The approximate homomorphism form of PFR

Here we apply PFR to show that almost homomorphisms f from a 2-group to a 2-group often coincide with a (shifted) actual homomorphisms. Here, approximate is in the sense that f(x+y)=f(x)+f(y) is true for a positive proportion of x,y.

Main result

• approx_hom_pfr : If $f:G\to G^{\prime }$ is a map between finite abelian elementary 2-groups such that $f(x+y)=f(x)+f(y)$ for at least $|G|/K$ values, then then there is a homomorphism $\varphi :G\to G^{\prime }$ and a constant $c$ such that $f(x)=\varphi (x)+c$ for a substantial set of values.

theorem approx_hom_pfr {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [Fintype G] [AddCommGroup G'] [Fintype G'] [Module (ZMod 2) G] [Module (ZMod 2) G'] (f : G → G') (K : ℝ) (hK : K > 0) (hf : ↑(Nat.card G) ^ 2 / K ≤ ↑(Nat.card ↑{x : G × G | f (x.1 + x.2) = f x.1 + f x.2})) : ∃ (φ : G →+ G') (c : G'), ↑(Nat.card ↑{x : G | f x = φ x + c}) ≥ ↑(Nat.card G) / (2 ^ 144 * K ^ 122)

Let $G,G^{\prime }$ be finite abelian $2$-groups. Let $f:G\to G^{\prime }$ be a function, and suppose that there are at least $|G{|}^2/K$ pairs $(x,y)\in G^2$ such that $$f(x+y)=f(x)+f(y).$$ Then there exists a homomorphism $\varphi :G\to G^{\prime }$ and a constant $c\in G^{\prime }$ such that $f(x)=\varphi (x)+c$ for at least $|G|/(2^{144}\ast K^{122})$ values of $x\in G$.