Diophantine equations and ergodic theorems

Abstract

Let ( X , μ) be a probability measure space and T 1 , . . . , T n be a family of commuting, measure preserving invertible transformations on X . Let Q ( m 1 , . . . , m n ) be a homogeneous, positive polynomial with integer coefficients, and let S λ = { m ∈ Z n : Q ( m ) = λ} denote the set of integer solutions m = ( m 1 , . . . 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 = ( T 1 , . . . , T n ) the images of the solution sets: Ω x ,λ = {( T m1 1 T m2 2 … T mn n x ): m ∈ S λ} 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 R n corresponding to the level surfaces of the polynomial Q ( x ).