Abstract :
We prove that (∫2π0 (Nθ(t)−D t2)2 dθ)1/2=O(t2/3), where Dθ is a rotation of a convex domain in image2 and Nθ(t)=#{image2∩tDθ}. It follows that for any δ>0, there exists a set of θʹs of measure 2π−δ, such that for tset membership, variantΛ, where Λ is any lacunary sequence, Nθ(t)−D t2less-than-or-equals, slantCΛt2/3 log(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)=#{image2∩tD+τ}, τset membership, variantimage2, the two-dimensional torus, andimagethe 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 greater-or-equal, slanted4 with one of its tangent lines.
