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Deterministic bound for the large-coefficient CNA term

Lax253009.LargeCoefficientSoundness · concepts/Lax253009/LargeCoefficientSoundness.lean · lax-253009

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    Natural Language Statement

    Theorem

    Suppose every retained support contains a distinguished point yy, has size at most ℓ<N\ell<N, and contains no other point with the same label as yy. For coefficients of magnitude at least δ>0\delta>0 and total squared mass at most one, the absolute normalized Fourier sum is at most ℓ/((N−ℓ)δ)\ell/((N-\ell)\delta). This supplies the deterministic cancellation estimate for the second CNA term in Lemma 4.8.

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    1import Lax253009.HighDegreeSoundness
    2import Lax253009.BalancedCancellation
    3
    4/-!
    5---
    6title: Deterministic bound for the large-coefficient CNA term
    7type: theorem
    8---
    9Suppose every retained support contains a distinguished point yy, has
    10size at most ℓ<N\ell<N, and contains no other point with the same label
    11as yy. For coefficients of magnitude at least δ>0\delta>0 and total
    12squared mass at most one, the absolute normalized Fourier sum is at most
    13ℓ/((N−ℓ)δ)\ell/((N-\ell)\delta). This supplies the deterministic cancellation
    14estimate for the second CNA term in Lemma 4.8.
    15-/
    16
    17namespace Lax253009.LargeCoefficientSoundness
    18
    19open HighDegreeSoundness
    20open scoped BigOperators
    21
    22axiom bound {ι κ : Type} [DecidableEq ι] [Fintype κ] [DecidableEq κ]
    23 (n : ℕ) (hN : Fintype.card κ = 2 * n)
    24 (supports : Finset (Finset ι)) (c : Finset ι → ℝ) (l : ℕ)
    25 (f : ι → κ) (y : ι) (δ : ℝ) (hδ : 0 < δ) (hl : l < Fintype.card κ)
    26 (hsupport : ∀ S ∈ supports, y ∈ S ∧ S.card ≤ l ∧ ∀ i ∈ S, i ≠ y → f i ≠ f y)
    27 (hlarge : ∀ S ∈ supports, δ ≤ |c S|) (henergy : ∑ S ∈ supports, c S ^ 2 ≤ 1) :
    28 |normalizedSum supports c n f| ≤ (l : ℝ) / ((Fintype.card κ : ℝ) - l) / δ
    29
    30end Lax253009.LargeCoefficientSoundness
    31
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