Research proof index

Reconstructing a concrete coset FFT domain from its class instance gives back the original domain.

Scaling the normalized FFT-domain image by ω 0 recovers the original coset FFT-domain image.

The existence of a nontrivial smooth FFT domain rules out characteristic 2.

In a smooth FFT domain of nonzero logarithmic size, -1 belongs to the domain.

Doubly stochastic Hölder inequality: for nonneg a, b, doubly stochastic w, and conjugate p, q > 1: ∑{ij} a_i * b_j * w{ij} ≤ (∑ a_i^p)^{1/p} * (∑ b_j^q)^{1/q}.

A dependent-random-choice estimate. For $p \ge 2$, a nonnegative function $f$, nonempty $A$, and intersecting sets $B_1,B_2$, the support hypothesis produces subsets $A_1 \subseteq B_1$ and $A_2 \subseteq B_2$ whose normalized difference convolution has controlled correlation with $f$. Both relative sizes $|A_i|/|B_i|$ are bounded below by the same explicit quantity, namely one quarter of a normalized $2p$-th power of the weighted $L^p$ norm of $1_A \mathbin{\circleddash} 1_A$.