On affine maps and an arithmetic limit on compact groups

Let $G$ be a compact connected metric abelian group equipped with its normalised Haar measure. Let $Tx = a + A(x)$ be a continuous surjective affine map of $G$ such that $\gamma \equiv 1$ is the only character of $G$ satisfying $\gamma \circ A^n = \gamma$ for some positive integer $n$. Then if $\phi$ is a polynomial with real coefficients mapping the natural numbers to themselves,$(p_l)_{l=1}^{\infty}$ is the sequence of rational primes and $f$ is in $L^p(G)$ for $p > 1$, we prove that $$\lim _{N \to \infty}{1\over N}\sum _{n=1}^Nf (T^{\phi (p_n)}x) = \int_Gf(g)dg$$ almost everywhere with respect to Haar measure on $G.$