A finite Gaussian-width route to moment extraction

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Standing

This is a sufficient extraction criterion at the Boolean-moment boundary. It avoids requiring a basis with simultaneously small Gram--Schmidt coordinates and bounded energy. The theta estimate (4) below is proved at the rounded critical boundary by the stable-lattice bridge, so the criterion supplies the extraction constant there. Arbitrary-color growth remains open; nothing here proves unbounded coloring or completes E0774.

1. A checked single-set extraction theorem

For a finite set A of n nonzero distinct characters of a compact abelian group, let alpha(A) be its maximum dissociated cardinality and put

$$W(A)=\mathbb E\left\|\sum_{\gamma\in A}Z_\gamma\gamma\right\|_\infty,$$

where the Z_gamma are independent complex Gaussians. There is an absolute C such that

$$\alpha(A)\geq W(A)^2/(Cn). \tag{1}$$

This is Rodriguez-Piazza's finite theorem, stated as Theorem 3.4, equation (3.3), printed page 10 of Lefevre--Li--Queffelec--Rodriguez-Piazza (arXiv:0902.2625). The author checked the primary abstract, then the definitions of the Gaussian norm on printed page 5 and of alpha and W on page 10, and the theorem itself. Gaussian normalization only changes C. The statement is for one finite set, not a conclusion inferred from a hypothesis on all subsets.

For moment configurations, flip transitivity gives eta=alpha/n, as already proved. Thus W(E_(d,k))>=c n for one c>0 would settle hereditary extraction along a parameter sequence. At the rounded critical boundary that width lower bound follows from (2) and (4) below with the stable-lattice bridge.

2. Fixed-scale phase packing implies linear width

Let T={(gamma(t))_(gamma in A):t in G}, a subset of (S^1)^n. Suppose T contains at least exp(a n) points separated by epsilon in the coordinate maximum metric, for fixed a>0 and 0<epsilon<=1. Then

$$W(A)\geq c(a,\epsilon)n,\qquad \alpha(A)\geq c'(a,\epsilon)n. \tag{2}$$

Here is the finite metric argument. Put M=ceil(8 pi/epsilon), and choose 0<beta<1/2 with h(beta)+beta log M<=a/2, where h is binary entropy using natural logarithms. Set r=epsilon sqrt(beta)/4. In any Euclidean ball of radius r sqrt(n), a point has at most beta n coordinates differing from its center by at least epsilon/4. Specify those coordinates and their labels in an M-arc partition of the circle with arc diameter at most epsilon/4. Two packing points with the same specification are less than epsilon apart everywhere, and so must coincide. The ball therefore contains at most

$$\sum_{j\leq\beta n}\binom njM^j \leq\exp(n[h(\beta)+\beta\log M])\leq e^{an/2}$$

packing points. Greedy deletion leaves Q>=exp(a n/2) points at pairwise Euclidean distance greater than r sqrt(n).

For independent standard real Gaussians g_j,h_j, their evaluation process X_t=sum_j(g_j Re t_j+h_j Im t_j) has increment variance ||t-s||^2. Compare it on these Q points to independent Gaussians Y_i of variance r^2 n/2. The Gaussian comparison needed here follows directly by differentiating the expected soft maximum F_b(z)=b^(-1)log sum_i exp(b z_i) along sqrt(u)X+sqrt(1-u)Y. Gaussian integration by parts makes the derivative

$$\frac b4\mathbb E\sum_{i,j}p_ip_j [\mathbb E(X_i-X_j)^2-\mathbb E(Y_i-Y_j)^2]\geq0,$$

where the p_i are the positive soft-max weights. Let b tend to infinity. The elementary Gaussian tail bound gives E max_(i<=Q)Y_i>=c r sqrt(n log Q). It follows that W(A)>=c epsilon sqrt(a beta) n, up to the immaterial complex-Gaussian normalization. Applying (1) proves (2).

3. A theta bound controlling the actual wrapped torus

Let V be a rational subspace of R^n, let T=exp(2 pi i V), and let K=V perpendicular intersect Z^n. For s>0 define

$$\vartheta_K(s)=\sum_{z\in K}e^{-\pi s^2\|z\|^2}.$$

For 0<epsilon<=1/2 let B_epsilon be the normalized-angle neighborhood of the identity in T: every coordinate has distance at most epsilon from an integer. Its Haar probability satisfies

$$\boxed{\mu_T(B_\epsilon) \leq e^{\pi n\epsilon^2/s^2}\,s^n\vartheta_K(s).} \tag{3}$$

Indeed, the one-dimensional periodic Gaussian g_s(t)=sum_(a in Z)exp(-pi(t-a)^2/s^2) has absolutely convergent Fourier series s sum_(j in Z)exp(-pi s^2 j^2)exp(2 pi i jt). This follows by integrating its Fourier coefficients and evaluating the Gaussian Fourier transform. Integrating the product of n such series over T retains exactly frequencies in K, giving s^n vartheta_K(s). On B_epsilon each factor is at least exp(-pi epsilon^2/s^2), proving (3).

Consequently, the following fixed-parameter estimate would suffice: there exist s>0 and b>0, independent of d, such that

$$\boxed{\log s+\frac1n\log\vartheta_K(s)\leq-b.} \tag{4}$$

Choose fixed epsilon>0 small enough that pi epsilon^2/s^2<=b/2. Then (3) is at most exp(-bn/2). A maximal epsilon-separated set in the normalized-angle metric covers T by its translated B_epsilon balls, and so has at least exp(bn/2) points. On this range angular separation epsilon gives a fixed positive chordal separation. Equation (2) applies.

For moments, K is exactly the integer relation lattice already proved semistable. The reverse-Minkowski theorem alone controls it at a parameter depending on log n and the normalized determinant; it does not give (4) with fixed s,b at the critical boundary. The weak stable-lattice free-energy theorem does, and the boundary extraction page applies it. The compression identities by themselves do not bound the theta function of the already compressed degree ball.

4. Why phase-period covolume cannot replace (3)

The covolume of V intersect Z^n alone is insufficient. For A_N={N+1,...,N+n}, that covolume is sqrt(sum_(j=1)^n(N+j)^2), since these consecutive integers are primitive. It can be arbitrarily large while n is fixed. Nevertheless any dissociated m-subset satisfies

$$2^m\leq(m+1)(mn+1).$$

For each subset cardinality j, subtract jN from its sum; at most mn+1 values remain, and there are m+1 cardinalities. Thus alpha(A_N)=O(log n) uniformly in N. Multiplication by exp(2 pi i Nt) also shows that the Gaussian sup norm is independent of N. These phase curves all lie on the two-parameter surface (exp(2 pi i(a+jb)))_j, whose fixed-scale coordinate covering number is O(n epsilon^(-2)).

For n>=3 this example does not refute a stronger use of semistability: its kernel contains {z:sum z_j=sum jz_j=0}, of rank n-2 and covolume n sqrt((n^2-1)/12), independent of N. The whole kernel's normalized covolume grows with N, so it is not semistable for sufficiently large N. The remaining bridge is therefore a genuine estimate such as (4), not another unqualified volume-to-packing inference.

All new finite arguments here were checked internally. No computation, external review, boundary extraction theorem, or counterexample is claimed.