| {"name":"approx_hom_pfr","declaration":"/-- Let $G, G'$ be finite abelian $2$-groups.\nLet $f : G \\to G'$ be a function, and suppose that there are at least\n$|G|^2 / K$ pairs $(x,y) \\in G^2$ such that $$ f(x+y) = f(x) + f(y).$$\nThen there exists a homomorphism $\\phi : G \\to G'$ and a constant $c \\in G'$ such that\n$f(x) = \\phi(x)+c$ for at least $|G| / (2 ^ {172} * K ^ {146})$ values of $x \\in G$. -/\ntheorem approx_hom_pfr {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [Fintype G] [AddCommGroup G'] [Fintype G'] [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (f : G → G') (K : ℝ) (hK : K > 0) (hf : ↑(Nat.card ↑{x | f (x.1 + x.2) = f x.1 + f x.2}) ≥ ↑(Nat.card G) ^ 2 / K) : ∃ φ c, ↑(Nat.card ↑{x | f x = φ x + c}) ≥ ↑(Nat.card G) / (2 ^ 172 * K ^ 146)"} | |