Lattice Points and Generalized Diophantine Conditions

Abstract

We prove that (∫2π0(|Nθ(t)−|D|t2|)2dθ)1/2=O(t2/3), where Dθ is a rotation of a convex domain in R2 and Nθ(t)=#{Z2∩tDθ}. It follows that for any δ>0, there exists a set of θ's of measure 2π−δ, such that for t∈Λ, where Λ is any lacunary sequence, |Nθ(t)−|D|t2|⩽CΛt2/3log(t). Moreover, we prove, under some additional assumptions, that for almost every θ,Nθ(t)−|D|t2=O(t2/3), (*)up to a small logarithmic transgression. We also prove that if D is convex and finite type, and also in some infinite type situations, Nτ(t)=#{Z2∩tD+τ}, τ∈T2, the two-dimensional torus, and‖Nτ(t)−t2|D|‖L2(T2)⩽Ct1/2, (**)the optimal bound, then N(0, 0)(t)=t2|D|+O(t2/3). We conclude that (**) cannot in general hold if the boundary of D has order of contact ⩾4 with one of its tangent lines.