Diophantine equations and ergodic theorems

Let (X, μ) be a probability measure space and T₁, . . . , T_n be a family of commuting, measure preserving invertible transformations on X. Let Q(m₁, . . . ,m_n) be a homogeneous, positive polynomial with integer coefficients, and let S_λ = {m ∈ Zⁿ: Q(m) = λ} denote the set of integer solutions m = (m₁, . . .m_n) of the diophantine equation Q(m) = λ. We prove that under a certain nondegeneracy condition on the polynomial Q(m) and an ergodic condition on the family of transformations T = (T₁, . . . , T_n) the images of the solution sets: Ω_x,λ = {(T^m1₁T^m2₂ … T^mn_nx): m ∈ Sλ} become uniformly distributed on X w.r.t. μ for a.e. x ∈ X as λ → ∞. That is the pointwise ergodic theorem holds when the standard averages are replaced by the ones, where the exponents satisfy a diophantine equation. The proof uses a variant of the Hardy-Littlewood method of exponential sums developed by Birch and Davenport and techniques from harmonic analysis. A key point is the corresponding maximal theorem, which is a discrete analogue of a maximal theorem on Rⁿ corresponding to the level surfaces of the polynomial Q(x).