-
Diophantine equations and ergodic theorems
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 124, Number 5, October 2002
- pp. 921-953
- 10.1353/ajm.2002.0029
- Article
- Additional Information
- Purchase/rental options available:
Let (X, μ) be a probability measure space and T1, . . . , Tn be a family of commuting, measure preserving invertible transformations on X. Let Q(m1, . . . ,mn) be a homogeneous, positive polynomial with integer coefficients, and let Sλ = {m ∈ Zn: Q(m) = λ} denote the set of integer solutions m = (m1, . . .mn) 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, . . . , Tn) the images of the solution sets: Ωx,λ = {(Tm11Tm22 … Tmnnx): 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).