

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). For any measurable \(A\subseteq \mathbb R^n\), real \(\delta \), and positive integer \(\ell \), \(s_A(\delta )\le \ell \, s_A(\delta /\ell )\).

Lemma 250.5 Disagreement between independent Gaussian endpoints

✓

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). If \(A\subseteq \mathbb R^n\) is measurable and \(\alpha =\gamma _n(A)\), then \(s_A(\pi /2)=2\alpha (1-\alpha )\).

Theorem 250.6 Balanced-set disagreement on an angular grid

✓

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). If \(A\subseteq \mathbb R^n\) is measurable with \(\gamma _n(A)=1/2\), then \(s_A(\pi /(2\ell ))\ge 1/(2\ell )\) for every positive integer \(\ell \).

Lemma 250.7 Continuity of disagreement at zero angle

✓

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). For any measurable \(A\subseteq \mathbb R^n\), \(s_A(t)\to 0\) as the real angle \(t\to 0\).