import Erdos1135.Tao.Probability.LogWindowPMF /-! # Source-Index Logarithmic Windows This module bridges Tao's source-index convention `N ↦ 2 * N + 1` with the odd-value logarithmic windows from `Probability.LogWindowPMF`. It keeps real-power endpoint choices for `N_y` out of this finite integer adapter. -/ namespace Erdos1135 namespace Tao open scoped BigOperators /-- Odd value represented by a source index. -/ def oddLogSourceIndexToValue (n : ℕ) : ℕ := 2 * n + 1 /-- Source index represented by an odd positive value. -/ def oddLogSourceIndexOfValue (m : ℕ) : ℕ := (m - 1) / 2 /-- Source indices whose odd values lie in the inclusive odd window `[lo, hi]`. -/ def oddLogSourceIndexWindow (lo hi : ℕ) : Finset ℕ := (Finset.range (hi + 1)).filter fun n => oddLogSourceIndexToValue n ∈ oddLogWindow lo hi theorem oddLogSourceIndexToValue_mod_two (n : ℕ) : oddLogSourceIndexToValue n % 2 = 1 := by unfold oddLogSourceIndexToValue omega theorem oddLogSourceIndexOfValue_oddLogSourceIndexToValue (n : ℕ) : oddLogSourceIndexOfValue (oddLogSourceIndexToValue n) = n := by unfold oddLogSourceIndexOfValue oddLogSourceIndexToValue omega theorem oddLogSourceIndexToValue_oddLogSourceIndexOfValue {m : ℕ} (hm : m % 2 = 1) : oddLogSourceIndexToValue (oddLogSourceIndexOfValue m) = m := by unfold oddLogSourceIndexToValue oddLogSourceIndexOfValue have hdecomp := Nat.div_add_mod m 2 omega theorem oddLogSourceIndexWindow_mem {lo hi n : ℕ} : n ∈ oddLogSourceIndexWindow lo hi ↔ oddLogSourceIndexToValue n ∈ oddLogWindow lo hi := by rw [oddLogSourceIndexWindow, Finset.mem_filter, Finset.mem_range] constructor · intro h exact h.2 · intro h have hhi : oddLogSourceIndexToValue n ≤ hi := (oddLogWindow_mem.mp h).2.1 exact ⟨by unfold oddLogSourceIndexToValue at hhi; omega, h⟩ theorem oddLogSourceIndexWindow_mem_bounds {lo hi n : ℕ} : n ∈ oddLogSourceIndexWindow lo hi ↔ lo ≤ oddLogSourceIndexToValue n ∧ oddLogSourceIndexToValue n ≤ hi := by rw [oddLogSourceIndexWindow_mem, oddLogWindow_mem] constructor · intro h exact ⟨h.1, h.2.1⟩ · intro h exact ⟨h.1, h.2, oddLogSourceIndexToValue_mod_two n⟩ /-- Equivalence between source indices and odd values inside the same finite window. -/ noncomputable def oddLogSourceIndexValueEquiv (lo hi : ℕ) : {n : ℕ // n ∈ oddLogSourceIndexWindow lo hi} ≃ {m : ℕ // m ∈ oddLogWindow lo hi} where toFun n := ⟨oddLogSourceIndexToValue n.1, oddLogSourceIndexWindow_mem.mp n.2⟩ invFun m := ⟨oddLogSourceIndexOfValue m.1, by rw [oddLogSourceIndexWindow_mem] have hodd : m.1 % 2 = 1 := (oddLogWindow_mem.mp m.2).2.2 simp [oddLogSourceIndexToValue_oddLogSourceIndexOfValue hodd, m.2]⟩ left_inv n := by apply Subtype.ext exact oddLogSourceIndexOfValue_oddLogSourceIndexToValue n.1 right_inv m := by apply Subtype.ext have hodd : m.1 % 2 = 1 := (oddLogWindow_mem.mp m.2).2.2 exact oddLogSourceIndexToValue_oddLogSourceIndexOfValue hodd theorem oddLogSourceIndexValueEquiv_apply {lo hi : ℕ} (n : {n : ℕ // n ∈ oddLogSourceIndexWindow lo hi}) : ((oddLogSourceIndexValueEquiv lo hi n : {m : ℕ // m ∈ oddLogWindow lo hi}) : ℕ) = oddLogSourceIndexToValue n.1 := rfl theorem oddLogSourceIndexValueEquiv_symm_apply {lo hi : ℕ} (m : {m : ℕ // m ∈ oddLogWindow lo hi}) : (((oddLogSourceIndexValueEquiv lo hi).symm m : {n : ℕ // n ∈ oddLogSourceIndexWindow lo hi}) : ℕ) = oddLogSourceIndexOfValue m.1 := rfl /-- Source-index logarithmic PMF with weights inherited from the represented odd values. -/ noncomputable def oddLogSourceIndexPMF (lo hi : ℕ) (hmass : 0 < logFinsetMass (oddLogWindow lo hi)) : PMF {n : ℕ // n ∈ oddLogSourceIndexWindow lo hi} := PMF.ofFintype (fun n : {n : ℕ // n ∈ oddLogSourceIndexWindow lo hi} => ENNReal.ofReal (logNatWeight (oddLogSourceIndexToValue n.1) / logFinsetMass (oddLogWindow lo hi))) (by classical rw [← ENNReal.ofReal_sum_of_nonneg] · have hsum_value : (∑ m : {m : ℕ // m ∈ oddLogWindow lo hi}, logNatWeight m.1) = logFinsetMass (oddLogWindow lo hi) := by rw [show (Finset.univ : Finset {m : ℕ // m ∈ oddLogWindow lo hi}) = (oddLogWindow lo hi).attach by exact Finset.univ_eq_attach (oddLogWindow lo hi)] unfold logFinsetMass exact Finset.sum_attach (oddLogWindow lo hi) (fun m => logNatWeight m) have hsum_index : (∑ n : {n : ℕ // n ∈ oddLogSourceIndexWindow lo hi}, logNatWeight (oddLogSourceIndexToValue n.1)) = logFinsetMass (oddLogWindow lo hi) := by have hsum_equiv : (∑ n : {n : ℕ // n ∈ oddLogSourceIndexWindow lo hi}, logNatWeight (oddLogSourceIndexToValue n.1)) = ∑ m : {m : ℕ // m ∈ oddLogWindow lo hi}, logNatWeight m.1 := by exact Fintype.sum_equiv (oddLogSourceIndexValueEquiv lo hi) (fun n : {n : ℕ // n ∈ oddLogSourceIndexWindow lo hi} => logNatWeight (oddLogSourceIndexToValue n.1)) (fun m : {m : ℕ // m ∈ oddLogWindow lo hi} => logNatWeight m.1) (by intro n; rfl) exact hsum_equiv.trans hsum_value have hsum_div : (∑ n : {n : ℕ // n ∈ oddLogSourceIndexWindow lo hi}, logNatWeight (oddLogSourceIndexToValue n.1) / logFinsetMass (oddLogWindow lo hi)) = 1 := by rw [← Finset.sum_div, hsum_index] exact div_self hmass.ne' rw [hsum_div] norm_num · intro n _hn exact div_nonneg (logNatWeight_nonneg (oddLogSourceIndexToValue n.1)) hmass.le) theorem oddLogSourceIndexPMF_apply_toReal {lo hi : ℕ} (hmass : 0 < logFinsetMass (oddLogWindow lo hi)) (n : {n : ℕ // n ∈ oddLogSourceIndexWindow lo hi}) : (oddLogSourceIndexPMF lo hi hmass n).toReal = logNatWeight (oddLogSourceIndexToValue n.1) / logFinsetMass (oddLogWindow lo hi) := by rw [oddLogSourceIndexPMF, PMF.ofFintype_apply] exact ENNReal.toReal_ofReal (div_nonneg (logNatWeight_nonneg (oddLogSourceIndexToValue n.1)) hmass.le) theorem oddLogSourceIndexPMF_map_value_apply_toReal {lo hi : ℕ} (hmass : 0 < logFinsetMass (oddLogWindow lo hi)) (m : {m : ℕ // m ∈ oddLogWindow lo hi}) : (((oddLogSourceIndexPMF lo hi hmass).map (oddLogSourceIndexValueEquiv lo hi)) m).toReal = logNatWeight m.1 / logFinsetMass (oddLogWindow lo hi) := by classical rw [PMF.map_apply] rw [tsum_eq_single ((oddLogSourceIndexValueEquiv lo hi).symm m)] · have hvalue : oddLogSourceIndexToValue (((oddLogSourceIndexValueEquiv lo hi).symm m).1) = m.1 := by exact congrArg Subtype.val (Equiv.apply_symm_apply (oddLogSourceIndexValueEquiv lo hi) m) have hhit : m = (oddLogSourceIndexValueEquiv lo hi) ((oddLogSourceIndexValueEquiv lo hi).symm m) := (Equiv.apply_symm_apply (oddLogSourceIndexValueEquiv lo hi) m).symm rw [if_pos hhit] rw [oddLogSourceIndexPMF_apply_toReal] simp [hvalue] · intro n hn have hne : ¬ m = (oddLogSourceIndexValueEquiv lo hi) n := by intro hm apply hn exact (Equiv.symm_apply_apply (oddLogSourceIndexValueEquiv lo hi) n).symm.trans (congrArg (oddLogSourceIndexValueEquiv lo hi).symm hm.symm) simp [hne] theorem oddLogSourceIndexPMF_map_value_toReal_eq_oddLogWindowPMF {lo hi : ℕ} (hmass : 0 < logFinsetMass (oddLogWindow lo hi)) (m : {m : ℕ // m ∈ oddLogWindow lo hi}) : (((oddLogSourceIndexPMF lo hi hmass).map (oddLogSourceIndexValueEquiv lo hi)) m).toReal = (oddLogWindowPMF lo hi hmass m).toReal := by rw [oddLogSourceIndexPMF_map_value_apply_toReal, oddLogWindowPMF_apply_toReal] theorem oddLogSourceIndexPMF_map_value_eq_oddLogWindowPMF {lo hi : ℕ} (hmass : 0 < logFinsetMass (oddLogWindow lo hi)) : (oddLogSourceIndexPMF lo hi hmass).map (oddLogSourceIndexValueEquiv lo hi) = oddLogWindowPMF lo hi hmass := by apply PMF.ext intro m exact (ENNReal.toReal_eq_toReal_iff' (PMF.apply_ne_top _ m) (PMF.apply_ne_top _ m)).mp (oddLogSourceIndexPMF_map_value_toReal_eq_oddLogWindowPMF hmass m) end Tao end Erdos1135