Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Three APFree w Inner one mu ddconv mu mu two smul mu

ThreeAPFree.wInner_one_mu_ddconv_mu_mu_two_smul_mu

additive combinatorics arithmetic progressions Fourier analysis three ap

Plain-language statement

For a finite group of odd order and a three-term-progression-free set $s$ , the normalized inner product between $\mu_s*\mu_s$ and the uniform measure on $2s$ is exactly $|s|^{-2}$ . The identity records the precise normalized count forced by the absence of nontrivial three-term progressions.

Exact Lean statement

lemma ThreeAPFree.wInner_one_mu_ddconv_mu_mu_two_smul_mu (hG : Odd (card G))
    (hs : ThreeAPFree (s : Set G)) :
    ⟪μ_[ℝ] s ∗ᵈ μ s, μ (s.image (2 • ·))⟫_[ℝ] = (#s ^ 2 : ℝ)⁻¹

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma ThreeAPFree.wInner_one_mu_ddconv_mu_mu_two_smul_mu (hG : Odd (card G))    (hs : ThreeAPFree (s : Set G)) :    ⟪μ_[ℝ] s ∗ᵈ μ s, μ (s.image (2 • ·))⟫_[ℝ] = (#s ^ 2 : ℝ)⁻¹ := by  obtain rfl | hs' := s.eq_empty_or_nonempty  · simp  simp only [wInner_one_eq_sum, inner_apply', sum_ddconv_mul, ← sum_product', RCLike.conj_to_real]  rw [← diag_union_offDiag univ, sum_union (disjoint_diag_offDiag _), sum_diag, ←    sum_add_sum_compl s, @sum_eq_card_nsmul _ _ _ _ _ (#s ^ 3 : ℝ)⁻¹, nsmul_eq_mul,    Finset.sum_eq_zero, Finset.sum_eq_zero, add_zero, add_zero, pow_succ', mul_inv,    mul_inv_cancel_left₀]  · exact Nat.cast_ne_zero.2 hs'.card_pos.ne'  · refine fun i hi ↦ not_ne_iff.1 fun h ↦ (mem_offDiag.1 hi).2.2 ?_    simp_rw [mul_ne_zero_iff, ← mem_support, support_mu, mem_coe, mem_image, two_smul] at h    obtain ⟨b, hb, hab⟩ := h.2    obtain rfl := hs h.1.1 hb h.1.2 hab.symm    simpa using hab  · simpa using fun _ ↦ Or.inl  · rintro a ha    simp only [mu_apply, ha, if_true, mul_one, mem_image, mul_ite, mul_zero]    rw [if_pos ⟨_, ha, two_smul _ _⟩, card_image_of_injective, pow_three', mul_inv, mul_inv]    rw [← Nat.card_eq_fintype_card] at hG    exact hG.coprime_two_right.nsmul_right_bijective.injective

Project: Arithmetic Progressions Almost Periodicity
License: Apache-2.0
Commit: afafc42a5326
Source: APAP/Prereqs/Convolution/ThreeAP.lean:20-41

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Project-declaredLean 4.32.0

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Plain-language statement

A boosted $L^\infty$ almost-periodicity estimate. Under $\sigma[A,S]\le K$ , it finds a large set $T$ , with the stated lower bound $|T|\ge K^{-4096\lceil\mathcal L(|C|/|B|)\rceil k^2/\varepsilon^2}|S|$ , such that averaging the target convolution against the $k$ -fold convolution of $\mu_T$ changes it by at most $\varepsilon$ in $L^\infty$ .

additive combinatoricsarithmetic progressionsFourier analysis

Source project: Arithmetic Progressions Almost Periodicity

Person-level attribution pending.

View proof record