claimroutine

Block functions are simplex-valued, low influence

Block functions are simplex-valued with mean 1/q1/q, low influence if pseudorandom
puts blocks in the form needed by rounding
Claim·§A.2 The finite ordering transfer·inside propFinite ordering transfer from Unique Games

Statement (paper)

For each left vertex aa, the block functions Fa=(Fa1,…,Faq)F_a=(F_a^1,\ldots,F_a^q) defined in satisfy Fa(z)∈SqF_a(z)\in\mathcal S_q for every zz, and EzFaj(z)=∣Ca,j∣hmR=1q\mathbb E_z F_a^j(z)=\frac{|C_{a,j}|}{hm^R}=\frac1q for every j∈[q]j\in[q]. If aa is pseudorandom at threshold tt, then Inf⁡ℓ(T1−κFaj)≤t\operatorname{Inf}_\ell(T_{1-\kappa}F_a^j)\le t for every block index jj and every ℓ∈[R]\ell\in[R].

In plain English

For each left vertex aa, the block functions Fa=(Fa1,…,Faq)F_a=(F_a^1,\ldots,F_a^q) take values in the simplex Sq\mathcal S_q, and each coordinate FajF_a^j has mean exactly 1/q1/q. If aa is pseudorandom at threshold tt, then every noised block function T1−κFajT_{1-\kappa}F_a^j has all coordinate influences Inf⁡ℓ\operatorname{Inf}_\ell, ℓ∈[R]\ell\in[R], at most tt.

Why it's here

This puts the block functions of a left vertex in the form needed to apply the Gaussian/noise-stability machinery: simplex-valued, with equal means 1/q1/q, and low influence when the vertex is pseudorandom. The local payoff bound for pseudorandom vertices uses it to compare the vertex's ordering payoff with the optimum opt⁡q(J)\operatorname{opt}_q(J) of the qq-block problem. The mean 1/q1/q comes from the blocks Ca,jC_{a,j} having equal size hmR/qhm^R/q, which follows from q∣hq\mid h.
Classified routine:
Direct verification from the definition of the block functions: the simplex values and the 1/q means follow from the blocks partitioning the ordering into equal sizes (q | h), and the influence bound is the definition of pseudorandomness at threshold t, as in standard Unique Games dictatorship-test reductions.

Uses

Nothing else in the paper

Used by

claimPseudorandom vertices have local payoff bound

Sources

PaperOpen
Section
§A.2 The finite ordering transfer
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appendices/finite-consequences.tex:707
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