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 T x = a + A(x) be a continuous surjective affine map of G such that γ ≡ 1 is the only character of G satisfying γ ◦ An = γ for some positive integer n. Then if φ is a polynomial with real coefficients mapping the natural numbers to themselves, (p l)∞ l=1 is the sequence of rational primes and f is in Lp(G) for p > 1, we prove that N 1 lim f T φ(pn)x = f (g)dg N→∞ N nX =1Z G almost everywhere with respect to Haar measure on G.