Discrete analogues of John’s theorem

The inverse theorem for the $U^3$ Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches

“…Let G be a finite additive group, let 0 < η ≤ 1/2, and let f : G → C be a 1-bounded function with f U 3 (G) ≥ η. Then there exists a regular Bohr set B(S , ρ) with |S | ≪ η −O (1) and exp(−η −O (1) ) ≪ ρ ≤ 1/100, a locally quadratic function φ : Bohr(S , ρ) → R/Z, and a function ξ : G → Ĝ such that (1) .…”

“…Let G be a finite additive group, let 0 < η ≤ 1/2, and let f : G → C be a 1-bounded function with f U 3 (G) ≥ η. Then there exists a natural number N ≪ η −O (1) , a polynomial map g : G → H(R N )/H(Z N ), and a Lipschitz function F :…”

“…4 In the earlier references an ostensibly weaker version of the conjecture is established in which one works with "non-periodic nilsequences" rather than "periodic nilsequences", but the two formulations can be shown to be equivalent: see [26] for further discussion. 5 For instance, as remarked previously, when G = F n p , the CFR nilspace can be chosen to be isomorphic to the disconnected nilmanifold 1 p j Z/Z; see for instance [37].…”

“…contains Z S as a finite index subgroup, and is thus a lattice. Applying the discrete John's theorem [34,Lemma 1.6] (see also [1] for recent refinements), we can find linearly independent vectors w ξ ∈ Γ and real numbers N ξ > 0 for ξ ∈ S such that…”

“…It is possible that the finite generation or finiteness hypotheses on G here can be relaxed, but we will not need to do so here. If |G| were odd, then one proceed by setting φ(x) ≔ B(x, 1 2 x) since one now has the ability to divide by two in G; this observation was implicit in the arguments in [14]. Similarly, if G = F N 2 , then (for part (i) at least) one could proceed by inspecting the matrix coefficients of B and extracting the upper triangular half to define φ; this observation was implicit in the arguments in [28].…”