Linear independence of additive convolutions of multiplicative coset indicators: an existence theorem for index-p cyclotomy

Jeff Pickhardt

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Linear independence of additive convolutions of multiplicative coset indicators: an existence theorem for index-p cyclotomy

Abstract

Let q = fp + 1 with p, q odd primes, let H ≤ F × q be the subgroup of order f (index p), and let Ai = gki H (i = 1, 2, 3) be three of its cosets. We study the 23 = 8 subset convolutions μS = ∗i∈S 1Ai (S ⊆ {1, 2, 3}) as elements of the group algebra Fq [Fq , +], and prove that there exist absolute constants C0, c > 0 such that for every prime pair q = f p + 1 with C0 ≤ f ≤ c p_1/2, labels k1, k2, k3 can be chosen so that the eight convolutions are linearly independent modulo the socle (the span of the all-ones vector). This is a positive- characteristic, modular analogue of a nonvanishing-minors problem for compressed Fourier transforms. It is distinct from the complex nonvanishing-minors theorem of Garcia–Karaali– Katz: a minor that is nonzero over C may vanish after reduction modulo q. At f ∈ {2, 4} genuine label failures show that no all-label version can hold. The proof is an elimination scheme driven by an exact characterization of failure as integer-affine rigidity of a shifted cyclotomic convolution, closed by a second-moment (diagonal-identity) contradiction; all counts are elementary and no character-sum input is used at all — no Weil bounds and no Stepanov-type subgroup-intersection bound. The window f ≤ c p_1/2 is the exact reach of the second-moment mechanism, which degenerates at f 2 ∼ p. We also prove an exact grid- multinomial collapse identity for the Fq -native frequency transform of the pair convolution, which subsumes the depth-2 binomial-jet structure and is proposed as the foundation for removing the window f ≤ c p_1/2.

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