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Quadratic character bias on affine subspaces

Lax342547.QuadraticBias · concepts/Lax342547/QuadraticBias.lean · lax-342547

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    Natural Language Statement

    Lemma

    Translation and character orthogonality give the actual quadratic sign bias from polar rank and its affine-subspace restriction.

    Concept map
    6 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 7 statements. Each proof establishes one of them relative to its assumptions.

    1 affine_restricted_polar proven

    2 affine_subspace_bias proven

    3 polar_sign_product proven

    4 quadratic_bias_rank_bound proven

    5 quadratic_bias_square_bound proven

    6 quadratic_sign_square proven

    7 square_translation proven

    Lean source view on GitHub

    1import Lax342547.Walsh
    2import Lax342547.FiniteLinearLaw
    3import Lax342547.RestrictionRank
    4import Lax342547.FormalQuadratic
    5
    6/-!
    7---
    8title: Quadratic character bias on affine subspaces
    9type: lemma
    10---
    11Translation and character orthogonality give the actual quadratic sign bias from polar rank and its affine-subspace restriction.
    12-/
    13
    14namespace Lax342547.QuadraticBias
    15
    16open Lax342547.MomentSpace Lax342547.Walsh
    17open scoped BigOperators
    18
    19axiom square_translation {V : Type} [AddCommGroup V] [Fintype V]
    20 (f : V → ℝ) :
    21 (∑ x, f x)^2 = ∑ h, ∑ x, f x*f (x+h)
    22
    23axiom polar_sign_product {V : Type} [AddCommGroup V] [Module Binary V]
    24 (Q : V → Binary) (B : LinearMap.BilinForm Binary V)
    25 (hpolar : ∀ x h, Q (x+h)+Q x+Q h+Q 0 = B h x) (x h : V) :
    26 sign (Q x)*sign (Q (x+h)) = sign (Q h+Q 0)*sign (B h x)
    27
    28axiom quadratic_sign_square {V : Type} [AddCommGroup V] [Module Binary V] [Fintype V]
    29 (Q : V → Binary) (B : LinearMap.BilinForm Binary V)
    30 (hpolar : ∀ x h, Q (x+h)+Q x+Q h+Q 0 = B h x) :
    31 by
    32 classical
    33 exact (∑ x, sign (Q x))^2 = (Fintype.card V : ℝ)*
    34 ∑ h ∈ Finset.univ.filter (fun h => B h = 0),sign (Q h+Q 0)
    35
    36axiom quadratic_bias_square_bound {V : Type} [AddCommGroup V] [Module Binary V]
    37 [Fintype V] [FiniteDimensional Binary V]
    38 (Q : V → Binary) (B : LinearMap.BilinForm Binary V)
    39 (hpolar : ∀ x h, Q (x+h)+Q x+Q h+Q 0 = B h x) :
    40 ((∑ x, sign (Q x))/(Fintype.card V : ℝ))^2 ≤
    41 1/(2 : ℝ)^Module.finrank Binary (LinearMap.range B)
    42
    43axiom quadratic_bias_rank_bound {V : Type} [AddCommGroup V] [Module Binary V]
    44 [Fintype V] [FiniteDimensional Binary V]
    45 (Q : V → Binary) (B : LinearMap.BilinForm Binary V) (r : ℕ)
    46 (hpolar : ∀ x h, Q (x+h)+Q x+Q h+Q 0 = B h x)
    47 (hrank : 2*r ≤ Module.finrank Binary (LinearMap.range B)) :
    48 |(∑ x, sign (Q x))/(Fintype.card V : ℝ)| ≤ 1/(2 : ℝ)^r
    49
    50axiom affine_restricted_polar {V : Type} [AddCommGroup V] [Module Binary V]
    51 (Q : V → Binary) (B : LinearMap.BilinForm Binary V)
    52 (hpolar : ∀ x h, Q (x+h)+Q x+Q h+Q 0 = B h x)
    53 (S : Submodule Binary V) (w : V) (x h : S) :
    54 Q (w+(x+h).val)+Q (w+x.val)+Q (w+h.val)+Q w = (B.restrict S) h x
    55
    56axiom affine_subspace_bias {V : Type} [AddCommGroup V] [Module Binary V]
    57 [FiniteDimensional Binary V]
    58 (Q : V → Binary) (B : LinearMap.BilinForm Binary V) (S : Submodule Binary V) [Fintype S]
    59 (w : V) (r c : ℕ)
    60 (hpolar : ∀ x h, Q (x+h)+Q x+Q h+Q 0 = B h x)
    61 (hrank : 2*r ≤ Module.finrank Binary (LinearMap.range B))
    62 (hcodim : Module.finrank Binary V ≤ Module.finrank Binary S+c) :
    63 |(∑ x : S, sign (Q (w+x.val)))/(Fintype.card S : ℝ)| ≤ 1/(2 : ℝ)^(r-c)
    64
    65end Lax342547.QuadraticBias
    66
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