add example

examples.lean

@@ -1,5 +1,6 @@
 import Mathlib.Probability.Notation
 import PFR.Homomorphism
+import PFR.ApproxHomomorphism
 import PFR.ImprovedPFR
 import PFR.ForMathlib.MeasureReal
 
@@ -32,7 +33,13 @@ example (f : G → G') (S : Set G') (hS : ∀ x y : G, f (x + y) - f x - f y ∈
   convert homomorphism_pfr f S hS
   norm_cast
 
--- TODO: the approximate homomorphism version of PFR
+-- Todo: replace the constants C₁, C₂, C₃, C₄ below with actual values
+
+/-- The approximate homomorphism version of PFR -/
+example (f : G → G') (K : ℝ) (hK: K > 0) (hf: Nat.card { x : G × G| f (x.1+x.2) = (f x.1) + (f x.2) } ≥ (Nat.card G)^2/ K) : ∃ (φ : G →+ G') (c : G'), Nat.card { x : G | f x = φ x + c } ≥ (Nat.card G) / (4 * C₁^25 * C₃^24 * K^(50 * C₄ + 48 * C₂)) := by
+  convert approx_hom_pfr f K hK hf
+
+
 
 end PFR